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Semi-analytical study of Pine Wilt Disease model with convex rate under Caputo–Febrizio fractional order derivative

2020-03-30
Kamal Shah , Manar A. Alqudah , Fahd Jarad +1 more

On more general forms of proportional fractional operators

2020-01-01
Fahd Jarad , Manar A. Alqudah , Thabet Abdeljawad
Abstract In this article, more general types of fractional proportional integrals and derivatives are proposed. Some properties of these operators are discussed. Abstract In this article, more general types of fractional proportional integrals and derivatives are proposed. Some properties of these operators are discussed.
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Study of fractional order dynamics of nonlinear mathematical model

2022-05-11
Kamal Shah , Amjad Ali , Salman Zeb +3 more
This manuscript is devoted to establishing some theoretical and numerical results for a nonlinear dynamical system under Caputo fractional order derivative. Further, the said system addresses an infectious disease like … This manuscript is devoted to establishing some theoretical and numerical results for a nonlinear dynamical system under Caputo fractional order derivative. Further, the said system addresses an infectious disease like COVID-19. The proposed system involves natural death rates of susceptible, infected and recovered classes respectively. By using nonlinear analysis feasible region and boundedness have been established first in this study. Global and Local stability analysis along with basic reproduction number have also addressed by using the next generation matrix method. Upon using the fixed point approach, existence and uniqueness of the approximate solution for the mentioned problem has also investigated. Some stability results of Hyers-Ulam (H-U) type have also discussed. Further for numerical treatment, we have exercised two numerical schemes including modified Euler method (MEM) and nonstandard finite difference (NSFD) method. Further the two numerical schemes have also compared with respect to CPU time. Graphical presentations have been displayed corresponding to different fractional order by using some real data.

Fear effect in a predator-prey model with additional food, prey refuge and harvesting on super predator

2022-04-23
Ashraf Adnan Thirthar , Salam J. Majeed , Manar A. Alqudah +2 more

On computational analysis of highly nonlinear model addressing real world applications

2022-03-26
S. Ali , Aziz Khan , Kamal Shah +3 more
This paper presents a numerical strategy for solving boundary value problems (BVPs) that is based on the Haar wavelets method (HWM). BVPs having high Prandtl numbers are discussed, Because they … This paper presents a numerical strategy for solving boundary value problems (BVPs) that is based on the Haar wavelets method (HWM). BVPs having high Prandtl numbers are discussed, Because they are very important in many practical problems of science and engineering. By using group-theoretic method, the considered model of partial differential equations (PDEs) are converted to system of nonlinear ordinary differential equations. By using HWM, the numerical results are established. Further, solutions obtained on a coarse resolution with low accuracy is refined towards higher accuracy by increasing the level of resolution. Superiority of the HWM has been established over the commercial software NDSolve and available numerical and approximated methods.

New results on Caputo fractional-order neutral differential inclusions without compactness

2019-12-01
Manar A. Alqudah , C. Ravichandran , Thabet Abdeljawad +1 more
Abstract This article deals with existence results of Caputo fractional neutral inclusions without compactness in Banach space using weak topology. In fact, for weakly sequentially closed maps we apply fixed … Abstract This article deals with existence results of Caputo fractional neutral inclusions without compactness in Banach space using weak topology. In fact, for weakly sequentially closed maps we apply fixed point theorems to obtain the existence of the solution. Furthermore, the results are manifested for fractional neutral system held by nonlocal conditions. To justify the application of the reported results an illustration is presented.
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On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative

2021-01-21
Mohammed S. ‬Abdo , Thabet Abdeljawad , Kishor D. Kucche +3 more
Abstract In this paper, we obtain sufficient conditions for the existence and uniqueness results of the pantograph fractional differential equations (FDEs) with nonlocal conditions involving Atangana–Baleanu–Caputo (ABC) derivative operator with … Abstract In this paper, we obtain sufficient conditions for the existence and uniqueness results of the pantograph fractional differential equations (FDEs) with nonlocal conditions involving Atangana–Baleanu–Caputo (ABC) derivative operator with fractional orders. Our approach is based on the reduction of FDEs to fractional integral equations and on some fixed point theorems such as Banach’s contraction principle and the fixed point theorem of Krasnoselskii. Further, Gronwall’s inequality in the frame of the Atangana–Baleanu fractional integral operator is applied to develop adequate results for different kinds of Ulam–Hyers stabilities. Lastly, the paper includes an example to substantiate the validity of the results.
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Numerical solutions of time fractional Burgers’ equation involving Atangana–Baleanu derivative via cubic B-spline functions

2022-01-31
Madiha Shafiq , Muhammad Abbas , Farah Aini Abdullah +3 more
The current paper uses the cubic B-spline functions and θ-weighted scheme to achieve numerical solutions of the time fractional Burgers' equation with Atangana–Baleanu derivative. A non-singular kernel is involved in … The current paper uses the cubic B-spline functions and θ-weighted scheme to achieve numerical solutions of the time fractional Burgers' equation with Atangana–Baleanu derivative. A non-singular kernel is involved in the Atangana–Baleanu fractional derivative. For discretization along temporal and spatial grids, the proposed numerical technique employs the finite difference approach and cubic B-spline functions, respectively. This scheme is unconditionally stable and second order convergent in spatial and temporal directions. The presented approach is endorsed by some numerical examples, which show that it is applicable and accurate.

Ulam stability of Caputo q-fractional delay difference equation: q-fractional Gronwall inequality approach

2019-11-29
Rabia Ilyas Butt , Thabet Abdeljawad , Manar A. Alqudah +1 more
Abstract In this article, we discuss the existence and uniqueness of solution of a delay Caputo q -fractional difference system. Based on the q -fractional Gronwall inequality, we analyze the … Abstract In this article, we discuss the existence and uniqueness of solution of a delay Caputo q -fractional difference system. Based on the q -fractional Gronwall inequality, we analyze the Ulam–Hyers stability and the Ulam–Hyers–Rassias stability. An example is provided to support the theoretical results.
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Cancer treatment by stem cells and chemotherapy as a mathematical model with numerical simulations

2020-01-10
Manar A. Alqudah
The main idea of this work, is to introduce a new mathematical model of stem cells and chemotherapy to treat cancer, as an autonomous system. Local stability of the equilibrium … The main idea of this work, is to introduce a new mathematical model of stem cells and chemotherapy to treat cancer, as an autonomous system. Local stability of the equilibrium points before and after the treatment are discussed. Numerical simulation and graphic for the concentration of cells along the time are explored the effects of the therapies on the decay rate of tumor cells and on the growth rate of effector cells to modify the immune system of the cancer patient and hence hope a cure by helping Medical scientists take highlight action to start the screening process.

Novel Numerical Investigations of Fuzzy Cauchy Reaction–Diffusion Models via Generalized Fuzzy Fractional Derivative Operators

2021-10-03
Manar A. Alqudah , Rehana Ashraf , Saima Rashid +3 more
The present research correlates with a fuzzy hybrid approach merged with a homotopy perturbation transform method known as the fuzzy Shehu homotopy perturbation transform method (SHPTM). With the aid of … The present research correlates with a fuzzy hybrid approach merged with a homotopy perturbation transform method known as the fuzzy Shehu homotopy perturbation transform method (SHPTM). With the aid of Caputo and Atangana–Baleanu under generalized Hukuhara differentiability, we illustrate the reliability of this scheme by obtaining fuzzy fractional Cauchy reaction–diffusion equations (CRDEs) with fuzzy initial conditions (ICs). Fractional CRDEs play a vital role in diffusion and instabilities may develop spatial phenomena such as pattern formation. By considering the fuzzy set theory, the proposed method enables the solution of the fuzzy linear CRDEs to be evaluated as a series of expressions in which the components can be efficiently identified and generating a pair of approximate solutions with the uncertainty parameter λ∈[0,1]. To demonstrate the usefulness and capabilities of the suggested methodology, several numerical examples are examined to validate convergence outcomes for the supplied problem. The simulation results reveal that the fuzzy SHPTM is a viable strategy for precisely and accurately analyzing the behavior of a proposed model.
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Implementation of the Exp-function approach for the solution of KdV equation with dual power law nonlinearity

2022-10-05
Naila Sajid , Zahida Perveen , Maasoomah Sadaf +4 more

Theory and Semi-Analytical Study of Micropolar Fluid Dynamics through a Porous Channel

2023-01-01
Aziz Khan , Sana Ullah , Kamal Shah +3 more
In this work, We are looking at the characteristics of micropolar flow in a porous channel that's being driven by suction or injection. The working of the fluid is described … In this work, We are looking at the characteristics of micropolar flow in a porous channel that's being driven by suction or injection. The working of the fluid is described in the flow model. We can reduce the governing nonlinear partial differential equations (PDEs) to a model of coupled systems of nonlinear ordinary differential equations using similarity variables (ODEs). In order to obtain the results of a coupled system of nonlinear ODEs, we discuss a method which is known as the differential transform method (DTM). The concern transform is an excellent mathematical tool to obtain the analytical series solution to the nonlinear ODEs. To observe beast agreement between analytical method and numerical method, we compare our result with the Rung-Kutta method of order four (RK4). We also provide simulation plots to the obtained result by using Mathematica. On these plots, we discuss the effect of different parameters which arise during the calculation of the flow model equations.
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Developing a fuzzy logic-based carbon emission cost-incorporated inventory model with memory effects

2024-03-23
Rituparna Pakhira , Bapin Mondal , Ashraf Adnan Thirthar +2 more
Inventory control is a widely discussed topic in the real world, and recently, it has become closely linked to concerns about carbon emissions and global warming. Global warming is a … Inventory control is a widely discussed topic in the real world, and recently, it has become closely linked to concerns about carbon emissions and global warming. Global warming is a pressing issue, mainly due to a lack of awareness and action. Traditional inventory models, which typically use integer-order differential equations, overlook the memory aspect of the system. Addressing inventory management is essential in our efforts to combat global warming. This paper introduces a novel approach by incorporating carbon emission costs within a fuzzy environment. Fractional Calculus, a powerful mathematical tool, is employed to capture the memory effect of the system. This approach distinguishes between long memory, characterized by a strong memory effect, and short memory, associated with a poor memory effect, through the use of fractional derivatives and integrals. Numerical results are analyzed based on these memory concepts. Entrepreneurs often find it difficult to determine exact values for known parameters in the real world. Therefore, this study considers the uncertain nature of ordering costs, the rate of deterioration, and demand rates as triangular fuzzy numbers. The optimal average cost and ordering intervals are determined using a solution methodology. A sensitivity analysis is performed to demonstrate how different system parameters influence the outcomes within a fuzzy environment. Notably, it is observed that profit tends to be higher under conditions of strong memory compared to poor memory effects. Moreover, it's worth emphasizing that profits are notably more favorable when employing the signed distance method compared to the graded mean integration method, especially in situations marked by strong memory effects. This study highlights the importance of considering sensitive parameters in the model, especially under conditions of strong memory effects. Such parameters require careful attention in the pursuit of effective inventory management strategies to mitigate carbon emissions and combat global warming.

New discrete inequalities of Hermite–Hadamard type for convex functions

2021-02-23
Pshtiwan Othman Mohammed , Thabet Abdeljawad , Manar A. Alqudah +1 more
Abstract We introduce new time scales on $\mathbb{Z}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math> . Based on this, we investigate the discrete inequality of Hermite–Hadamard type for discrete convex functions. Finally, we improve our … Abstract We introduce new time scales on $\mathbb{Z}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math> . Based on this, we investigate the discrete inequality of Hermite–Hadamard type for discrete convex functions. Finally, we improve our result to investigate the discrete fractional inequality of Hermite–Hadamard type for the discrete convex functions involving the left nabla and right delta fractional sums.
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Existence of results and computational analysis of a fractional order two strain epidemic model

2022-06-03
Aziz Khan , Kamal Shah , Thabet Abdeljawad +1 more
In this paper, we investigate fractional order two strain epidemic model in the sense of ABC operator. This study includes existence and uniqueness of solution, stability and numerical simulations of … In this paper, we investigate fractional order two strain epidemic model in the sense of ABC operator. This study includes existence and uniqueness of solution, stability and numerical simulations of the model under consideration. Fixed point postulates are used for the existence and uniqueness of solution. A theoretical approach is employed to investigate sufficient results for Hyers–Ulam's stability to the model under study. For the numerical demonstration Lagrange's interpolation polynomial technique is utilized. Graphical presentations against different fractional orders are displayed.

Mathematical analysis of fractional order alcoholism model

2023-07-25
Sher Muhammad , Kamal Shah , Muhammad Sarwar +2 more
In this manuscript, we are going to study a novel model of the dynamics of alcohol consumption under induced complications. The mentioned model is considered under the concept of conformable … In this manuscript, we are going to study a novel model of the dynamics of alcohol consumption under induced complications. The mentioned model is considered under the concept of conformable fractional order derivative (CFOD). Currently, most of real-world problems are considered under fractional order derivatives because of their stable and global behavior. First, we will investigate the model for qualitative theory including existence and uniqueness of solution and Ulam-Hyers stability. For qualitative theory, we will use fixed point theory. In addition, we use a numerical method to find the approximate solution of the proposed model. In the final part of the paper, we give a detailed discussion of its numerical results and its graphical presentation.

Linear conformable differential system and its controllability

2020-08-27
Awais Younas , Thabet Abdeljawad , Rida Batool +2 more
Abstract This article deals with the sequential conformable linear equations. We have focused on the solution techniques of these equations and particularly on the controllability conditions of the time-invariant system. … Abstract This article deals with the sequential conformable linear equations. We have focused on the solution techniques of these equations and particularly on the controllability conditions of the time-invariant system. For the controllability conditions and results, we have defined the conformable controllability Gramian matrix, the conformable fundamental matrix, and the conformable controllability matrix.
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Hermite–Hadamard integral inequalities on coordinated convex functions in quantum calculus

2021-05-20
Manar A. Alqudah , Artion Kashuri , Pshtiwan Othman Mohammed +4 more
Abstract At first, we recall the q -operators in the context of q -calculus and by examining these operators we will introduce new definitions of the partial q -operators. Then, … Abstract At first, we recall the q -operators in the context of q -calculus and by examining these operators we will introduce new definitions of the partial q -operators. Then, we investigate some new refinements inequalities of Hermite–Hadamard ( $H-H$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> <mml:mo>−</mml:mo> <mml:mi>H</mml:mi> </mml:math> ) type on the coordinated convex functions involving the new defined partial q -operators. From our main results, we establish several specific inequalities and we point out the existing results which had already been obtained in the literature.
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On a class of Langevin equations in the frame of Caputo function-dependent-kernel fractional derivatives with antiperiodic boundary conditions

2021-01-01
Abdelatif Boutiara , Mohammed S. ‬Abdo , Manar A. Alqudah +1 more
<abstract> In this manuscript, we consider a class of nonlinear Langevin equations involving two different fractional orders in the frame of Caputo fractional derivative with respect to another monotonic function … <abstract> In this manuscript, we consider a class of nonlinear Langevin equations involving two different fractional orders in the frame of Caputo fractional derivative with respect to another monotonic function $ \vartheta $ with antiperiodic boundary conditions. The existence and uniqueness results are proved for the suggested problem. Our approach is relying on properties of $ \vartheta $-Caputo's derivative, and implementation of Krasnoselskii's and Banach's fixed point theorem. At last, we discuss the Ulam-Hyers stability criteria for a nonlinear fractional Langevin equation. Some examples justifying the results gained are provided. The results are novel and provide extensions to some of the findings known in the literature. </abstract>

Analytical Solutions of a Modified Predator-Prey Model through a New Ecological Interaction

2019-10-16
Noufe H. Aljahdaly , Manar A. Alqudah
The predator-prey model is a common tool that researchers develop continuously to predict the dynamics of the animal population within a certain phenomenon. Due to the sexual interaction of the … The predator-prey model is a common tool that researchers develop continuously to predict the dynamics of the animal population within a certain phenomenon. Due to the sexual interaction of the predator in the mating period, the males and females feed together on one or more preys. This scenario describes the ecological interaction between two predators and one prey. In this study, the nonlinear diffusive predator-prey model is presented where this type of interaction is accounted for. The influence of this interaction on the population of predators and preys is predicted through analytical solutions of the dynamical system. The solutions are obtained by using two reliable and simple methods and are presented in terms of hyperbolic functions. In addition, the biological relevance of the solutions is discussed.
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Neural networking study of worms in a wireless sensor model in the sense of fractal fractional

2023-01-01
Aziz Khan , Thabet Abdeljawad , Manar A. Alqudah
&lt;abstract&gt;&lt;p&gt;We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in … &lt;abstract&gt;&lt;p&gt;We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in the context of fractal fractional differential operators. In addition, the Banach contraction technique is utilized to achieve the existence and unique outcomes of the given model. Further, the stability of the proposed model is analyzed through functional analysis and the Ulam-Hyers (UH) stability technique. In the last, a numerical scheme is established to check the dynamical behavior of the fractional fractal order WSN model.&lt;/p&gt;&lt;/abstract&gt;

Complex dynamics in a two species system with Crowley–Martin response function: Role of cooperation, additional food and seasonal perturbations

2024-03-16
Bapin Mondal , Ashraf Adnan Thirthar , Nazmul Sk +2 more

Stability analysis and simulation of the novel Corornavirus mathematical model via the Caputo fractional-order derivative: A case study of Algeria

2021-05-23
Yacine El hadj Moussa , Ahmed Boudaoui , Saif Ullah +3 more
The novel coronavirus infectious disease (or COVID-19) almost spread widely around the world and causes a huge panic in the human population. To explore the complex dynamics of this novel … The novel coronavirus infectious disease (or COVID-19) almost spread widely around the world and causes a huge panic in the human population. To explore the complex dynamics of this novel infection, several mathematical epidemic models have been adopted and simulated using the statistical data of COVID-19 in various regions. In this paper, we present a new nonlinear fractional order model in the Caputo sense to analyze and simulate the dynamics of this viral disease with a case study of Algeria. Initially, after the model formulation, we utilize the well-known least square approach to estimate the model parameters from the reported COVID-19 cases in Algeria for a selected period of time. We perform the existence and uniqueness of the model solution which are proved via the Picard-Lindelöf method. We further compute the basic reproduction numbers and equilibrium points, then we explore the local and global stability of both the disease-free equilibrium point and the endemic equilibrium point. Finally, numerical results and graphical simulation are given to demonstrate the impact of various model parameters and fractional order on the disease dynamics and control.

INVESTIGATION OF INTEGRAL BOUNDARY VALUE PROBLEM WITH IMPULSIVE BEHAVIOR INVOLVING NON-SINGULAR DERIVATIVE

2022-09-05
Kamal Shah , Thabet Abdeljawad , Arshad Ali +1 more
This paper is devoted to investigating a class of impulsive fractional order differential equations (FODEs) with integral boundary condition. For the proposed paper, we use non-singular type derivative of fractional … This paper is devoted to investigating a class of impulsive fractional order differential equations (FODEs) with integral boundary condition. For the proposed paper, we use non-singular type derivative of fractional order which has been introduced by Atangana, Baleanu and Caputo (ABC). The aforesaid type problems have numerous applications in fluid mechanics and hydrodynamics to model various problems of flow phenomenons. We establish some sufficient conditions for the existence and uniqueness of solution to the proposed problem by using classical fixed point results due to Banach and Krasnoselskii. Further, on using tools of the nonlinear analysis, sufficient conditions are developed for Hyers–Ulam (HU) type stability results. A pertinent example is given to justify our results.

A solution of a nonlinear Volterra integral equation with delay via a faster iteration method

2022-09-28
Godwin Amechi Okeke , Austine Efut Ofem , Thabet Abdeljawad +2 more
&lt;abstract&gt;&lt;p&gt;The purpose of this article is to study the convergence, stability and data dependence results of an iterative method for contractive-like mappings. The concept of stability considered in this study … &lt;abstract&gt;&lt;p&gt;The purpose of this article is to study the convergence, stability and data dependence results of an iterative method for contractive-like mappings. The concept of stability considered in this study is known as $ w^2 $-stability, which is larger than the simple notion of stability considered by several prominent authors. Some illustrative examples on $ w^2 $-stability of the iterative method have been presented for different choices of parameters and initial guesses. As an application of our results, we establish the existence, uniqueness and approximation results for solutions of a nonlinear Volterra integral equation with delay. Finally, we provide an illustrative example to support the application of our results. The novel results of this article extend and generalize several well known results in existing literature.&lt;/p&gt;&lt;/abstract&gt;

AN ECOSYSTEM MODEL WITH MEMORY EFFECT CONSIDERING GLOBAL WARMING PHENOMENA AND AN EXPONENTIAL FEAR FUNCTION

2023-01-01
Ashraf Adnan Thirthar , Prabir Panja , Aziz Khan +2 more
Global warming is becoming a big concern for the environment since it is causing serious and often unexpected impacts on species, affecting their abundance, genetic composition, behavior and survival. So, … Global warming is becoming a big concern for the environment since it is causing serious and often unexpected impacts on species, affecting their abundance, genetic composition, behavior and survival. So, the modeling study is necessary to investigate the effects of global warming in predator–prey dynamics. This research paper analyzed the memory effect evaluated by Caputo fractional derivative on predator–prey interaction using an exponential fear function with a Holling-type II function in the presence of global warming effect on prey and predator species. It is assumed that the densities of prey and predator species decrease due to the increase of global warming. It is considered that both prey and predator species are contributing to the increase of global warming. Also, it is considered that global warming is increasing constantly and decreasing due to the natural decay rate. All possible equilibria of the system are determined, and the stability of the system around all equilibria points is investigated. Around the interior equilibrium point, the Hopf bifurcation is also theoretically and numerically studied. A number of numerical simulation results are presented to demonstrate the impacts of fear, fractional order and global warming on the behavior of the model. It is observed that the global warming effect on predator species may destabilize the system but ultimately the system may become stable. Again, it is obtained that the natural decay rate of global warming can stabilize the system initially but a higher decay rate may destabilized the system. It is also found that the fractional-order model is determined to be more stable than the integer-order model.

Utilizing memory effects to enhance resilience in disease-driven prey-predator systems under the influence of global warming

2023-11-30
Ashraf Adnan Thirthar , Nazmul Sk , Bapin Mondal +2 more
Abstract This research paper presents an eco-epidemiological model that investigates the intricate dynamics of a predator–prey system, considering the impact of fear-induced stress, hunting cooperation, global warming, and memory effects … Abstract This research paper presents an eco-epidemiological model that investigates the intricate dynamics of a predator–prey system, considering the impact of fear-induced stress, hunting cooperation, global warming, and memory effects on species interactions. The model employs fractional-order derivatives to account for temporal dependencies and memory in ecological processes. By incorporating these factors, we aim to provide a more comprehensive understanding of the underlying mechanisms that govern the stability and behavior of ecological systems. Mathematically we investigate system’s existence, equilibria and their stability. Moreover, global stability and hopf bifurcation also analyzed in this study. Numerical simulations have been performed to validate the analytical results. We find that the coexistence equilibrium is stable under specific conditions, along with the predator equilibrium and the disease-free equilibrium. Bifurcation analyses demonstrate the intricate behavior of species densities in response to changes in model parameters. Fear and global warming are found to stabilize the system, while cooperation and additional food for predators lead to destabilization. Additionally, the influence of species memory has been explored. We observe that memory tends to stabilize the system as species memory levels increase.
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ON FRACTIONAL DIFFERENTIAL INCLUSION PROBLEMS INVOLVING FRACTIONAL ORDER DERIVATIVE WITH RESPECT TO ANOTHER FUNCTION

2020-05-06
Samiha Belmor , Fahd Jarad , Thabet Abdeljawad +1 more
In this research work, we investigate the existence of solutions for a class of nonlinear boundary value problems for fractional-order differential inclusion with respect to another function. Endpoint theorem for … In this research work, we investigate the existence of solutions for a class of nonlinear boundary value problems for fractional-order differential inclusion with respect to another function. Endpoint theorem for [Formula: see text]-weak contractive maps is the main tool in determining our results. An example is presented in aim to illustrate the results.

EXISTENCE AND STABILITY RESULTS FOR COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS INVOLVING AB-CAPUTO DERIVATIVE

2023-01-01
Nayyar Mehmood , AHSAN ABBAS , Ali Akgül +2 more
In this paper, we use Krasnoselskii’s fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo … In this paper, we use Krasnoselskii’s fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo fractional derivative [Formula: see text] with boundary conditions [Formula: see text] We discuss uniqueness with the help of the Banach contraction principle. The criteria for Hyers–Ulam stability of given AB-Caputo fractional-coupled boundary value problem (BVP) is also discussed. Some examples are provided to validate our results. In Example 1, we find a unique and stable solution of AB-Caputo fractional-coupled BVP. In Example 2, the analysis of approximate and exact solutions with errors of nonlinear integral equations is elaborated with graphs.

A FRACTAL-FRACTIONAL ORDER MODEL TO STUDY MULTIPLE SCLEROSIS: A CHRONIC DISEASE

2023-11-02
Kamal Shah , Bahaaeldin Abdalla , Thabet Abdeljawad +1 more
A mathematical model of progressive disease of the nervous system also called multiple sclerosis (MS) is studied in this paper. The proposed model is investigated under the concept of the … A mathematical model of progressive disease of the nervous system also called multiple sclerosis (MS) is studied in this paper. The proposed model is investigated under the concept of the fractal-fractional order derivative (FFOD) in the Caputo sense. In addition, the tools of nonlinear functional analysis are applied to prove some qualitative results including the existence theory, stability, and numerical analysis. For the recommended results of the existence theory, Banach and Krassnoselski’s fixed point theorems are used. Additionally, Hyers–Ulam (HU) concept is used to derive some results for stability analysis. Additionally, for numerical illustration of approximate solutions of various compartments of the considered model, the modified Euler method is utilized. The aforementioned results are displayed graphically for various values of fractal-fractional orders.

Global stability and numerical simulation of a mathematical model of stem cells therapy of HIV-1 infection

2020-06-26
Manar A. Alqudah , Noufe H. Aljahdaly

Solution of Singular Integral Equations via Riemann–Liouville Fractional Integrals

2020-09-30
Manar A. Alqudah , Pshtiwan Othman Mohammed , Thabet Abdeljawad
In this attempt, we introduce a new technique to solve main generalized Abel’s integral equations and generalized weakly singular Volterra integral equations analytically. This technique is based on the Adomian … In this attempt, we introduce a new technique to solve main generalized Abel’s integral equations and generalized weakly singular Volterra integral equations analytically. This technique is based on the Adomian decomposition method, Laplace transform method, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi mathvariant="normal">Ψ</mml:mi></mml:math>-Riemann–Liouville fractional integrals. Finally, some examples are proposed and they illustrate the rapidness of our new technical method.
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Solving the Modified Regularized Long Wave Equations via Higher Degree B-Spline Algorithm

2021-03-03
Pshtiwan Othman Mohammed , Manar A. Alqudah , Y. S. Hamed +2 more
The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mtext>MRLW</a:mtext> </a:math> ) equation. … The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mtext>MRLW</a:mtext> </a:math> ) equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:msub> <c:mrow> <c:mi mathvariant="script">L</c:mi> </c:mrow> <c:mrow> <c:mn>2</c:mn> </c:mrow> </c:msub> </c:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M3"> <f:msub> <f:mrow> <f:mi mathvariant="script">L</f:mi> </f:mrow> <f:mrow> <f:mo>∞</f:mo> </f:mrow> </f:msub> </f:math> to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.
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ADAPTED HOMOTOPY PERTURBATION METHOD WITH SHEHU TRANSFORM FOR SOLVING CONFORMABLE FRACTIONAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

2023-01-01
Muhammad Imran Liaqat , Aziz Khan , Manar A. Alqudah +1 more
There is considerable literature on solutions to the gas-dynamic equation (GDE) and Fokker–Planck equation (FPE), where the fractional derivative is expressed in terms of the Caputo fractional derivative. There is … There is considerable literature on solutions to the gas-dynamic equation (GDE) and Fokker–Planck equation (FPE), where the fractional derivative is expressed in terms of the Caputo fractional derivative. There is hardly any work on analytical and numerical GDE and FPE solutions involving conformable fractional derivative (CFD). For the reasons stated above, we are required to solve the GDE and FPE in the form of CFD. The main goal of this research is to offer a novel combined method by employing the conformable Shehu transform (CST) and the homotopy perturbation method (HPM) for extracting analytical and numerical solutions of the time-fractional conformable GDE and FPE. The proposed method is called the conformable Shehu homotopy perturbation method (CSHPM). To evaluate its efficiency and consistency, relative and absolute errors among the approximate and exact solutions to three nonlinear problems of GDE and FPE are considered numerically and graphically. Moreover, fifth-term approximate and exact solutions are also compared by 2D and 3D graphs. This method has the benefit of not requiring any minor or major physical parameter assumptions in the problem. As a result, it may be used to solve both weakly and strongly nonlinear problems, overcoming some of the inherent constraints of classic perturbation approaches. Second, while addressing nonlinear problems, the CSHPM does not require Adomian polynomials. Therefore, to solve nonlinear GDE and FPE, just a few computations are necessary. As a consequence, it outperforms homotopy analysis and Adomian decomposition approaches significantly. It does not require discretization or linearization, unlike traditional numerical methods. The convergence and error analysis of the series solutions are also presented.

Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

2021-09-08
Saima Rashid , Rehana Ashraf , Ahmet Ocak Akdemi̇r +3 more
This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian … This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.
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On fractional impulsive system for methanol detoxification in human body

2022-06-01
Qura Tul Ain , Aziz Khan , Muhammad Ullah +2 more

Existence theory and approximate solution to prey–predator coupled system involving nonsingular kernel type derivative

2020-09-24
Manar A. Alqudah , Thabet Abdeljawad , Eiman +3 more
Abstract This manuscript considers a nonlinear coupled system under nonsingular kernel type derivative. The considered problem is investigated from two aspects including existence theory and approximate analytical solution. For the … Abstract This manuscript considers a nonlinear coupled system under nonsingular kernel type derivative. The considered problem is investigated from two aspects including existence theory and approximate analytical solution. For the concerned qualitative theory, some fixed point results are used. While for approximate solution, the Laplace transform coupled with Adomian method is applied. Finally, by a pertinent example of prey–predator system, we support our results. Some graphical presentations are given using Matlab.
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Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials

2022-01-31
Zulfiqar Ahmad Noor , Imran Talib , Thabet Abdeljawad +1 more
In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational … In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational matrix of the fractional-order derivatives in the Caputo sense, which is then used with spectral tau and spectral collocation methods to reduce the FDEs to a system of algebraic equations. Several numerical examples are given to show the accuracy of this method. These examples show that the obtained results have good agreement with the analytical solutions in both linear and non-linear FDEs. In addition to this, the numerical results obtained by using our method are compared with the numerical results obtained otherwise in the literature.
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Effect of Weather on the Spread of COVID-19 Using Eigenspace Decomposition

2021-01-01
Manar A. Alqudah , Thabet Abdeljawad , Anwar Zeb +2 more
Since the end of 2019, the world has suffered from a pandemic of the disease called COVID-19. WHO reports show approximately 113 M confirmed cases of infection and 2.5 M … Since the end of 2019, the world has suffered from a pandemic of the disease called COVID-19. WHO reports show approximately 113 M confirmed cases of infection and 2.5 M deaths. All nations are affected by this nightmare that continues to spread. Widespread fear of this pandemic arose not only from the speed of its transmission: a rapidly changing "normal life" became a fear for everyone. Studies have mainly focused on the spread of the virus, which showed a relative decrease in high temperature, low humidity, and other environmental conditions. Therefore, this study targets the effect of weather in considering the spread of the novel coronavirus SARS-CoV-2 for some confirmed cases in Iraq. The eigenspace decomposition technique was used to analyze the effect of weather conditions on the spread of the disease. Our theoretical findings showed that the average number of confirmed COVID-19 cases has cyclic trends related to temperature, humidity, wind speed, and pressure. We supposed that the dynamic spread of COVID-19 exists at a temperature of 130 F. The minimum transmission is at 120 F, while steady behavior occurs at 160 F. On the other hand, during the spread of COVID-19, an increase in the rate of infection was seen at 125% humidity, where the minimum spread was achieved at 200%. Furthermore, wind speed showed the most significant effect on the spread of the virus. The spread decreases with a wind speed of 45 KPH, while an increase in the infectious spread appears at 50 KPH.

Future implications of COVID-19 through Mathematical modeling

2021-12-25
Muhammad Zamir , Fawad Nadeem , Manar A. Alqudah +1 more
COVID-19 is a pandemic respiratory illness. The disease spreads from human to human and is caused by a novel coronavirus SARS-CoV-2. In this study, we formulate a mathematical model of … COVID-19 is a pandemic respiratory illness. The disease spreads from human to human and is caused by a novel coronavirus SARS-CoV-2. In this study, we formulate a mathematical model of COVID-19 and discuss the disease free state and endemic equilibrium of the model. Based on the sensitivity indexes of the parameters, control strategies are designed. The strategies reduce the densities of the infected classes but do not satisfy the criteria/threshold condition of the global stability of disease free equilibrium. On the other hand, the endemic equilibrium of the disease is globally asymptotically stable. Therefore it is concluded that the disease cannot be eradicated with present resources and the human population needs to learn how to live with corona. For validation of the results, numerical simulations are obtained using fourth order Runge-Kutta method.

Numerical Approximations for the Solutions of Fourth Order Time Fractional Evolution Problems Using a Novel Spline Technique

2022-03-19
Ghazala Akram , Muhammad Abbas , Hira Tariq +3 more
Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems … Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline technique (SPST). These equations are frequently applied in a wide variety of real-world applications, such as strain gradient elasticity, phase separation in binary mixtures, and modelling of thin beams and plates, all of which are key parts of mechanical engineering. The SPST can be used for space discretization, whereas the backward Euler formula can be used for time discretization. For the temporal discretization, the method’s convergence and stability are assessed. To show the accuracy and applicability of the proposed technique, numerical simulations are employed.
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Mathematical study of SIR epidemic model under convex incidence rate

2020-01-01
Rahim Ud Din , Kamal Shah , Manar A. Alqudah +2 more
In this manuscript, we examine the SIR model under convex incidence rate. We first formulate the famous SIR model under the aforesaid incidence rate. Further, we develop some sufficient analysis … In this manuscript, we examine the SIR model under convex incidence rate. We first formulate the famous SIR model under the aforesaid incidence rate. Further, we develop some sufficient analysis to examine the dynamical behavior of the model under consideration. We compute the basic reproductive number $\mathcal{R}_0.$ Also we study the global attractivity results via using Dulac function theory. Further, we also provide some information about the stability of the endemic and disease free equilibria for the considered model. In addition, we use nonstandard finite difference scheme to perform numerical simulation of the considered model via using Matlab. We provide different numerical plots for two different values of contact rate and taking various initial values for compartments involved in the considered model.

Unique Solution of Integral Equations via Intuitionistic Extended Fuzzy b-Metric-Like Spaces

2022-06-23
Naeem Saleem , Khalil Javed , Fahim Ud Din +4 more
In this manuscript, our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces. We establish some fixed point theorems in this setting. Also, we plot some graphs … In this manuscript, our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces. We establish some fixed point theorems in this setting. Also, we plot some graphs of an example of obtained result for better understanding. We use the concepts of continuous triangular norms and continuous triangular conorms in an intuitionistic fuzzy metric-like space. Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions. Triangular conorms are known as dual operations of triangular norms. The obtained results boost the approaches of existing ones in the literature and are supported by some examples and applications.
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ANALYSIS OF THE CONFORMABLE TEMPORAL-FRACTIONAL SWIFT–HOHENBERG EQUATION USING A NOVEL COMPUTATIONAL TECHNIQUE

2023-01-01
Aziz Khan , Muhammad Imran Liaqat , Manar A. Alqudah +1 more
The main objective of this study is to provide a new computational procedure for extracting approximate and exact solutions of the temporal-fractional Swift–Hohenberg (S–H) equations in the context of conformable … The main objective of this study is to provide a new computational procedure for extracting approximate and exact solutions of the temporal-fractional Swift–Hohenberg (S–H) equations in the context of conformable derivatives using the conformable natural transform (CNT) and Daftardar–Jafari method (DJM). We refer to it as the “natural conformable Daftardar–Jafari method” (CNDJM). The three types of errors are assessed in order to gauge the efficiency and consistency of the proposed method. Furthermore, 2D and 3D graphics are used to compare the exact and approximate solutions. This method offers a considerable benefit over homotopy analysis and Adomian decomposition methods in terms of computational work because it does not require Adomian and He’s polynomials. The procedure is quick and easy to use.

NUMERICAL ANALYSIS OF SOME FRACTIONAL ORDER DIFFERENTIAL EQUATIONS VIA LEGENDRE SPECTRAL METHOD

2023-01-01
Aziz Khan , Hafsa Naz , Muhammad Sarwar +3 more
In this research paper, we find the numerical solutions of fractional order scalers and coupled system of differential equations under initial conditions using shifted Legendre polynomials. By using the properties … In this research paper, we find the numerical solutions of fractional order scalers and coupled system of differential equations under initial conditions using shifted Legendre polynomials. By using the properties of shifted Legendre polynomials, we establish operational matrices of integration and differentiation in order to simplify our considered problems under initial conditions. In order to check the accuracy of the proposed model, some test problems are solved along with the graphical representations. For coupled system, we applied the algorithm to the Pharmacokinetic two-compartment model. As the proposed method is computer-oriented, we use therefore the MATLAB for required calculations. Numerical results are shown graphically.

Existence Results of Langevin Equations with Caputo–Hadamard Fractional Operator

2023-06-23
Sombir Dhaniya , Anoop Kumar , Aziz Khan +2 more
In this manuscript, we deal with a nonlinear Langevin fractional differential equation that involves the Caputo–Hadamard and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. The results presented … In this manuscript, we deal with a nonlinear Langevin fractional differential equation that involves the Caputo–Hadamard and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. The results presented in this study establish the existence, uniqueness, and Hyers–Ulam (HU) stability of the solution to the proposed equation. We achieved our main result by using the Banach contraction mapping principle and Krasonoselskii’s fixed point theorem. Furthermore, we introduce an application to demonstrate the validity of the results of our findings.
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Boundary value problem of weighted fractional derivative of a function with a respect to another function of variable order

2023-10-09
Kheireddine Benia , Mohammed Said Souıd , Fahd Jarad +2 more
Abstract This study aims to resolve weighted fractional operators of variable order in specific spaces. We establish an investigation on a boundary value problem of weighted fractional derivative of one … Abstract This study aims to resolve weighted fractional operators of variable order in specific spaces. We establish an investigation on a boundary value problem of weighted fractional derivative of one function with respect to another variable order function. It is essential to keep in mind that the symmetry of a transformation for differential equations is connected to local solvability, which is synonymous with the existence of solutions. As a consequence, existence requirements for weighted fractional derivative of a function with respect to another function of constant order are necessary. Moreover, the stability with in Ulam–Hyers–Rassias sense is reviewed. The outcomes are derived using the Kuratowski measure of non-compactness. A model illustrates the trustworthiness of the observed results.
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Analysis of a class of fractal hybrid fractional differential equation with application to a biological model

2024-08-15
Thabet Abdeljawad , Sher Muhammad , Kamal Shah +4 more
Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. … Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer's (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end.

Modeling the pandemic trend of 2019 Coronavirus with optimal control analysis

2020-12-09
BiBi Fatima , Gul Zaman , Manar A. Alqudah +1 more
In this work, we propose a mathematical model to analyze the outbreak of the Coronavirus disease (COVID-19). The proposed model portrays the multiple transmission pathways in the infection dynamics and … In this work, we propose a mathematical model to analyze the outbreak of the Coronavirus disease (COVID-19). The proposed model portrays the multiple transmission pathways in the infection dynamics and stresses the role of the environmental reservoir in the transmission of the disease. The basic reproduction number R0 is calculated from the model to assess the transmissibility of the COVID-19. We discuss sensitivity analysis to clarify the importance of epidemic parameters. The stability theory is used to discuss the local as well as the global properties of the proposed model. The problem is formulated as an optimal control one to minimize the number of infected people and keep the intervention cost as low as possible. Medical mask, isolation, treatment, detergent spray will be involved in the model as time dependent control variables. Finally, we present and discuss results by using numerical simulations.

Investigating Jerk Phenomenon by Using Fractal-Fractional Analysis

2025-05-28
Kamal Shah , A.Bharathy et. al , Rohul Amin +4 more

A mathematical analysis of human papilloma virus (HPV) disease with new perspectives of fractional calculus

2025-04-23
Thabet Abdeljawad , Nadeem Alam Khan , Bahaaeldin Abdalla +3 more

Solving one- and two-dimensional advection-diffusion type initial boundary value problems with a wavelet hybrid scheme

2025-03-04
Aslam Khan , Abdul Ghafoor , Emel Khan +3 more
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Study a Nonlinear Fractional Order Conformable Model of Infectious Disease Using Updated Analytical Techniques

2025-02-14
Manar A. Alqudah , Kamal Shah , Thabet Abdeljawad +1 more

New Generalized Results for Modified Atangana-Baleanu Fractional Derivatives and Integral Operators

2025-01-31
Gauhar Rahman , Muhammad Samraiz , Çetin Yıldız +2 more
In this current study, first we establish the modified power Atangana-Baleanu fractional derivative operators (MPC) in both the Caputo and Riemann-Liouville (MPRL) senses. Using the convolution approach and Laplace transformation, … In this current study, first we establish the modified power Atangana-Baleanu fractional derivative operators (MPC) in both the Caputo and Riemann-Liouville (MPRL) senses. Using the convolution approach and Laplace transformation, the so-called modified power fractional Caputo and R-L derivative operators with non-singular kernels are introduced. We establish theboundedness of the modified Caputo fractional derivative operator in this study. The fractional differential equations are solved with the generalised Laplace transform (GLT). In addition, the corresponding form of the fractional integral operator is defined. Also, we prove the boundedness and Laplace transform of the fractional integral operator. The composition of power fractionalderivative and integral operators is given in the study. Additionally, several examples related to our findings along with their graphical representation are presented.

Advanced Analytical Approaches for Nonlinear Fractional PDEs Reduced Differential Transform Method and Elzaki Transform Homotopy Perturbation Method

2025-01-01
Zahid Ullah , Jianghao Hao , Ali Akgül +3 more
This study presents a comparative analysis of two advanced analytical methods—the Elzaki Transform Homotopy Perturbation Method (ETHPM) and the Fractional Reduced Differential Transform Method (FRDTM)-for solving nonlinear fractional partial differential … This study presents a comparative analysis of two advanced analytical methods—the Elzaki Transform Homotopy Perturbation Method (ETHPM) and the Fractional Reduced Differential Transform Method (FRDTM)-for solving nonlinear fractional partial differential equations (FPDEs) arising in biological population dynamics. After establishing the mathematical foundations of fractional calculus, the Elzaki transform, and homotopy perturbation theory, we demonstrate the applicability of both methods to FPDEs modelling population growth and swarm behaviour. Our results reveal that ETHPM and FRDTM yield highly accurate approximate solutions, underscoring their efficacy as computational tools for complex biological systems. The study highlights the broader implications of these fractional-order models in ecology and population dynamics, bridging theoretical mathematics with practical applications in life sciences. Through systematic comparisons, we provide insights into the strengths and limitations of each method, offering valuable guidance for researchers working with nonlinear fractional systems in biological contexts.

Analysis of a class of fractal hybrid fractional differential equation with application to a biological model

2024-08-15
Thabet Abdeljawad , Sher Muhammad , Kamal Shah +4 more
Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. … Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer's (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end.

A comprehensive analysis of COVID-19 nonlinear mathematical model by incorporating the environment and social distancing

2024-05-28
Muhammad Bilal Riaz , Kamal Shah , Thabet Abdeljawad +3 more
Abstract This research conducts a detailed analysis of a nonlinear mathematical model representing COVID-19, incorporating both environmental factors and social distancing measures. It thoroughly analyzes the model’s equilibrium points, computes … Abstract This research conducts a detailed analysis of a nonlinear mathematical model representing COVID-19, incorporating both environmental factors and social distancing measures. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. The study develops a sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V–L) matrices method. To understand the dynamic behavior of COVID-19, numerical simulations are essential. For this purpose, the study employs a robust numerical technique known as the non-standard finite difference (NSFD) method, introduced by Mickens. Various results are visually presented through graphical representations across different parameter values to illustrate the impact of environmental factors and social distancing measures.
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Developing a fuzzy logic-based carbon emission cost-incorporated inventory model with memory effects

2024-03-23
Rituparna Pakhira , Bapin Mondal , Ashraf Adnan Thirthar +2 more
Inventory control is a widely discussed topic in the real world, and recently, it has become closely linked to concerns about carbon emissions and global warming. Global warming is a … Inventory control is a widely discussed topic in the real world, and recently, it has become closely linked to concerns about carbon emissions and global warming. Global warming is a pressing issue, mainly due to a lack of awareness and action. Traditional inventory models, which typically use integer-order differential equations, overlook the memory aspect of the system. Addressing inventory management is essential in our efforts to combat global warming. This paper introduces a novel approach by incorporating carbon emission costs within a fuzzy environment. Fractional Calculus, a powerful mathematical tool, is employed to capture the memory effect of the system. This approach distinguishes between long memory, characterized by a strong memory effect, and short memory, associated with a poor memory effect, through the use of fractional derivatives and integrals. Numerical results are analyzed based on these memory concepts. Entrepreneurs often find it difficult to determine exact values for known parameters in the real world. Therefore, this study considers the uncertain nature of ordering costs, the rate of deterioration, and demand rates as triangular fuzzy numbers. The optimal average cost and ordering intervals are determined using a solution methodology. A sensitivity analysis is performed to demonstrate how different system parameters influence the outcomes within a fuzzy environment. Notably, it is observed that profit tends to be higher under conditions of strong memory compared to poor memory effects. Moreover, it's worth emphasizing that profits are notably more favorable when employing the signed distance method compared to the graded mean integration method, especially in situations marked by strong memory effects. This study highlights the importance of considering sensitive parameters in the model, especially under conditions of strong memory effects. Such parameters require careful attention in the pursuit of effective inventory management strategies to mitigate carbon emissions and combat global warming.

Complex dynamics in a two species system with Crowley–Martin response function: Role of cooperation, additional food and seasonal perturbations

2024-03-16
Bapin Mondal , Ashraf Adnan Thirthar , Nazmul Sk +2 more

Khasminskii Approach for $$\psi $$-Caputo Fractional Stochastic Pantograph Problem

2024-02-06
Manar A. Alqudah , Hamid Boulares , Bahaaeldin Abdalla +1 more
Abstract In this manuscript, we study an averaging principle for fractional stochastic pantograph differential equations (FSDPEs) in the $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> -sense accompanied by Brownian movement. Under … Abstract In this manuscript, we study an averaging principle for fractional stochastic pantograph differential equations (FSDPEs) in the $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> -sense accompanied by Brownian movement. Under certain assumptions, we are able to approximate solutions for FSPEs by solutions to averaged stochastic systems in the sense of mean square. Analysis of system solutions before and after the average allows extending the classical Khasminskii approach to random fractional differential equations in the sense of $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> -Caputo. For clarity, we present at the end an applied example to facilitate the clarification of the theoretical results obtained
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An extension of Schweitzer's inequality to Riemann-Liouville fractional integral

2024-01-01
Thabet Abdeljawad , Badreddine Meftah , Abdelghani Lakhdari +1 more
Abstract This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental … Abstract This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.

Utilizing memory effects to enhance resilience in disease-driven prey-predator systems under the influence of global warming

2023-11-30
Ashraf Adnan Thirthar , Nazmul Sk , Bapin Mondal +2 more
Abstract This research paper presents an eco-epidemiological model that investigates the intricate dynamics of a predator–prey system, considering the impact of fear-induced stress, hunting cooperation, global warming, and memory effects … Abstract This research paper presents an eco-epidemiological model that investigates the intricate dynamics of a predator–prey system, considering the impact of fear-induced stress, hunting cooperation, global warming, and memory effects on species interactions. The model employs fractional-order derivatives to account for temporal dependencies and memory in ecological processes. By incorporating these factors, we aim to provide a more comprehensive understanding of the underlying mechanisms that govern the stability and behavior of ecological systems. Mathematically we investigate system’s existence, equilibria and their stability. Moreover, global stability and hopf bifurcation also analyzed in this study. Numerical simulations have been performed to validate the analytical results. We find that the coexistence equilibrium is stable under specific conditions, along with the predator equilibrium and the disease-free equilibrium. Bifurcation analyses demonstrate the intricate behavior of species densities in response to changes in model parameters. Fear and global warming are found to stabilize the system, while cooperation and additional food for predators lead to destabilization. Additionally, the influence of species memory has been explored. We observe that memory tends to stabilize the system as species memory levels increase.
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A FRACTAL-FRACTIONAL ORDER MODEL TO STUDY MULTIPLE SCLEROSIS: A CHRONIC DISEASE

2023-11-02
Kamal Shah , Bahaaeldin Abdalla , Thabet Abdeljawad +1 more
A mathematical model of progressive disease of the nervous system also called multiple sclerosis (MS) is studied in this paper. The proposed model is investigated under the concept of the … A mathematical model of progressive disease of the nervous system also called multiple sclerosis (MS) is studied in this paper. The proposed model is investigated under the concept of the fractal-fractional order derivative (FFOD) in the Caputo sense. In addition, the tools of nonlinear functional analysis are applied to prove some qualitative results including the existence theory, stability, and numerical analysis. For the recommended results of the existence theory, Banach and Krassnoselski’s fixed point theorems are used. Additionally, Hyers–Ulam (HU) concept is used to derive some results for stability analysis. Additionally, for numerical illustration of approximate solutions of various compartments of the considered model, the modified Euler method is utilized. The aforementioned results are displayed graphically for various values of fractal-fractional orders.

Boundary value problem of weighted fractional derivative of a function with a respect to another function of variable order

2023-10-09
Kheireddine Benia , Mohammed Said Souıd , Fahd Jarad +2 more
Abstract This study aims to resolve weighted fractional operators of variable order in specific spaces. We establish an investigation on a boundary value problem of weighted fractional derivative of one … Abstract This study aims to resolve weighted fractional operators of variable order in specific spaces. We establish an investigation on a boundary value problem of weighted fractional derivative of one function with respect to another variable order function. It is essential to keep in mind that the symmetry of a transformation for differential equations is connected to local solvability, which is synonymous with the existence of solutions. As a consequence, existence requirements for weighted fractional derivative of a function with respect to another function of constant order are necessary. Moreover, the stability with in Ulam–Hyers–Rassias sense is reviewed. The outcomes are derived using the Kuratowski measure of non-compactness. A model illustrates the trustworthiness of the observed results.
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A fractional approach to solar heating model using extended ODE system

2023-09-20
Muhammad Ullah , Qura Tul Ain , Aziz Khan +2 more
In contrast to fossil fuels, the sun has more than enough energy to supply the entire world's energy needs. The sole constraint on solar energy as a renewable resource is … In contrast to fossil fuels, the sun has more than enough energy to supply the entire world's energy needs. The sole constraint on solar energy as a renewable resource is our capacity to efficiently and economically convert it to electricity. In this work, we take advantage of the fractional derivative, we introduce the dynamics of the solar heating model. We incorporated the Atangana-Baleanu derivative (ABC) in our analysis. Then the solutions of our fractional system are investigated for existence and uniqueness. In order to visualize the fractional order model solution, we used a novel and trendy numerical method to represent the dynamics of different parameters of the nonlinear ordinary differential equation system. It is shown that the proposed ODE models are valid and efficient for fractional order data for a functioning solar heating system.

The Volterra-Lyapunov matrix theory and nonstandard finite difference scheme to study a dynamical system

2023-09-01
Muhammad Bilal Riaz , Kamal Shah , Aman Ullah +2 more
A compartmental model is considered to study the transmission dynamics of COVID-19. The proposed model is investigated for different results by using Volterra-Lyapunov (V-L) matrix theory. In this regard, first … A compartmental model is considered to study the transmission dynamics of COVID-19. The proposed model is investigated for different results by using Volterra-Lyapunov (V-L) matrix theory. In this regard, first we presented a modified form of SEIR model by incorporating three new compartments C (protected), D (death due to corona) and Q (quarantined). Both equilibrium points are computed together with basic reproductive number. In addition, local stability of both equilibrium points for our proposed model is examined by assuming that wearing of mask, testing of the unaware infected individuals and medical care of the individuals that got infected should constantly be maintained. Hence, subsequently by combining the V-L stable matrix theory with the traditional methodology of constructing the Lyapunov functions, a procedure for the global stability analysis of COVID-19 is presented. Furthermore, based on LaSalle and Lipschitz invariance principle, the global stability of disease free equilibrium point is also examined. The technique we introduced in this paper will provide the more profound comprehension to understand the basic structure of COVID-19. Moreover, for numerical interpretation of our proposed model non-standard finite difference (NSFD) scheme is utilized for simulations. Different graphical illustrations are provided to understand the transmission dynamics.

Mathematical analysis of fractional order alcoholism model

2023-07-25
Sher Muhammad , Kamal Shah , Muhammad Sarwar +2 more
In this manuscript, we are going to study a novel model of the dynamics of alcohol consumption under induced complications. The mentioned model is considered under the concept of conformable … In this manuscript, we are going to study a novel model of the dynamics of alcohol consumption under induced complications. The mentioned model is considered under the concept of conformable fractional order derivative (CFOD). Currently, most of real-world problems are considered under fractional order derivatives because of their stable and global behavior. First, we will investigate the model for qualitative theory including existence and uniqueness of solution and Ulam-Hyers stability. For qualitative theory, we will use fixed point theory. In addition, we use a numerical method to find the approximate solution of the proposed model. In the final part of the paper, we give a detailed discussion of its numerical results and its graphical presentation.

Existence Results of Langevin Equations with Caputo–Hadamard Fractional Operator

2023-06-23
Sombir Dhaniya , Anoop Kumar , Aziz Khan +2 more
In this manuscript, we deal with a nonlinear Langevin fractional differential equation that involves the Caputo–Hadamard and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. The results presented … In this manuscript, we deal with a nonlinear Langevin fractional differential equation that involves the Caputo–Hadamard and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. The results presented in this study establish the existence, uniqueness, and Hyers–Ulam (HU) stability of the solution to the proposed equation. We achieved our main result by using the Banach contraction mapping principle and Krasonoselskii’s fixed point theorem. Furthermore, we introduce an application to demonstrate the validity of the results of our findings.
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ANALYSIS OF THE CONFORMABLE TEMPORAL-FRACTIONAL SWIFT–HOHENBERG EQUATION USING A NOVEL COMPUTATIONAL TECHNIQUE

2023-01-01
Aziz Khan , Muhammad Imran Liaqat , Manar A. Alqudah +1 more
The main objective of this study is to provide a new computational procedure for extracting approximate and exact solutions of the temporal-fractional Swift–Hohenberg (S–H) equations in the context of conformable … The main objective of this study is to provide a new computational procedure for extracting approximate and exact solutions of the temporal-fractional Swift–Hohenberg (S–H) equations in the context of conformable derivatives using the conformable natural transform (CNT) and Daftardar–Jafari method (DJM). We refer to it as the “natural conformable Daftardar–Jafari method” (CNDJM). The three types of errors are assessed in order to gauge the efficiency and consistency of the proposed method. Furthermore, 2D and 3D graphics are used to compare the exact and approximate solutions. This method offers a considerable benefit over homotopy analysis and Adomian decomposition methods in terms of computational work because it does not require Adomian and He’s polynomials. The procedure is quick and easy to use.

Theory and Semi-Analytical Study of Micropolar Fluid Dynamics through a Porous Channel

2023-01-01
Aziz Khan , Sana Ullah , Kamal Shah +3 more
In this work, We are looking at the characteristics of micropolar flow in a porous channel that's being driven by suction or injection. The working of the fluid is described … In this work, We are looking at the characteristics of micropolar flow in a porous channel that's being driven by suction or injection. The working of the fluid is described in the flow model. We can reduce the governing nonlinear partial differential equations (PDEs) to a model of coupled systems of nonlinear ordinary differential equations using similarity variables (ODEs). In order to obtain the results of a coupled system of nonlinear ODEs, we discuss a method which is known as the differential transform method (DTM). The concern transform is an excellent mathematical tool to obtain the analytical series solution to the nonlinear ODEs. To observe beast agreement between analytical method and numerical method, we compare our result with the Rung-Kutta method of order four (RK4). We also provide simulation plots to the obtained result by using Mathematica. On these plots, we discuss the effect of different parameters which arise during the calculation of the flow model equations.
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EXISTENCE RESULTS FOR MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS INVOLVING ATANGANA–BALEANU DERIVATIVE

2023-01-01
AHSAN ABBAS , Nayyar Mehmood , Ali Akgül +2 more
In this paper, the existence results for the solutions of the multi-term ABC-fractional differential boundary value problem (BVP) [Formula: see text] of order [Formula: see text] with nonlocal boundary conditions … In this paper, the existence results for the solutions of the multi-term ABC-fractional differential boundary value problem (BVP) [Formula: see text] of order [Formula: see text] with nonlocal boundary conditions have been derived by using Krasnoselskii’s fixed point theorem. The uniqueness of the solution is obtained with the help of Banach contraction principle. Examples are provided to confirm our obtained results.

NUMERICAL ANALYSIS OF SOME FRACTIONAL ORDER DIFFERENTIAL EQUATIONS VIA LEGENDRE SPECTRAL METHOD

2023-01-01
Aziz Khan , Hafsa Naz , Muhammad Sarwar +3 more
In this research paper, we find the numerical solutions of fractional order scalers and coupled system of differential equations under initial conditions using shifted Legendre polynomials. By using the properties … In this research paper, we find the numerical solutions of fractional order scalers and coupled system of differential equations under initial conditions using shifted Legendre polynomials. By using the properties of shifted Legendre polynomials, we establish operational matrices of integration and differentiation in order to simplify our considered problems under initial conditions. In order to check the accuracy of the proposed model, some test problems are solved along with the graphical representations. For coupled system, we applied the algorithm to the Pharmacokinetic two-compartment model. As the proposed method is computer-oriented, we use therefore the MATLAB for required calculations. Numerical results are shown graphically.

EXISTENCE AND STABILITY RESULTS FOR COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS INVOLVING AB-CAPUTO DERIVATIVE

2023-01-01
Nayyar Mehmood , AHSAN ABBAS , Ali Akgül +2 more
In this paper, we use Krasnoselskii’s fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo … In this paper, we use Krasnoselskii’s fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo fractional derivative [Formula: see text] with boundary conditions [Formula: see text] We discuss uniqueness with the help of the Banach contraction principle. The criteria for Hyers–Ulam stability of given AB-Caputo fractional-coupled boundary value problem (BVP) is also discussed. Some examples are provided to validate our results. In Example 1, we find a unique and stable solution of AB-Caputo fractional-coupled BVP. In Example 2, the analysis of approximate and exact solutions with errors of nonlinear integral equations is elaborated with graphs.

ADAPTED HOMOTOPY PERTURBATION METHOD WITH SHEHU TRANSFORM FOR SOLVING CONFORMABLE FRACTIONAL NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

2023-01-01
Muhammad Imran Liaqat , Aziz Khan , Manar A. Alqudah +1 more
There is considerable literature on solutions to the gas-dynamic equation (GDE) and Fokker–Planck equation (FPE), where the fractional derivative is expressed in terms of the Caputo fractional derivative. There is … There is considerable literature on solutions to the gas-dynamic equation (GDE) and Fokker–Planck equation (FPE), where the fractional derivative is expressed in terms of the Caputo fractional derivative. There is hardly any work on analytical and numerical GDE and FPE solutions involving conformable fractional derivative (CFD). For the reasons stated above, we are required to solve the GDE and FPE in the form of CFD. The main goal of this research is to offer a novel combined method by employing the conformable Shehu transform (CST) and the homotopy perturbation method (HPM) for extracting analytical and numerical solutions of the time-fractional conformable GDE and FPE. The proposed method is called the conformable Shehu homotopy perturbation method (CSHPM). To evaluate its efficiency and consistency, relative and absolute errors among the approximate and exact solutions to three nonlinear problems of GDE and FPE are considered numerically and graphically. Moreover, fifth-term approximate and exact solutions are also compared by 2D and 3D graphs. This method has the benefit of not requiring any minor or major physical parameter assumptions in the problem. As a result, it may be used to solve both weakly and strongly nonlinear problems, overcoming some of the inherent constraints of classic perturbation approaches. Second, while addressing nonlinear problems, the CSHPM does not require Adomian polynomials. Therefore, to solve nonlinear GDE and FPE, just a few computations are necessary. As a consequence, it outperforms homotopy analysis and Adomian decomposition approaches significantly. It does not require discretization or linearization, unlike traditional numerical methods. The convergence and error analysis of the series solutions are also presented.

Multivariate Mittag-Leffler function and related fractional integral operators

2023-01-01
Gauhar Rahman , Muhammad Samraiz , Manar A. Alqudah +1 more
&lt;abstract&gt;&lt;p&gt;In this paper, we describe a new generalization of the multivariate Mittag-Leffler (M-L) function in terms of generalized Pochhammer symbol and study its properties. We provide a few differential and … &lt;abstract&gt;&lt;p&gt;In this paper, we describe a new generalization of the multivariate Mittag-Leffler (M-L) function in terms of generalized Pochhammer symbol and study its properties. We provide a few differential and fractional integral formulas for the generalized multivariate M-L function. Furthermore, by using the generalized multivariate M-L function in the kernel, we present a new generalization of the fractional integral operator. Finally, we describe some fundamental characteristics of generalized fractional integrals.&lt;/p&gt;&lt;/abstract&gt;

AN ECOSYSTEM MODEL WITH MEMORY EFFECT CONSIDERING GLOBAL WARMING PHENOMENA AND AN EXPONENTIAL FEAR FUNCTION

2023-01-01
Ashraf Adnan Thirthar , Prabir Panja , Aziz Khan +2 more
Global warming is becoming a big concern for the environment since it is causing serious and often unexpected impacts on species, affecting their abundance, genetic composition, behavior and survival. So, … Global warming is becoming a big concern for the environment since it is causing serious and often unexpected impacts on species, affecting their abundance, genetic composition, behavior and survival. So, the modeling study is necessary to investigate the effects of global warming in predator–prey dynamics. This research paper analyzed the memory effect evaluated by Caputo fractional derivative on predator–prey interaction using an exponential fear function with a Holling-type II function in the presence of global warming effect on prey and predator species. It is assumed that the densities of prey and predator species decrease due to the increase of global warming. It is considered that both prey and predator species are contributing to the increase of global warming. Also, it is considered that global warming is increasing constantly and decreasing due to the natural decay rate. All possible equilibria of the system are determined, and the stability of the system around all equilibria points is investigated. Around the interior equilibrium point, the Hopf bifurcation is also theoretically and numerically studied. A number of numerical simulation results are presented to demonstrate the impacts of fear, fractional order and global warming on the behavior of the model. It is observed that the global warming effect on predator species may destabilize the system but ultimately the system may become stable. Again, it is obtained that the natural decay rate of global warming can stabilize the system initially but a higher decay rate may destabilized the system. It is also found that the fractional-order model is determined to be more stable than the integer-order model.

On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives

2023-01-01
Ridha Dida , Hamid Boulares , Bahaaeldin Abdalla +2 more
&lt;abstract&gt;&lt;p&gt;Our main interest in this manuscript is to explore the main positive solutions (PS) and the first implications of their existence and uniqueness for a type of fractional pantograph differential … &lt;abstract&gt;&lt;p&gt;Our main interest in this manuscript is to explore the main positive solutions (PS) and the first implications of their existence and uniqueness for a type of fractional pantograph differential equation using Caputo fractional derivatives with a kernel depending on a strictly increasing function $ \Psi $ (shortly $ \Psi $-Caputo). Such function-dependent kernel fractional operators unify and generalize several types of fractional operators such as Riemann-Liouvile, Caputo and Hadamard etc. Hence, our investigated qualitative concepts in this work generalise and unify several existing results in literature. Using Schauder's fixed point theorem (SFPT), we prove the existence of PS to this equation with the addition of the upper and lower solution method (ULS). Furthermore using the Banach fixed point theorem (BFPT), we are able to prove the existence of a unique PS. Finally, we conclude our work and give a numerical example to explain our theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Neural networking study of worms in a wireless sensor model in the sense of fractal fractional

2023-01-01
Aziz Khan , Thabet Abdeljawad , Manar A. Alqudah
&lt;abstract&gt;&lt;p&gt;We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in … &lt;abstract&gt;&lt;p&gt;We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in the context of fractal fractional differential operators. In addition, the Banach contraction technique is utilized to achieve the existence and unique outcomes of the given model. Further, the stability of the proposed model is analyzed through functional analysis and the Ulam-Hyers (UH) stability technique. In the last, a numerical scheme is established to check the dynamical behavior of the fractional fractal order WSN model.&lt;/p&gt;&lt;/abstract&gt;

Spectral analysis of variable-order multi-terms fractional differential equations

2023-01-01
Kamal Shah , Thabet Abdeljawad , Mdi Begum Jeelani +1 more
Abstract In this work, a numerical scheme based on shifted Jacobi polynomials (SJPs) is deduced for variable-order fractional differential equations (FDEs). We find numerical solution of consider problem of fractional … Abstract In this work, a numerical scheme based on shifted Jacobi polynomials (SJPs) is deduced for variable-order fractional differential equations (FDEs). We find numerical solution of consider problem of fractional order. The proposed numerical scheme is based on operational matrices of variable-order differentiation and integration. To create the mentioned operational matrices for variable-order integration and differentiation, SJPs are used. Using the aforementioned operational matrices, we change the problem under consideration into matrix equation. The resultant matrix equation is solved by using Matlab, which executes the Gauss elimination method to provide the necessary numerical solution. The technique is effective and produced reliable outcomes. To determine the effectiveness of the suggested method, the results are compared to the outcomes of some other numerical procedure. Additional examples are included in this article to further clarify the process. For various scale levels and fractional-order values, absolute errors are also recorded.
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A Theoretical Investigation of the SARS-CoV-2 Model via Fractional Order Epidemiological Model

2022-10-27
Tahir Khan , Rahman Ullah , Thabet Abdeljawad +2 more
We propose a theoretical study investigating the spread of the novel coronavirus (COVID-19) reported in Wuhan City of China in 2019. We develop a mathematical model based on the novel … We propose a theoretical study investigating the spread of the novel coronavirus (COVID-19) reported in Wuhan City of China in 2019. We develop a mathematical model based on the novel corona virus's characteristics and then use fractional calculus to fractionalize it. Various fractional order epidemic models have been formulated and analyzed using a number of iterative and numerical approaches while the complications arise due to singular kernel. We use the well-known Caputo-Fabrizio operator for the purposes of fictionalization because this operator is based on the non-singular kernel. Moreover, to analyze the existence and uniqueness, we will use the well-known fixed point theory. We also prove that the considered model has positive and bounded solutions. We also draw some numerical simulations to verify the theoretical work via graphical representations. We believe that the proposed epidemic model will be helpful for health officials to take some positive steps to control contagious diseases.
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On Cerone's and Bellman's generalization of Steffensen's integral inequality via conformable sense

2022-10-27
Mohammed S. El-Khatib , Atta A. K. Abu Hany , Mohammed M. Matar +2 more
&lt;abstract&gt;&lt;p&gt;By making use of the conformable integrals, we establish some new results on Cerone's and Bellman's generalization of Steffensen's integral inequality. In fact, we provide a variety of generalizations of … &lt;abstract&gt;&lt;p&gt;By making use of the conformable integrals, we establish some new results on Cerone's and Bellman's generalization of Steffensen's integral inequality. In fact, we provide a variety of generalizations of Steffensen's integral inequality by using conformable calculus.&lt;/p&gt;&lt;/abstract&gt;

Implementation of the Exp-function approach for the solution of KdV equation with dual power law nonlinearity

2022-10-05
Naila Sajid , Zahida Perveen , Maasoomah Sadaf +4 more

A solution of a nonlinear Volterra integral equation with delay via a faster iteration method

2022-09-28
Godwin Amechi Okeke , Austine Efut Ofem , Thabet Abdeljawad +2 more
&lt;abstract&gt;&lt;p&gt;The purpose of this article is to study the convergence, stability and data dependence results of an iterative method for contractive-like mappings. The concept of stability considered in this study … &lt;abstract&gt;&lt;p&gt;The purpose of this article is to study the convergence, stability and data dependence results of an iterative method for contractive-like mappings. The concept of stability considered in this study is known as $ w^2 $-stability, which is larger than the simple notion of stability considered by several prominent authors. Some illustrative examples on $ w^2 $-stability of the iterative method have been presented for different choices of parameters and initial guesses. As an application of our results, we establish the existence, uniqueness and approximation results for solutions of a nonlinear Volterra integral equation with delay. Finally, we provide an illustrative example to support the application of our results. The novel results of this article extend and generalize several well known results in existing literature.&lt;/p&gt;&lt;/abstract&gt;

On Fuzzy Conformable Double Laplace Transform with Applications to燩artial Differential Equations

2022-09-20
Thabet Abdeljawad , Awais Younus , Manar A. Alqudah +1 more
The Laplace transformation is a very important integral transform, and it is extensively used in solving ordinary differential equations, partial differential equations, and several types of integro-differential equations. Our purpose … The Laplace transformation is a very important integral transform, and it is extensively used in solving ordinary differential equations, partial differential equations, and several types of integro-differential equations. Our purpose in this study is to introduce the notion of fuzzy double Laplace transform, fuzzy conformable double Laplace transform (FCDLT). We discuss some basic properties of FCDLT. We obtain the solutions of fuzzy partial differential equations (both one-dimensional and two-dimensional cases) through the double Laplace approach. We demonstrate through numerical examples that our proposed method is very successful and convenient for resolving partial differential equations.
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INVESTIGATION OF INTEGRAL BOUNDARY VALUE PROBLEM WITH IMPULSIVE BEHAVIOR INVOLVING NON-SINGULAR DERIVATIVE

2022-09-05
Kamal Shah , Thabet Abdeljawad , Arshad Ali +1 more
This paper is devoted to investigating a class of impulsive fractional order differential equations (FODEs) with integral boundary condition. For the proposed paper, we use non-singular type derivative of fractional … This paper is devoted to investigating a class of impulsive fractional order differential equations (FODEs) with integral boundary condition. For the proposed paper, we use non-singular type derivative of fractional order which has been introduced by Atangana, Baleanu and Caputo (ABC). The aforesaid type problems have numerous applications in fluid mechanics and hydrodynamics to model various problems of flow phenomenons. We establish some sufficient conditions for the existence and uniqueness of solution to the proposed problem by using classical fixed point results due to Banach and Krasnoselskii. Further, on using tools of the nonlinear analysis, sufficient conditions are developed for Hyers–Ulam (HU) type stability results. A pertinent example is given to justify our results.

Unique Solution of Integral Equations via Intuitionistic Extended Fuzzy b-Metric-Like Spaces

2022-06-23
Naeem Saleem , Khalil Javed , Fahim Ud Din +4 more
In this manuscript, our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces. We establish some fixed point theorems in this setting. Also, we plot some graphs … In this manuscript, our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces. We establish some fixed point theorems in this setting. Also, we plot some graphs of an example of obtained result for better understanding. We use the concepts of continuous triangular norms and continuous triangular conorms in an intuitionistic fuzzy metric-like space. Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions. Triangular conorms are known as dual operations of triangular norms. The obtained results boost the approaches of existing ones in the literature and are supported by some examples and applications.
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Existence of results and computational analysis of a fractional order two strain epidemic model

2022-06-03
Aziz Khan , Kamal Shah , Thabet Abdeljawad +1 more
In this paper, we investigate fractional order two strain epidemic model in the sense of ABC operator. This study includes existence and uniqueness of solution, stability and numerical simulations of … In this paper, we investigate fractional order two strain epidemic model in the sense of ABC operator. This study includes existence and uniqueness of solution, stability and numerical simulations of the model under consideration. Fixed point postulates are used for the existence and uniqueness of solution. A theoretical approach is employed to investigate sufficient results for Hyers–Ulam's stability to the model under study. For the numerical demonstration Lagrange's interpolation polynomial technique is utilized. Graphical presentations against different fractional orders are displayed.

On fractional impulsive system for methanol detoxification in human body

2022-06-01
Qura Tul Ain , Aziz Khan , Muhammad Ullah +2 more

Study of fractional order dynamics of nonlinear mathematical model

2022-05-11
Kamal Shah , Amjad Ali , Salman Zeb +3 more
This manuscript is devoted to establishing some theoretical and numerical results for a nonlinear dynamical system under Caputo fractional order derivative. Further, the said system addresses an infectious disease like … This manuscript is devoted to establishing some theoretical and numerical results for a nonlinear dynamical system under Caputo fractional order derivative. Further, the said system addresses an infectious disease like COVID-19. The proposed system involves natural death rates of susceptible, infected and recovered classes respectively. By using nonlinear analysis feasible region and boundedness have been established first in this study. Global and Local stability analysis along with basic reproduction number have also addressed by using the next generation matrix method. Upon using the fixed point approach, existence and uniqueness of the approximate solution for the mentioned problem has also investigated. Some stability results of Hyers-Ulam (H-U) type have also discussed. Further for numerical treatment, we have exercised two numerical schemes including modified Euler method (MEM) and nonstandard finite difference (NSFD) method. Further the two numerical schemes have also compared with respect to CPU time. Graphical presentations have been displayed corresponding to different fractional order by using some real data.

Fear effect in a predator-prey model with additional food, prey refuge and harvesting on super predator

2022-04-23
Ashraf Adnan Thirthar , Salam J. Majeed , Manar A. Alqudah +2 more

Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models

2022-04-18
Abdul Ghafoor , Sirajul Haq , Manzoor Hussain +2 more
In this work, an efficient and robust numerical scheme is proposed to solve the variable coefficients' fourth-order partial differential equations (FOPDEs) that arise in Euler-Bernoulli beam models. When partial differential … In this work, an efficient and robust numerical scheme is proposed to solve the variable coefficients' fourth-order partial differential equations (FOPDEs) that arise in Euler-Bernoulli beam models. When partial differential equations (PDEs) are of higher order and invoke variable coefficients, then the numerical solution is quite a tedious and challenging problem, which is our main concern in this paper. The current scheme is hybrid in nature in which the second-order finite difference is used for temporal discretization, while spatial derivatives and solutions are approximated via the Haar wavelet. Next, the integration and Haar matrices are used to convert partial differential equations (PDEs) to the system of linear equations, which can be handled easily. Besides this, we derive the theoretical result for stability via the Lax-Richtmyer criterion and verify it computationally. Moreover, we address the computational convergence rate, which is near order two. Several test problems are given to measure the accuracy of the suggested scheme. Computations validate that the present scheme works well for such problems. The calculated results are also compared with the earlier work and the exact solutions. The comparison shows that the outcomes are in good agreement with both the exact solutions and the available results in the literature.
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On computational analysis of highly nonlinear model addressing real world applications

2022-03-26
S. Ali , Aziz Khan , Kamal Shah +3 more
This paper presents a numerical strategy for solving boundary value problems (BVPs) that is based on the Haar wavelets method (HWM). BVPs having high Prandtl numbers are discussed, Because they … This paper presents a numerical strategy for solving boundary value problems (BVPs) that is based on the Haar wavelets method (HWM). BVPs having high Prandtl numbers are discussed, Because they are very important in many practical problems of science and engineering. By using group-theoretic method, the considered model of partial differential equations (PDEs) are converted to system of nonlinear ordinary differential equations. By using HWM, the numerical results are established. Further, solutions obtained on a coarse resolution with low accuracy is refined towards higher accuracy by increasing the level of resolution. Superiority of the HWM has been established over the commercial software NDSolve and available numerical and approximated methods.

Numerical Approximations for the Solutions of Fourth Order Time Fractional Evolution Problems Using a Novel Spline Technique

2022-03-19
Ghazala Akram , Muhammad Abbas , Hira Tariq +3 more
Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems … Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline technique (SPST). These equations are frequently applied in a wide variety of real-world applications, such as strain gradient elasticity, phase separation in binary mixtures, and modelling of thin beams and plates, all of which are key parts of mechanical engineering. The SPST can be used for space discretization, whereas the backward Euler formula can be used for time discretization. For the temporal discretization, the method’s convergence and stability are assessed. To show the accuracy and applicability of the proposed technique, numerical simulations are employed.
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Numerical approximations based on sextic B-spline functions for solving fourth-order singular problems

2022-02-04
M. Esther Rani , Farah Aini Abdullah , Irum Samreen +4 more
In this work, a new access for solving fourth-order singular boundary value problem (SBVP) by using sextic B-spline is presented. The SBVP yields in mathematical modelling of several real-world development … In this work, a new access for solving fourth-order singular boundary value problem (SBVP) by using sextic B-spline is presented. The SBVP yields in mathematical modelling of several real-world development essentially electro-hydrodynamics, chemical reactions, thermal explosions, fluid dynamics, aerodynamics and atomic nuclear reactions. It must be indicated that the original non-linear boundary value problem (BVP) is first remodelled into a linear BVP by quasilinearization technique. Convergence analysis of the method is explored. Three trial evidences are examined to determine the accuracy and suitability of the described technique and to validate the analytical results. The proposed method is simple, robust and straightforward for solving the non-linear SBVP.

Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials

2022-01-31
Zulfiqar Ahmad Noor , Imran Talib , Thabet Abdeljawad +1 more
In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational … In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational matrix of the fractional-order derivatives in the Caputo sense, which is then used with spectral tau and spectral collocation methods to reduce the FDEs to a system of algebraic equations. Several numerical examples are given to show the accuracy of this method. These examples show that the obtained results have good agreement with the analytical solutions in both linear and non-linear FDEs. In addition to this, the numerical results obtained by using our method are compared with the numerical results obtained otherwise in the literature.
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Numerical solutions of time fractional Burgers’ equation involving Atangana–Baleanu derivative via cubic B-spline functions

2022-01-31
Madiha Shafiq , Muhammad Abbas , Farah Aini Abdullah +3 more
The current paper uses the cubic B-spline functions and θ-weighted scheme to achieve numerical solutions of the time fractional Burgers' equation with Atangana–Baleanu derivative. A non-singular kernel is involved in … The current paper uses the cubic B-spline functions and θ-weighted scheme to achieve numerical solutions of the time fractional Burgers' equation with Atangana–Baleanu derivative. A non-singular kernel is involved in the Atangana–Baleanu fractional derivative. For discretization along temporal and spatial grids, the proposed numerical technique employs the finite difference approach and cubic B-spline functions, respectively. This scheme is unconditionally stable and second order convergent in spatial and temporal directions. The presented approach is endorsed by some numerical examples, which show that it is applicable and accurate.

Fixed point theory in complex valued controlled metric spaces with an application

2022-01-01
Muhammad Suhail Aslam , Mohammad S. R. Chowdhury , Liliana Guran +2 more
&lt;abstract&gt;&lt;p&gt;In this article we have introduced a metric named complex valued controlled metric type space, more generalized form of controlled metric type spaces. This concept is a new extension of … &lt;abstract&gt;&lt;p&gt;In this article we have introduced a metric named complex valued controlled metric type space, more generalized form of controlled metric type spaces. This concept is a new extension of the concept complex valued $ b $-metric type space and this one is different from complex valued extended $ b $-metric space. Using the idea of this new metric, some fixed point theorems involving Banach, Kannan and Fisher contractions type are proved. Some examples togetheran application are described to sustain our primary results.&lt;/p&gt;&lt;/abstract&gt;

Generalized exponential function and initial value problem for conformable dynamic equations

2022-01-01
Awais Younus , Khizra Bukhsh , Manar A. Alqudah +1 more
&lt;abstract&gt;&lt;p&gt;In this article, we define the generalized exponential function on arbitrary time scales in the conformable setting and develop its fundamental characteristics. We address the fundamental theory of a conformable … &lt;abstract&gt;&lt;p&gt;In this article, we define the generalized exponential function on arbitrary time scales in the conformable setting and develop its fundamental characteristics. We address the fundamental theory of a conformable fractional dynamic equation on time scales, subject to the local and non-local initial conditions. We generalized the Grönwall type inequalities in a conformable environment. The generalized exponential function and the Grönwall's inequalities are indispensable for the study of the qualitative aspects of the local initial value problem. We developed some criteria related to global existence, extension and boundedness, as well as stability of solutions.&lt;/p&gt;&lt;/abstract&gt;

An existence result involving both the generalized proportional Riemann-Liouville and Hadamard fractional integral equations through generalized Darbo's fixed point theorem

2022-01-01
Rahul Rahul , Nihar Kumar Mahato , Sumati Kumari Panda +2 more
&lt;abstract&gt;&lt;p&gt;In this paper, we propose and prove an extension and generalization, which extends and generalizes the Darbo's fixed point theorem (DFPT) in the context of measure of noncompactness (MNC). Thereafter, … &lt;abstract&gt;&lt;p&gt;In this paper, we propose and prove an extension and generalization, which extends and generalizes the Darbo's fixed point theorem (DFPT) in the context of measure of noncompactness (MNC). Thereafter, we use DFPT to investigate the existence of solutions to mixed-type fractional integral equations (FIE), which include both the generalized proportional $ (\kappa, \tau) $-Riemann-Liouville and Hadamard fractional integral equations. We've included a suitable example to strengthen the article.&lt;/p&gt;&lt;/abstract&gt;
Coauthor Papers Together
Thabet Abdeljawad 76
Kamal Shah 19
Aziz Khan 12
Fahd Jarad 11
Ashraf Adnan Thirthar 5
Bahaaeldin Abdalla 5
Pshtiwan Othman Mohammed 5
Muhammad Abbas 4
Muhammad Sarwar 3
Artion Kashuri 3
Hamid Boulares 3
Ali Akgül 3
Asma Al-Jaser 3
Bapin Mondal 3
Y. S. Hamed 3
Rehana Ashraf 2
Fatma Bozkurt 2
Saima Rashid 2
Muhammad Ullah 2
Prabir Panja 2
AHSAN ABBAS 2
Nayyar Mehmood 2
Mohammed S. ‬Abdo 2
Matloob Anwar 2
Farah Aini Abdullah 2
Imran Talib 2
Muhammad Bilal Riaz 2
Nazmul Sk 2
Muhammad Samraiz 2
Qura Tul Ain 2
Muhammad Imran Liaqat 2
Muhammad Raees 2
Mdi Begum Jeelani 2
Ghazala Akram 2
Noufe H. Aljahdaly 2
Abdul Majeed 2
Inas Amacha 2
Maasoomah Sadaf 2
Awais Younus 2
Gauhar Rahman 2
Gul Zaman 2
Sher Muhammad 2
Manzoor Hussain 1
Hafsa Naz 1
Samiha Belmor 1
Saif Ullah 1
Zahida Perveen 1
Hira Tariq 1
Madiha Shafiq 1
Atta A. K. Abu Hany 1

New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model

2016-01-01
Abdon Atangana , Dumitru Băleanu
In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional … In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional heat transfer model.
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A new definition of fractional derivative

2014-01-16
Roshdi Khalil , Mohammed Al Horani , A. Yousef +1 more
We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The … We give a new definition of fractional derivative and fractional integral. The form of the definition shows that it is the most natural definition, and the most fruitful one. The definition for 0≤α<1 coincides with the classical definitions on polynomials (up to a constant). Further, if α=1, the definition coincides with the classical definition of first derivative. We give some applications to fractional differential equations.

Stability analysis and numerical solutions of fractional order HIV/AIDS model

2019-03-25
Aziz Khan , J.F. Gómez‐Aguilar , Tahir Khan +1 more

Theory and Applications of Fractional Differential Equations

2006-01-01
Anatoly A. Kilbas , H. M. Srivastava , Juan J. Trujillo

On conformable fractional calculus

2014-10-29
Thabet Abdeljawad

Applications of Fractional Calculus in Physics

2000-03-01
R. Hilfer

Fractional Differential Equations - An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications

1999-01-01

Generalized fractional derivatives and Laplace transform

2019-03-21
Fahd Jarad , Thabet Abdeljawad
In this article, we study generalized fractional derivatives that contain kernels depending on a function on the space of absolute continuous functions. We generalize the Laplace transform in order to … In this article, we study generalized fractional derivatives that contain kernels depending on a function on the space of absolute continuous functions. We generalize the Laplace transform in order to be applicable for the generalized fractional integrals and derivatives and apply this transform to solve some ordinary differential equations in the frame of the fractional derivatives under discussion.

ON FRACTAL-FRACTIONAL WATERBORNE DISEASE MODEL: A STUDY ON THEORETICAL AND NUMERICAL ASPECTS OF SOLUTIONS VIA SIMULATIONS

2023-01-01
Hasib Khan , Jehad Alzabut , Anwar Shah +4 more
Waterborne diseases are illnesses caused by pathogenic bacteria that spread through water and have a negative influence on human health. Due to the involvement of most countries in this vital … Waterborne diseases are illnesses caused by pathogenic bacteria that spread through water and have a negative influence on human health. Due to the involvement of most countries in this vital issue, accurate analysis of mathematical models of such diseases is one of the first priorities of researchers. In this regard, in this paper, we turn to a waterborne disease model for solution’s existence, HU-stability, and computational analysis. We transform the model to an analogous fractal-fractional integral form and study its qualitative analysis using an iterative convergent sequence and fixed-point technique to see whether there is a solution. We use Lagrange’s interpolation to construct numerical algorithms for the fractal-fractional waterborne disease model in terms of computations. The approach is then put to the test in a case study, yielding some interesting outcomes.

Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel

2017-03-13
Thabet Abdeljawad , Dumitru Băleanu
In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the newly suggested nonlocal fractional derivative with Mittag-Leffler kernel.Then, we obtain the related integration … In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the newly suggested nonlocal fractional derivative with Mittag-Leffler kernel.Then, we obtain the related integration by parts formula.We use the Q-operator to confirm our results.The related Euler-Lagrange equations are reported and one illustrative example is discussed.
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Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system

2017-04-25
Abdon Atangana

On the Stability of the Linear Functional Equation

1941-04-15
D. H. Hyers
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Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels

2016-09-05
Thabet Abdeljawad , Dumitru Băleanu
In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration … In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration by parts formulas are proved. Then a discrete variational problem is considered with an illustrative example. Finally, some more tools for these derivatives and their discrete versions have been obtained.

Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel

2019-07-20
Aziz Khan , Hasib Khan , J.F. Gómez‐Aguilar +1 more

On the generalized fractional derivatives and their Caputo modification

2017-05-12
Fahd Jarad , Thabet Abdeljawad , Dumitru Băleanu
In this manuscript, we define the generalized fractional derivative onWe present some of the properties of generalized fractional derivatives of these functions and then we define their Caputo version. In this manuscript, we define the generalized fractional derivative onWe present some of the properties of generalized fractional derivatives of these functions and then we define their Caputo version.
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Results concerning to approximate controllability of non‐densely defined Sobolev‐type Hilfer fractional neutral delay differential system

2021-07-08
Kottakkaran Sooppy Nisar , V. Vijayakumar
In this manuscript, our primary focus on the approximate controllability outcomes for non‐densely defined Sobolev‐type Hilfer fractional neutral differential system with infinite delay. By applying the findings and facts associated … In this manuscript, our primary focus on the approximate controllability outcomes for non‐densely defined Sobolev‐type Hilfer fractional neutral differential system with infinite delay. By applying the findings and facts associated with fractional theory and the fixed‐point method, the principal discussions are demonstrated. First, we focus on the existence and then discuss the approximate controllability. At last, we provide an example for the demonstration of theory.

The hunting cooperation of a predator under two prey's competition and fear-effect in the prey-predator fractional-order model

2022-01-01
Ali Yousef , Ashraf Adnan Thirthar , Abdesslem Larmani Alaoui +2 more
&lt;abstract&gt;&lt;p&gt;This paper investigates a fractional-order mathematical model of predator-prey interaction in the ecology considering the fear of the prey, which is generated in addition by competition of two prey species, … &lt;abstract&gt;&lt;p&gt;This paper investigates a fractional-order mathematical model of predator-prey interaction in the ecology considering the fear of the prey, which is generated in addition by competition of two prey species, to the predator that is in cooperation with its species to hunt the preys. At first, we show that the system has non-negative solutions. The existence and uniqueness of the established fractional-order differential equation system were proven using the Lipschitz Criteria. In applying the theory of Routh-Hurwitz Criteria, we determine the stability of the equilibria based on specific conditions. The discretization of the fractional-order system provides us information to show that the system undergoes Neimark-Sacker Bifurcation. In the end, a series of numerical simulations are conducted to verify the theoretical part of the study and authenticate the effect of fear and fractional order on our model's behavior.&lt;/p&gt;&lt;/abstract&gt;

A Caputo fractional derivative of a function with respect to another function

2016-09-13
Ricardo Almeida
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New approach to a generalized fractional integral

2011-04-15
Udita N. Katugampola
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Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

2019-03-21
Hasib Khan , Yongjin Li , Aftab Khan +1 more
In the literature, many researchers have studied Lotka‐Volterra (L‐V) models for different types of studies. In order to continue the study, we consider a fractional‐order L‐V model involving three different … In the literature, many researchers have studied Lotka‐Volterra (L‐V) models for different types of studies. In order to continue the study, we consider a fractional‐order L‐V model involving three different species in the Atangana‐Baleanu‐Caputo (ABC) sense of fractional derivative. This new model has potentials for a large number of research‐oriented studies. The first point that arises is whether the new model has a solution or not. Therefore, to answer this question, we consider the existence and uniqueness (EU) of the solutions and then Hyers‐Ulam (HU) stability for the proposed L‐V model.

On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative

2018-10-10
Fahd Jarad , Thabet Abdeljawad , Zakia Hammouch

Existence result for a neutral fractional integro-di erential equation with state dependent delay

2018-12-01
K. Jothimani , N. Valliammal , C. Ravichandran

North-Holland Mathematics Studies 202

2005-01-01

A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel

2017-06-05
Thabet Abdeljawad
In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order [Formula: see text] to higher arbitrary order and we … In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order [Formula: see text] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ([Formula: see text]) and Riemann ([Formula: see text]) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order [Formula: see text] in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

A fractional-order epidemic model with time-delay and nonlinear incidence rate

2019-06-08
Fathalla A. Rihan , Qasem M. Al‐Mdallal , Hebatallah J. Alsakaji +1 more

The Analysis of Fractional Differential Equations

2010-01-01
Kai Diethelm

A fractional order HIV‐TB coinfection model with nonsingular Mittag‐Leffler Law

2020-01-18
Hasib Khan , J.F. Gómez‐Aguilar , Abdulwasea Alkhazzan +1 more
The biological models for the study of human immunodeficiency virus (HIV) and its advanced stage acquired immune deficiency syndrome (AIDS) have been widely studied in last two decades. HIV virus … The biological models for the study of human immunodeficiency virus (HIV) and its advanced stage acquired immune deficiency syndrome (AIDS) have been widely studied in last two decades. HIV virus can be transmitted by different means including blood, semen, preseminal fluid, rectal fluid, breast milk, and many more. Therefore, initiating HIV treatment with the TB treatment development has some advantages including less HIV‐related losses and an inferior risk of HIV spread also having difficulties including incidence of immune reconstitution inflammatory syndrome (IRIS) because of a large pill encumbrance. It has been analyzed that patients with HIV have more weaker immune system and are susceptible to infections, for example, tuberculosis (TB). Keeping the importance of the HIV models, we are interested to consider an analysis of HIV‐TB coinfected model in the Atangana‐Baleanu fractional differential form. The model is studied for the existence, uniqueness of solution, Hyers‐Ulam (HU) stability and numerical simulations with assumption of specific parameters.

Haar wavelet collocation approach for the solution of fractional order COVID-19 model using Caputo derivative

2020-09-14
Kamal Shah , Zareen A. Khan , Amjad Ali +3 more
This article is devoted to study a compartmental mathematical model for the transmission dynamics of the novel Coronavirus-19 under Caputo fractional order derivative. By using fixed point theory of Schauder's … This article is devoted to study a compartmental mathematical model for the transmission dynamics of the novel Coronavirus-19 under Caputo fractional order derivative. By using fixed point theory of Schauder's and Banach we establish some necessary conditions for existence of at least one solution to model under investigation and its uniqueness. After the existence a general numerical algorithm based on Haar collocation method is established to compute the approximate solution of the model. Using some real data we simulate the results for various fractional order using Matlab to reveal the transmission dynamics of the current disease due to Coronavirus-19 through graphs.

Stability analysis of a dynamical model of tuberculosis with incomplete treatment

2020-09-17
Ihsan Ullah , Saeed Ahmad , Qasem M. Al‐Mdallal +3 more
Abstract A simple deterministic epidemic model for tuberculosis is addressed in this article. The impact of effective contact rate, treatment rate, and incomplete treatment versus efficient treatment is investigated. We … Abstract A simple deterministic epidemic model for tuberculosis is addressed in this article. The impact of effective contact rate, treatment rate, and incomplete treatment versus efficient treatment is investigated. We also analyze the asymptotic behavior, spread, and possible eradication of the TB infection. It is observed that the disease transmission dynamics is characterized by the basic reproduction ratio $\Re _{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℜ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> ; if $\Re _{0}&lt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℜ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math> , there is only a disease-free equilibrium which is both locally and globally asymptotically stable. Moreover, for $\Re _{0}&gt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℜ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:math> , a unique positive endemic equilibrium exists which is globally asymptotically stable. The global stability of the equilibria is shown via Lyapunov function. It is also obtained that incomplete treatment of TB causes increase in disease infection while efficient treatment results in a reduction in TB. Finally, for the estimated parameters, some numerical simulations are performed to verify the analytical results. These numerical results indicate that decrease in the effective contact rate λ and increase in the treatment rate γ play a significant role in the TB infection control.
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Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications

2017-10-17
Ricardo Almeida , Agnieszka B. Malinowska , M. Teresa T. Monteiro
This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo‐type fractional derivative with respect to another function. Existence and uniqueness … This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo‐type fractional derivative with respect to another function. Existence and uniqueness results for the problem are established by means of the some standard fixed point theorems. Next, we develop the Picard iteration method for solving numerically the problem and obtain results on the long‐term behavior of solutions. Finally, we analyze a population growth model and a gross domestic product model with governing equations being fractional differential equations that we have introduced in this work.
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On Fractional Derivatives with Exponential Kernel and their Discrete Versions

2017-08-01
Thabet Abdeljawad , Dumitru Băleanu
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Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination?

2020-05-28
Abdon Atangana
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Computational and theoretical modeling of the transmission dynamics of novel COVID-19 under Mittag-Leffler Power Law

2020-07-15
Sher Muhammad , Kamal Shah , Zareen A. Khan +2 more
In the current article, we studied the novel corona virus (2019-nCoV or COVID-19) which is a threat to the whole world nowadays. We consider a fractional order epidemic model which … In the current article, we studied the novel corona virus (2019-nCoV or COVID-19) which is a threat to the whole world nowadays. We consider a fractional order epidemic model which describes the dynamics of COVID-19 under nonsingular kernel type of fractional derivative. An attempt is made to discuss the existence of the model using the fixed point theorem of Banach and Krasnoselskii's type. We will also discuss the Ulam-Hyers type of stability of the mentioned problem. For semi analytical solution of the problem the Laplace Adomian decomposition method (LADM) is suggested to obtain the required solution. The results are simulated via Matlab by graphs. Also we have compare the simulated results with some reported real data for Commutative class at classical order.

Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative

2020-03-14
Muhammad Altaf Khan , Abdon Atangana
The present paper describes the mathematical modeling and dynamics of a novel corona virus (2019-nCoV). We describe the brief details of interaction among the bats and unknown hosts, then among … The present paper describes the mathematical modeling and dynamics of a novel corona virus (2019-nCoV). We describe the brief details of interaction among the bats and unknown hosts, then among the peoples and the infections reservoir (seafood market). The seafood marked are considered the main source of infection when the bats and the unknown hosts (may be wild animals) leaves the infection there. The purchasing of items from the seafood market by peoples have the ability to infect either asymptomatically or symptomatically. We reduced the model with the assumptions that the seafood market has enough source of infection that can be effective to infect people. We present the mathematical results of the model and then formulate a fractional model. We consider the available infection cases for January 21, 2020, till January 28, 2020 and parameterized the model. We compute the basic reproduction number for the data is R0≈2.4829. The fractional model is then solved numerically by presenting many graphical results, which can be helpful for the infection minimization.

Fractional order mathematical modeling of COVID-19 transmission

2020-09-02
Shabir Ahmad , Aman Ullah , Qasem M. Al‐Mdallal +3 more

Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness

2021-04-15
K. Kavitha , V. Vijayakumar , R. Udhayakumar +1 more
Abstract This manuscript mainly focuses on the controllability of Hilfer fractional differential system with infinite delay. We study our primary outcomes by employing fractional calculus, measures of noncompactness, and fixed‐point … Abstract This manuscript mainly focuses on the controllability of Hilfer fractional differential system with infinite delay. We study our primary outcomes by employing fractional calculus, measures of noncompactness, and fixed‐point approach. Then, we extend our results to the concept of nonlocal conditions. Lastly, an example is given for illustration of theory.

Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study

2020-01-31
Joseph T. Wu , Kathy Leung , GM Leung
Since Dec 31, 2019, the Chinese city of Wuhan has reported an outbreak of atypical pneumonia caused by the 2019 novel coronavirus (2019-nCoV). Cases have been exported to other Chinese … Since Dec 31, 2019, the Chinese city of Wuhan has reported an outbreak of atypical pneumonia caused by the 2019 novel coronavirus (2019-nCoV). Cases have been exported to other Chinese cities, as well as internationally, threatening to trigger a global outbreak. Here, we provide an estimate of the size of the epidemic in Wuhan on the basis of the number of cases exported from Wuhan to cities outside mainland China and forecast the extent of the domestic and global public health risks of epidemics, accounting for social and non-pharmaceutical prevention interventions.We used data from Dec 31, 2019, to Jan 28, 2020, on the number of cases exported from Wuhan internationally (known days of symptom onset from Dec 25, 2019, to Jan 19, 2020) to infer the number of infections in Wuhan from Dec 1, 2019, to Jan 25, 2020. Cases exported domestically were then estimated. We forecasted the national and global spread of 2019-nCoV, accounting for the effect of the metropolitan-wide quarantine of Wuhan and surrounding cities, which began Jan 23-24, 2020. We used data on monthly flight bookings from the Official Aviation Guide and data on human mobility across more than 300 prefecture-level cities in mainland China from the Tencent database. Data on confirmed cases were obtained from the reports published by the Chinese Center for Disease Control and Prevention. Serial interval estimates were based on previous studies of severe acute respiratory syndrome coronavirus (SARS-CoV). A susceptible-exposed-infectious-recovered metapopulation model was used to simulate the epidemics across all major cities in China. The basic reproductive number was estimated using Markov Chain Monte Carlo methods and presented using the resulting posterior mean and 95% credibile interval (CrI).In our baseline scenario, we estimated that the basic reproductive number for 2019-nCoV was 2·68 (95% CrI 2·47-2·86) and that 75 815 individuals (95% CrI 37 304-130 330) have been infected in Wuhan as of Jan 25, 2020. The epidemic doubling time was 6·4 days (95% CrI 5·8-7·1). We estimated that in the baseline scenario, Chongqing, Beijing, Shanghai, Guangzhou, and Shenzhen had imported 461 (95% CrI 227-805), 113 (57-193), 98 (49-168), 111 (56-191), and 80 (40-139) infections from Wuhan, respectively. If the transmissibility of 2019-nCoV were similar everywhere domestically and over time, we inferred that epidemics are already growing exponentially in multiple major cities of China with a lag time behind the Wuhan outbreak of about 1-2 weeks.Given that 2019-nCoV is no longer contained within Wuhan, other major Chinese cities are probably sustaining localised outbreaks. Large cities overseas with close transport links to China could also become outbreak epicentres, unless substantial public health interventions at both the population and personal levels are implemented immediately. Independent self-sustaining outbreaks in major cities globally could become inevitable because of substantial exportation of presymptomatic cases and in the absence of large-scale public health interventions. Preparedness plans and mitigation interventions should be readied for quick deployment globally.Health and Medical Research Fund (Hong Kong, China).
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A contribution to the mathematical theory of epidemics

1927-08-01
W. O. Kermack , A. G. McKendrick
(1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude … (1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important probems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population. It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.

A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative

2020-01-31
Behzad Ghanbari , Sunil Kumar , Ranbir Kumar

On a Special Functional Equation

1940-04-01
Kurt Mahler

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

2002-11-01
P. van den Driessche , James Watmough

A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative

2020-02-25
Dumitru Băleanu , Amin Jajarmi , Hakimeh Mohammadi +1 more

On the ψ -Hilfer fractional derivative

2018-01-09
J. Vanterler da C. Sousa , E. Capelas de Oliveira

On a Functional Differential Equation

1971-01-01
L. Fox , D. F. Mayers , J. R. Ockendon +1 more
This paper considers some analytical and numerical aspects of the problem defined by an equation or systems of equations of the type (d/dt)y(t) = ay(λt)+by(t), with a given initial condition … This paper considers some analytical and numerical aspects of the problem defined by an equation or systems of equations of the type (d/dt)y(t) = ay(λt)+by(t), with a given initial condition y(0) = 1. Series, integral representations and asymptotic expansions for y are obtained and discussed for various ranges of the parameters a, b and λ(> 0), and for all positive values of the argument t. A perturbation solution is constructed for ∣1−λ∣ ≪ 1, and confirmed by direct computation. For λ > 1 the solution is not unique, and an analysis is included of the eigensolutions for which y(0) = 0. Two numerical methods are analysed and illustrated. The first, using finite differences, is applicable for λ < 1, and two techniques are demonstrated for accelerating the convergence of the finite-difference solution towards the true solution. The second, an adaptation of the Lanczos τ method, is applicable for any λ > 0, though an error analysis is available only for λ < 1. Numerical evidence suggests that for λ > 1 the method still gives good approximations to some solution of the problem.

Recent applications of fractional calculus to science and engineering

2003-01-01
Lokenath Debnath
This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional‐order multipoles in electromagnetism, electrochemistry, tracer in … This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional‐order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons in biology. Special attention is given to numerical computation of fractional derivatives and integrals.
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A singular ABC-fractional differential equation with p-Laplacian operator

2019-09-03
Hasib Khan , Fahd Jarad , Thabet Abdeljawad +1 more

On a Fractional Operator Combining Proportional and Classical Differintegrals

2020-03-06
Dumitru Băleanu , Arran Fernandez , Ali Akgül
The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t … The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.
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Solutions of the Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space

2019-05-18
Thabet Abdeljawad , Ravi P. Agarwal , Erdal Karapınar +1 more
The present paper aims to define three new notions: Θ e -contraction, a Hardy–Rogers-type Θ -contraction, and an interpolative Θ -contraction in the framework of extended b-metric space. Further, some … The present paper aims to define three new notions: Θ e -contraction, a Hardy–Rogers-type Θ -contraction, and an interpolative Θ -contraction in the framework of extended b-metric space. Further, some fixed point results via these new notions and the study endeavors toward a feasible solution would be suggested for nonlinear Volterra–Fredholm integral equations of certain types, as well as a solution to a nonlinear fractional differential equation of the Caputo type by using the obtained results. It also considers a numerical example to indicate the effectiveness of this new technique.
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Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)

2006-02-01
Anatoly A. Kilbas , H. M. Srivastava , Juan J. Trujillo

New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models

2017-10-01
Toufik Mekkaoui , Abdon Atangana