Kamal Shah

Smoking is one of the major cause of health problems around the globe. The present article deals with the dynamics of giving up smoking model of fractional order. We study … Smoking is one of the major cause of health problems around the globe. The present article deals with the dynamics of giving up smoking model of fractional order. We study analytical solution (approximate solution) of the concerned model with the help of Laplace transformation. The solution of the model will be obtained in form of infinite series which converges rapidly to its exact value. Moreover, we compare our results with the results obtained by Runge-Kutta method. Some plots are presented to show the reliability and simplicity of the method.

Evaluation of one dimensional fuzzy fractional partial differential equations

2020-06-26

Kamal Shah , Aly R. Seadawy , Muhammad Arfan

This manuscript is related to investigate analytical solutions to some linear fractional partial fuzzy differential equations under certain conditions. For the concerned investigation, we utilize Laplace transform along with some … This manuscript is related to investigate analytical solutions to some linear fractional partial fuzzy differential equations under certain conditions. For the concerned investigation, we utilize Laplace transform along with some decomposition method to compute series type solution under fuzzy concept. Some examples are solved to demonstrate the proposed method.

On Ulam’s Stability for a Coupled Systems of Nonlinear Implicit Fractional Differential Equations

2018-04-30

Zeeshan Ali , Akbar Zada , Kamal Shah

On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative

2020-03-14

Kamal Shah , Fahd Jarad , Thabet Abdeljawad

In this manuscript, we investigate epidemic model of dengue fever disease under Caputo and Fabrizio fractional derivative abbreviated as (CFFD). The respective investigation is devoted to qualitative theory of existence … In this manuscript, we investigate epidemic model of dengue fever disease under Caputo and Fabrizio fractional derivative abbreviated as (CFFD). The respective investigation is devoted to qualitative theory of existence of solution for the model under consideration by using fixed point theory. After the establishing the qualitative aspect, we apply Laplace transform coupled with Adomian decomposition method to develop an algorithm for semi analytical solution under CFFD. In same line, we also develop the semi analytical solution for the considered model under usual Caputo fractional derivative (CFD). By using Matlab, we present both type of solutions via graphs and hence give some comparative remarks about the nature of the solutions of both derivatives.

Investigating a nonlinear dynamical model of COVID-19 disease under fuzzy caputo, random and ABC fractional order derivative

2020-08-25

Mati ur Rahman , Muhammad Arfan , Kamal Shah +1 more

Study of transmission dynamics of COVID-19 mathematical model under ABC fractional order derivative

2020-10-12

Sabri T. M. Thabet , Mohammed S. ‬Abdo , Kamal Shah +1 more

The current research work is devoted to address some results related to the existence and stability as well as numerical finding of a novel Coronavirus disease (COVID-19) by using a … The current research work is devoted to address some results related to the existence and stability as well as numerical finding of a novel Coronavirus disease (COVID-19) by using a mathematical model. By using fixed point results we establish existence results for the proposed model under Atangana-Baleanu-Caputo (ABC) derivative with fractional order. Further, using the famous numerical technique due to Adams Bashforth, we simulate the concerned results for two famous cities of China known as Wuhan and Huanggang which are interconnected cities. The graphical presentations are given to observe the transmission dynamics from February 1 a=2020 to April 20, 2020 through various fractional order. The concerned dynamics is global in nature due to the various values of fractional order.

Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations

2015-06-23

Kamal Shah , Hammad Khalil , Rahmat Ali Khan

Qualitative Analysis of a Mathematical Model in the Time of COVID-19

2020-05-27

Kamal Shah , Thabet Abdeljawad , Ibrahim Mahariq +1 more

In this article, a qualitative analysis of the mathematical model of novel corona virus named COVID-19 under nonsingular derivative of fractional order is considered. The concerned model is composed of … In this article, a qualitative analysis of the mathematical model of novel corona virus named COVID-19 under nonsingular derivative of fractional order is considered. The concerned model is composed of two compartments, namely, healthy and infected. Under the new nonsingular derivative, we, first of all, establish some sufficient conditions for existence and uniqueness of solution to the model under consideration. Because of the dynamics of the phenomenon when described by a mathematical model, its existence must be guaranteed. Therefore, via using the classical fixed point theory, we establish the required results. Also, we present the results of stability of Ulam’s type by using the tools of nonlinear analysis. For the semianalytical results, we extend the usual Laplace transform coupled with Adomian decomposition method to obtain the approximate solutions for the corresponding compartments of the considered model. Finally, in order to support our study, graphical interpretations are provided to illustrate the results by using some numerical values for the corresponding parameters of the model.

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Mathematical analysis of SIRD model of COVID-19 with Caputo fractional derivative based on real data

2020-12-29

Kottakkaran Sooppy Nisar , Shabir Ahmad , Aman Ullah +3 more

We discuss a fractional-order SIRD mathematical model of the COVID-19 disease in the sense of Caputo in this article. We compute the basic reproduction number through the next-generation matrix. We … We discuss a fractional-order SIRD mathematical model of the COVID-19 disease in the sense of Caputo in this article. We compute the basic reproduction number through the next-generation matrix. We derive the stability results based on the basic reproduction number. We prove the results of the solution existence and uniqueness via fixed point theory. We utilize the fractional Adams–Bashforth method for obtaining the approximate solution of the proposed model. We illustrate the obtained numerical results in plots to show the COVID-19 transmission dynamics. Further, we compare our results with some reported real data against confirmed infected and death cases per day for the initial 67 days in Wuhan city.

Analytical Solutions of Fractional Order Diffusion Equations by Natural Transform Method

2016-10-27

Kamal Shah , Hammad Khalil , Rahmat Ali Khan

Existence and Hyers‐Ulam stability of fractional nonlinear impulsive switched coupled evolution equations

2018-01-19

JinRong Wang , Kamal Shah , Amjad Ali

In this paper, we study a class of fractional nonlinear impulsive switched coupled evolution equations. Existence and uniqueness of solutions as well as Hyers‐Ulam stability results are presented. An example … In this paper, we study a class of fractional nonlinear impulsive switched coupled evolution equations. Existence and uniqueness of solutions as well as Hyers‐Ulam stability results are presented. An example is provided for the verification of our results.

Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative

2019-01-01

Sajjad Ali Khan , Kamal Shah , Gul Zaman +1 more

In this paper, taking fractional derivative due to Caputo and Fabrizo, we have investigated a biological model of smoking type. By using Sumudu transform and Picard successive iterative technique, we … In this paper, taking fractional derivative due to Caputo and Fabrizo, we have investigated a biological model of smoking type. By using Sumudu transform and Picard successive iterative technique, we develop the iterative solutions for the considered model. Furthermore, some results related to uniqueness of the equilibrium solution and its stability are discussed utilizing the techniques of nonlinear functional analysis. The dynamics of iterative solutions for various compartments of the model are plotted with the help of Matlab.

ON THE WEIGHTED FRACTIONAL OPERATORS OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION

2020-05-06

Fahd Jarad , Thabet Abdeljawad , Kamal Shah

The primary goal of this study is to define the weighted fractional operators on some spaces. We first prove that the weighted integrals are bounded in certain spaces. Afterwards, we … The primary goal of this study is to define the weighted fractional operators on some spaces. We first prove that the weighted integrals are bounded in certain spaces. Afterwards, we discuss the weighted fractional derivatives defined on absolute continuous-like spaces. At the end, we present a modified Laplace transform that can be applied perfectly to such operators.

An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet

2020-06-01

Rohul Amin , Kamal Shah , Muhammad Asif +2 more

Computational study on the dynamics of fractional order differential equations with applications

2022-03-15

Kamal Shah , Muhammad Arfan , Aman Ullah +3 more

Fractal-Fractional Mathematical Model Addressing the Situation of Corona Virus in Pakistan

2020-11-12

Kamal Shah , Muhammad Arfan , Ibrahim Mahariq +3 more

This work is the consideration of a fractal fractional mathematical model on the transmission and control of corona virus (COVID-19), in which the total population of an infected area is … This work is the consideration of a fractal fractional mathematical model on the transmission and control of corona virus (COVID-19), in which the total population of an infected area is divided into susceptible, infected and recovered classes. We consider a fractal-fractional order SIR type model for investigation of Covid-19. To realize the transmission and control of corona virus in a much better way, first we study the stability of the corresponding deterministic model using next generation matrix along with basic reproduction number. After this, we study the qualitative analysis using "fixed point theory" approach. Next, we use fractional Adams-Bashforth approach for investigation of approximate solution to the considered model. At the end numerical simulation are been given by matlab to provide the validity of mathematical system having the arbitrary order and fractal dimension.

Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method

2020-03-30

Thabet Abdeljawad , Rohul Amin , Kamal Shah +2 more

This manuscript deals a numerical technique based on Haar wavelet collocation which is developed for the approximate solution of some systems of linear and nonlinear fractional order differential equations (FODEs). … This manuscript deals a numerical technique based on Haar wavelet collocation which is developed for the approximate solution of some systems of linear and nonlinear fractional order differential equations (FODEs). Based on these techniques, we find the numerical solution to various systems of FODEs. We compare the obtain solution with the exact solution of the considered problems at integer orders. Also, we compute the maximum absolute error to demonstrate the efficiency and accuracy of the proposed method. For the illustration of our results we provide four test examples. The experimental rates of convergence for different number of collocation point is calculated which is approximately equal to 2. Fractional derivative is defined in the Caputo sense.

Fractional order mathematical modeling of typhoid fever disease

2021-12-14

Muhammad Sinan , Kamal Shah , Poom Kumam +4 more

This manuscript is devoted to focusing on the modeling and numerical solution of the dynamical model of Typhoid Fever. We use the Atangana–Baleanu operator with the Mittag–Leffler function in Caputo … This manuscript is devoted to focusing on the modeling and numerical solution of the dynamical model of Typhoid Fever. We use the Atangana–Baleanu operator with the Mittag–Leffler function in Caputo sense to study the behavior of the model. Both local and global stability analysis are studied. Further, for global stability, we use the Lyapunov function at both disease-free and endemic equilibrium points. As a result, the model of Typhoid fever is locally and globally stable around disease-free and endemic equilibrium points and also possesses a unique solution. The strong numerical Adams–Bashforth method is used for the numerical solution and graphical representation to justify the results. It is observed that increasing the interaction rate among the susceptible and infected population increases basic reproduction number which means that the spread of disease increases by increasing interaction rate and disease transmission can be controlled by decreasing the interaction.

Hyers‐Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions

2018-09-27

Kamal Shah , Arshad Ali , Samia Bushnaq

In this article, we deal with the existence and Hyers‐Ulam stability of solution to a class of implicit fractional differential equations (FDEs), having some initial and impulsive conditions. Some adequate … In this article, we deal with the existence and Hyers‐Ulam stability of solution to a class of implicit fractional differential equations (FDEs), having some initial and impulsive conditions. Some adequate conditions for the required results are obtained by utilizing fixed point theory and nonlinear functional analysis. At the end, we provide an illustrative example to demonstrate the applications of our obtained results.

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.

Existence and stability of solution to a toppled systems of differential equations of non-integer order

2017-01-25

Amjad Ali , Bessem Samet , Kamal Shah +1 more

The aim of this paper is developing conditions that guarantee the existence of a solution to a toppled system of differential equations of noninteger order with fractional integral boundary conditions … The aim of this paper is developing conditions that guarantee the existence of a solution to a toppled system of differential equations of noninteger order with fractional integral boundary conditions where the nonlinear functions involved in the considered system are continuous and satisfy some growth conditions. We convert the system of differential equations to a system of fixed point problems for condensing mapping. With the help of techniques of the topological degree theory, we establish adequate conditions that ensure the existence and uniqueness of positive solutions to a toppled system under consideration. Moreover, some conditions are also developed for the Hyers-Ullam stability of the solution to the system under consideration. Finally, to demonstrate the obtained results, we provide an example.

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Qualitative analysis of fractal-fractional order COVID-19 mathematical model with case study of Wuhan

2020-10-05

Zeeshan Ali , Faranak Rabiei , Kamal Shah +1 more

In this manuscript, a qualitative analysis of the mathematical model of novel coronavirus (COVID-19) involving anew devised fractal-fractional operator in the Caputo sense having the fractional-order q and the fractal … In this manuscript, a qualitative analysis of the mathematical model of novel coronavirus (COVID-19) involving anew devised fractal-fractional operator in the Caputo sense having the fractional-order q and the fractal dimension p is considered. The concerned model is composed of eight compartments: susceptible, exposed, infected, super-spreaders, asymptomatic, hospitalized, recovery and fatality. When, choosing the fractal order one we obtain fractional order, and when choosing the fractional order one a fractal system is obtained. Considering both the operators together we present a model with fractal-fractional. Under the new derivative the existence and uniqueness of the solution for considered model are proved using Schaefer's and Banach type fixed point approaches. Additionally, with the help of nonlinear functional analysis, the condition for Ulam's type of stability of the solution to the considered model is established. For numerical simulation of proposed model, a fractional type of two-step Lagrange polynomial known as fractional Adams-Bashforth (AB) method is applied to simulate the results. At last, the results are tested with real data from COVID-19 outbreak in Wuhan City, Hubei Province of China from 4 January to 9 March 2020, taken from a source (Ndaïrou, 2020). The Numerical results are presented in terms of graphs for different fractional-order q and fractal dimensions p to describe the transmission dynamics of disease infection.

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Study of transmission dynamics of novel COVID-19 by using mathematical model

2020-07-01

Rahim Ud Din , Kamal Shah , Imtiaz Ahmad +1 more

Abstract In this research work, we present a mathematical model for novel coronavirus-19 infectious disease which consists of three different compartments: susceptible, infected, and recovered under convex incident rate involving … Abstract In this research work, we present a mathematical model for novel coronavirus-19 infectious disease which consists of three different compartments: susceptible, infected, and recovered under convex incident rate involving immigration rate. We first derive the formulation of the model. Also, we give some qualitative aspects for the model including existence of equilibriums and its stability results by using various tools of nonlinear analysis. Then, by means of the nonstandard finite difference scheme (NSFD), we simulate the results for the data of Wuhan city against two different sets of values of immigration parameter. By means of simulation, we show how protection, exposure, death, and cure rates affect the susceptible, infected, and recovered population with the passage of time involving immigration. On the basis of simulation, we observe the dynamical behavior due to immigration of susceptible and infected classes or one of these two.

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Fractal-fractional mathematical modeling and forecasting of new cases and deaths of COVID-19 epidemic outbreaks in India

2020-12-16

Mansour A. Abdulwasaa , Mohammed S. ‬Abdo , Kamal Shah +4 more

Fractional-order derivative-based modeling is very significant to describe real-world problems with forecasting and analyze the realistic situation of the proposed model. The aim of this work is to predict future … Fractional-order derivative-based modeling is very significant to describe real-world problems with forecasting and analyze the realistic situation of the proposed model. The aim of this work is to predict future trends in the behavior of the COVID-19 epidemic of confirmed cases and deaths in India for October 2020, using the expert modeler model and statistical analysis programs (SPSS version 23 & Eviews version 9). We also generalize a mathematical model based on a fractal fractional operator to investigate the existing outbreak of this disease. Our model describes the diverse transmission passages in the infection dynamics and affirms the role of the environmental reservoir in the transmission and outbreak of this disease. We give an itemized analysis of the proposed model including, the equilibrium points analysis, reproductive number R0, and the positiveness of the model solutions. Besides, the existence, uniqueness, and Ulam-Hyers stability results are investigated of the suggested model via some fixed point technique. The fractional Adams Bashforth method is applied to solve the fractal fractional model. Finally, a brief discussion of the graphical results using the numerical simulation (Matlab version 16) is shown.

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.

A fractional order pine wilt disease model with Caputo–Fabrizio derivative

2018-11-07

Muhammad Altaf Khan , Saif Ullah , Kazeem O. Okosun +1 more

A Caputo–Fabrizio type fractional order mathematical model for the dynamics of pine wilt disease (FPWD) is presented. The basic properties of the model are investigated. The existence and uniqueness of … A Caputo–Fabrizio type fractional order mathematical model for the dynamics of pine wilt disease (FPWD) is presented. The basic properties of the model are investigated. The existence and uniqueness of the solution for the proposed FPWD model are given via the fixed point theorem. The numerical simulations for the model are obtained by using particular parameter values. The non-integer order derivative provides more flexible and deeper information about the complexity of the dynamics of the proposed FPWD model than the integer order models established before.

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Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem

2018-11-20

Zeeshan Ali , Akbar Zada , Kamal Shah

In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann–Liouville … In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann–Liouville fractional derivative. Applying the Banach contraction principle and the Leray–Schauder result of cone type, uniqueness and existence results are proved for the proposed toppled system. Stability is investigated by using the classical technique of nonlinear functional analysis. The results obtained are well illustrated with the aid of an example.

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Stability analysis and a numerical scheme for fractional Klein‐Gordon equations

2018-11-18

Hasib Khan , Aziz Khan , Wen Chen +1 more

Fractional order nonlinear Klein‐Gordon equations (KGEs) have been widely studied in the fields like; nonlinear optics, solid state physics, and quantum field theory. In this article, with help of the … Fractional order nonlinear Klein‐Gordon equations (KGEs) have been widely studied in the fields like; nonlinear optics, solid state physics, and quantum field theory. In this article, with help of the Sumudu decomposition method (SDM), a numerical scheme is developed for the solution of fractional order nonlinear KGEs involving the Caputo's fractional derivative. The coupled method provides us very efficient numerical scheme in terms of convergent series. The iterative scheme is applied to illustrative examples for the demonstration and applications.

Mathematical analysis of COVID-19 via new mathematical model

2020-12-26

Abdullah Abdullah , Saeed Ahmad , Saud Owyed +4 more

Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions

2015-01-01

Kamal Shah , Rahmat Ali Khan

In this article, we study sufficient conditions for existence and uniqueness of positive solutions to the following coupled system of fractional order differential equations with antiperiodic boundary conditionsare continuous functions … In this article, we study sufficient conditions for existence and uniqueness of positive solutions to the following coupled system of fractional order differential equations with antiperiodic boundary conditionsare continuous functions and D stands for Caputo derivative.We use Banach and Schauder fixed point theorems to develop sufficient conditions for existence and uniqueness of positive solutions.We also study sufficient conditions for existence of multiple positive solutions and conditions for non existence of solutions.We provide several examples to show the applicability of our results.We also link our analysis for the problem to equivalent integral equations.

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Numerical treatment for traveling wave solutions of fractional Whitham-Broer-Kaup equations

2017-05-13

Amjad Ali , Kamal Shah , Rahmat Ali Khan

In the concerned article, we present the numerical solution of nonlinear coupled system of Whitham-Broer-Kaup equations (WBK) of fractional order. With the help of Laplace transform coupled with Adomian decomposition … In the concerned article, we present the numerical solution of nonlinear coupled system of Whitham-Broer-Kaup equations (WBK) of fractional order. With the help of Laplace transform coupled with Adomian decomposition method, an iterative procedure is established to investigate approximate solution to the proposed coupled system of nonlinear partial fractional differential equations. The concerned techniques are demonstrated by some numerical examples. Also, we compared the results of our proposed method with the results of other well known numerical methods such as Variation iteration method (VIM), Adomian decomposition method (ADM) and Homotopy perturbation method (HPM). For computation, we use Maple 18 and Matlab.

Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

2020-01-07

Idris Ahmed , Poom Kumam , Kamal Shah +3 more

This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the … This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer’s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.

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Numerical analysis of fractional order model of HIV-1 infection of CD4+ T-cells

2017-01-01

Fazal Haq , Kamal Shah , Ghaus-UR Rahman +1 more

In this article, we present a fractional order HIV-1 infection model of CD4+ T-cell. We analyze the effect of the changing the average number of the viral particle N with … In this article, we present a fractional order HIV-1 infection model of CD4+ T-cell. We analyze the effect of the changing the average number of the viral particle N with initial conditions of the presented model. The Laplace Adomian decomposition method is applying to check the analytical solution of the problem. We obtain the solutions of the fractional order HIV-1 model in the form of infinite series. The concerned series rapidly converges to its exact value. Moreover, we compare our results with the results obtained by Runge-Kutta method in case of integer order derivative.

On a new conceptual mathematical model dealing the current novel coronavirus-19 infectious disease

2020-11-06

Anwarud Din , Kamal Shah , Aly R. Seadawy +2 more

The present paper describes a three compartment mathematical model to study the transmission of the current infection due to the novel coronavirus (2019-nCoV or COVID-19). We investigate the aforesaid dynamical … The present paper describes a three compartment mathematical model to study the transmission of the current infection due to the novel coronavirus (2019-nCoV or COVID-19). We investigate the aforesaid dynamical model by using Atangana, Baleanu and Caputo (ABC) derivative with arbitrary order. We derive some existence results together with stability of Hyers-Ulam type. Further for numerical simulations, we use Adams–Bashforth (AB) method with fractional differentiation. The mentioned method is a powerful tool to investigate nonlinear problems for their respective simulation. Some discussion and future remarks are also given.

Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations

2019-03-08

Arshad Ali , Kamal Shah , Fahd Jarad +2 more

In this paper, we study a coupled system of implicit impulsive boundary value problems (IBVPs) of fractional differential equations (FODEs). We use the Schaefer fixed point and Banach contraction theorems … In this paper, we study a coupled system of implicit impulsive boundary value problems (IBVPs) of fractional differential equations (FODEs). We use the Schaefer fixed point and Banach contraction theorems to obtain conditions for the existence and uniqueness of positive solutions. We discuss Hyers–Ulam (HU) type stability of the concerned solutions and provide an example for illustration of the obtained results.

Fuzzy fractional-order model of the novel coronavirus

2020-09-05

Shabir Ahmad , Aman Ullah , Kamal Shah +3 more

Abstract In this paper, a novel coronavirus infection system with a fuzzy fractional differential equation defined in Caputo’s sense is developed. By using the fuzzy Laplace method coupled with Adomian … Abstract In this paper, a novel coronavirus infection system with a fuzzy fractional differential equation defined in Caputo’s sense is developed. By using the fuzzy Laplace method coupled with Adomian decomposition transform, numerical results are obtained for better understanding of the dynamical structures of the physical behavior of COVID-19. Such behavior on the general properties of RNA in COVID-19 is also investigated for the governing model. The results demonstrate the efficiency of the proposed approach to address the uncertainty condition in the pandemic situation.

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On analysis of the fractional mathematical model of rotavirus epidemic with the effects of breastfeeding and vaccination under Atangana-Baleanu (AB) derivative

2020-08-30

Shabir Ahmad , Aman Ullah , Muhammad Arfan +1 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.

Analysis of multipoint impulsive problem of fractional-order differential equations

2023-01-03

Kamal Shah , Bahaaeldin Abdalla , Thabet Abdeljawad +1 more

Abstract This manuscript is related to establishing appropriate results for the existence and uniqueness of solutions to a class of nonlinear impulsive implicit fractional-order differential equations (FODEs). It is remarkable … Abstract This manuscript is related to establishing appropriate results for the existence and uniqueness of solutions to a class of nonlinear impulsive implicit fractional-order differential equations (FODEs). It is remarkable that impulsive differential equations have attracted great popularity due to various important applications in the mathematical modeling of real-world phenomena/processes, particularly in biological or biomedical engineering domains as well as in control theory. The mentioned problem is considered under four-point nonlocal boundary conditions and the derivative is taken in the Caputo sense. Our results are based on fixed-point theorems due to Banach and Schaefer. To justify our results, two suitable examples are given.

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Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions

2021-10-13

Sayed Saifullah , Amir Ali , M. Irfan +1 more

In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the … In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional operators. The suggested method is the composition of the double Laplace transform and decomposition method. The convergence of the considered method is demonstrated for the considered model. It is observed that the obtained solutions converge to the exact solution of the proposed model. For validity, we consider two particular examples with appropriate initial conditions and derived the series solution in the sense of both operators for the considered model. From numerical solutions, it is observed that the considered model admits pulse-shaped solitons. It is also observed that the wave amplitude enhances with variations in time, which infers the coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>α</mi> </math> significantly increases the wave amplitude and affects the nonlinearity/dispersion effects, therefore may admit monotonic shocks. The physical behavior of the considered numerical examples is illustrated explicitly which reveals the evolution of localized shock excitations.

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Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems

2016-02-11

Kamal Shah , Amjad Ali , Rahmat Ali Khan

In this article, we investigate existence and uniqueness of positive solutions to coupled systems of multi-point boundary value problems for fractional order differential equations of the form $$ \left \{ … In this article, we investigate existence and uniqueness of positive solutions to coupled systems of multi-point boundary value problems for fractional order differential equations of the form $$ \left \{ \textstyle\begin{array}{l} D^{\alpha} x(t)=\phi(t,x(t),y(t)), \quad t\in I=[0,1], \\ D^{\beta} y(t)=\psi(t,x(t),y(t)),\quad t\in I=[0,1], \\ x(0)=g(x),\qquad x(1)=\delta x(\eta),\quad 0< \eta< 1, \\ y(0)=h(y),\qquad y(1)=\gamma y(\xi),\quad 0< \xi< 1, \end{array}\displaystyle \right . $$ where $\alpha, \beta\in(1,2]$ , D denotes the Caputo fractional derivative, $0<\delta, \gamma<1$ are parameters such that $\delta\eta^{\alpha}<1$ , $\gamma\xi^{\beta}<1$ , $h, g\in C(I,\mathbb{R})$ are boundary functions and $\phi,\psi:I\times\mathbb{R} \times \mathbb{R} \rightarrow\mathbb{R}$ are continuous. We use the technique of topological degree theory to obtain sufficient conditions for existence and uniqueness of positive solutions to the system. Finally, we provide an example to illustrate our main results.

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Mathematical analysis of the Cauchy type dynamical system under piecewise equations with Caputo fractional derivative

2022-07-07

Kamal Shah , Thabet Abdeljawad , Arshad Ali

On Ulam's type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions

2017-09-12

Arshad Ali , Faranak Rabiei , Kamal Shah

In this manuscript, using Schaefer's fixed point theorem, we derive some sufficient conditions for the existence of solutions to a class of fractional differential equations (FDEs).The proposed class is devoted … In this manuscript, using Schaefer's fixed point theorem, we derive some sufficient conditions for the existence of solutions to a class of fractional differential equations (FDEs).The proposed class is devoted to the impulsive FDEs with nonlinear integral boundary condition.Further, using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss various kinds of Ulam-Hyers stability.Finally to illustrate the established results, we provide an example.

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Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative

2018-01-01

Samia Bushnaq , Sajjad Ali Khan , Kamal Shah +1 more

This manuscript is concerned to the existence and stability of HIV/AIDS infection model with fractional order derivative. The corresponding derivative is taken in Caputo-Fabrizio sense, which is a new approach … This manuscript is concerned to the existence and stability of HIV/AIDS infection model with fractional order derivative. The corresponding derivative is taken in Caputo-Fabrizio sense, which is a new approach for such type of biological models. With the help of Sumudu transform, some new results are handled. Further for the corresponding results, existence theory and uniqueness for the equilibrium solution are provided via using nonlinear functional analysis and fixed point theory due to Banach.

On fractional order model of tumor dynamics with drug interventions under nonlocal fractional derivative

2021-01-10

Muhammad Arfan , Kamal Shah , Aman Ullah +3 more

In our research article, the mathematical model of fractional non-linear cancer dynamical system with the interaction of tumor cells, immune, and drugs reaction systems is taken in ABC derivative sense … In our research article, the mathematical model of fractional non-linear cancer dynamical system with the interaction of tumor cells, immune, and drugs reaction systems is taken in ABC derivative sense of fractional order. Qualitative results including existence and uniqueness has been analyzed by applying some theorems of fixed point theory. The approximate results for the considered system are obtained by numerical method due to Adams–Bashforth (AB) in non-integer order. Simulation of the Obtained iterative scheme are then drawn against the real data for different non-integer order. The suitability of the proposed model for our results are applied to study dynamics of tumor cells which help in investigating the dynamical interaction among tumor cells, immune cells and drugs reaction of the disease.

A computational algorithm for the numerical solution of fractional order delay differential equations

2021-02-24

Rohul Amin , Kamal Shah , Muhammad Asif +1 more

Using treatment and vaccination strategies to investigate transmission dynamics of influenza mathematical model

2025-05-30

Zakirullah Zakirullah , Liang Li , Kamal Shah +2 more

Investigating Jerk Phenomenon by Using Fractal-Fractional Analysis

2025-05-28

Kamal Shah , A.Bharathy et. al , Rohul Amin +4 more

Numerical solution of fractional order Bagley-Torvik equation via Hermite wavelet method

2025-05-02

Muhammad Yousaf , Aman Ullah , Salman Zeb +2 more

Abstract In this article, we studied an approximating technique namely the Hermite wavelet method (HWM) for the solution of the fractional order Bagley-Torvik equation (BTE). The proposed HWM method uses … Abstract In this article, we studied an approximating technique namely the Hermite wavelet method (HWM) for the solution of the fractional order Bagley-Torvik equation (BTE). The proposed HWM method uses polynomials known as the Hermite polynomial which are the basis for this method. This method involves using the collocation points to convert the differential equation under consideration into an algebraic system of equations that are solved numerically. The approximate solution for the given fractional order differential equation is also visualized graphically. Two test problems are considered and evaluated for justification of our results. The results of the proposed method are found to be easily obtained and to be almost identical to the exact solution. In addition to being simple computationally and less complicated than other numerical techniques, the suggested methodology also yields more accurate results.

Investigation of Existence and Ulam's type Stability for Coupled Fractal Fractional Differential Equations

2025-05-01

Amjad Ali , Farha Bibi , Zeeshan Ali +3 more

In this study, we investigate the coupled system of fractal fractional differential equations (FFDEs) from an existence and stability point of view. The area related to coupled FFDEs is significant … In this study, we investigate the coupled system of fractal fractional differential equations (FFDEs) from an existence and stability point of view. The area related to coupled FFDEs is significant because its permitting us to analyze and predict the relationships between several variables throughout the real-world phenomena. Coupled systems have numerous applications in different fields, such as modeling of brain activities and spreads of disease in biology and medicine, modeling of mechanical systems, electrical circuits in engineering, and modeling of population dynamics, predator-prey models in ecology and financial mathematical in economics. Furthermore, the afore mentioned area is significant for environmental science in pollution control, climate change modeling, artificial intelligence, and control systems in technology. We analysis a coupled system and make more informed choices in a variety of fields through coupled system of FFDEs. Keeping these informative aspects of coupled systems in mind, this article aims to explore the qualitative analysis such as existence, uniqueness, and stability for the solution of underlying coupled systems of FFDEs. The tools of functional analysis (FA) and fixed point theory (FPT) have been applied to deduce our required results. We have used the Banach contraction principle (BCP) and the Krasnoselskii fixed point theorem (KFPT) to demonstrate the conditions for existence and uniqueness of a solutions (EUS). Additionally, the results related to stability have demonstrated by using Ulam's concept. Subsequently, a suitable example is given to illustrates the validity of this work.

A Mathematical Model of COVID-19 Using Piecewise Derivative of Fractional Order

2025-05-01

Shabana Naz , Muhammad Sarwar , Kamal Shah +2 more

Currently the dynamical systems of infectious disease were studied by using various deﬁnitions of fractional calculus. Because the mentioned area has the ability to demonstrate the short and long memory … Currently the dynamical systems of infectious disease were studied by using various deﬁnitions of fractional calculus. Because the mentioned area has the ability to demonstrate the short and long memory terms involve in the physical dynamics of numerous real world problems. In this work, we consider a seven compartmental model for the transmission dynamics of COVID-19 including susceptible (S ), vaccinated (V), exposed(E), infected (I), quarantined (Q), recovered (R), and death (D) classes. We ﬁrst revisit the fundamental outcomes related to equilibrium points, basic reproduction numbers, sensitivity analysis and equilibrium points including disease free (DF), and endemic equilibrium points (EE). In addition, our main goal to investigate the consider model under the new aspect of fractional calculus known as piecewise fractional order operators. The mentioned operators have the ability to demonstrate the multi phase behaviours of the dynamical problems. The said characteristics cannot be described by using the traditional operators. We apply the tools of nonlinear analysis to deduce suﬃcient conditions for the existence of at least one solution and its uniqueness. Additionally, we also investigate the results related to stability analysis of Ulam-Hyers (U-H)type. Finally, we extend the concepts of RK2 method to form a sophisticated algorithms to simulate the results graphically. We present the numerical results for various fractional order.

Modeling Virus Mutation Dynamics Using Piecewise Fractional Derivatives

2025-05-01

Eiman , Kamal Shah , Muhammad Sarwar +1 more

A virus mutation model under piecewise fractional order derivatives involving Mittag-Leffler type kernel has been studied in this manuscript. As mutation is an important phenomenon for the survival of virus. … A virus mutation model under piecewise fractional order derivatives involving Mittag-Leffler type kernel has been studied in this manuscript. As mutation is an important phenomenon for the survival of virus. The concerned study aims to detect the crossover behavior of the dynamics of virus mutation. The considered problem explains a compartmental model with pre and post mutation of virus. Fundamental results related to local and global stability of equilibrium points have been studied by using tools of nonlinear functional analysis. We have deduced positivity and feasibility of for the mentioned model using the fractional order derivatives. In addition, both trivial and non-trivial equilibrium points are computed and reproductive number is also derived. With the help of the considered numerical scheme based on Adam Bashforth method, we have simulated our results graphically for using different values of fractional order.

Adaptive Multi-Criteria group Decision-Making for optimized crop selection using the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si307.svg"><mml:mrow><mml:mrow><mml:mi mathvariant="bold-italic">p</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant="bold-italic">q</mml:mi></mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">-</mml:mo></mml:mrow></mml:math> quasirung orthopair fuzzy hybrid aggregation operator

2025-05-01

Muhammad Rahim , Kamal Shah , Haifa Alqahtani +2 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

Numerical Scheme for the Computational Study of Two Dimensional Diffusion and Burgers’ Systems with Stability and Error Estimate

2025-04-22

Muhammad Bilal , Abdul Ghafoor , Manzoor Hussain +2 more

Mathematical Modelling of Breast Cancer with Four Stages

2025-04-10

Thabet Abdeljawad , Rahim Ud Din , Nahid Fatima +3 more

Mathematical insights into chaos in fractional-order fishery model

2025-04-03

Zakirullah Zakirullah , Lu Chen , Liang Li +3 more

Computational scheme for the numerical solution of fractional order pantograph delay-integro-differential equations via the Bernstein approach

2025-04-01

E. Aourir , H. Laeli Dastjerdi , Kamal Shah +1 more

The self-protection behavior changes effect in SEIR-DM of COVID-19 pandemic model with numerical simulation and controlling strategies presentation

2025-03-29

Razia Begum , Sajjad Ali , Thabet Abdeljawad +1 more

A novel fractal fractional mathematical model for HIV/AIDS transmission stability and sensitivity with numerical analysis

2025-03-18

Mukhtiar Khan , Nadeem Alam Khan , Ibad Ullah +3 more

Understanding the complex dynamics of HIV/AIDS transmission requires models that capture real-world progression and intervention impacts. This study introduces an innovative mathematical framework using fractal-fractional calculus to analyze HIV/AIDS dynamics, … Understanding the complex dynamics of HIV/AIDS transmission requires models that capture real-world progression and intervention impacts. This study introduces an innovative mathematical framework using fractal-fractional calculus to analyze HIV/AIDS dynamics, emphasizing memory effects and nonlocal interactions critical to disease spread. By dividing populations into four distinct compartments-susceptible individuals, infected individuals, those undergoing treatment, and individuals in advanced AIDS stages-the model reflects key phases of infection and therapeutic interventions. Unlike conventional approaches, the proposed nonlinear transmission function, $$\frac{\nabla (\mathscr {I}+\alpha _1\mathscr {T}+\alpha _2\mathscr {A})}{\mathscr {N}}$$ , accounts for varying infectivity levels across stages (where $$\mathscr {N}$$ is the total population and $$\nabla$$ denotes the effective contact rate), offering a nuanced view of how treatment efficacy ( $$\alpha _1$$ ) and progression to AIDS ( $$\alpha _2$$ ) shape transmission. The analytical framework combines rigorous mathematical exploration with practical insights. We derive the basic reproduction number $$\mathscr {R}_0$$ to assess outbreak potential and employ Lyapunov theory to establish global stability conditions. Using the Schauder fixed-point theorem, we prove the existence and uniqueness of solutions, while bifurcation analysis via center manifold theory reveals critical thresholds for disease persistence or elimination. We use a computational scheme that combines the Adams-Bashforth method with an interpolation-based correction technique to ensure numerical precision and confirm theoretical results. Sensitivity analysis highlights medication accessibility and delaying the spread of AIDS as a vital control strategy by identifying ( $$\alpha _1$$ ) and ( $$\alpha _2$$ ) as critical parameters. The numerical simulations illustrate the predictive ability of the model, which shows how fractal-fractional order affects outbreak trajectories and long-term disease burden. The framework outperforms conventional integer order models and produces more accurate epidemiological predictions by integrating memory-dependent transmission with fractional order flexibility. These findings demonstrate the model's value in developing targeted public health initiatives, particularly in environments with limited resources where disease monitoring and balancing treatment allocation is essential. In the end, our work provides a tool to better predict and manage the evolving challenges of HIV/AIDS by bridging the gap between theoretical mathematics and actual disease control.

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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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Analyzing wave dynamics of Burger-Poisson fractional partial differential equation

2025-03-01

Zeeshan Ali , Abdullah , Kamal Shah +2 more

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

Author Correction: On modified Mittag–Leffler coupled hybrid fractional system constrained by Dhage hybrid fixed point in Banach algebra

2025-02-14

Mohammed A. ‬Almalahi , Khaled Aldwoah , Faez A. Alqarni +3 more

Numerical technique based on Bernstein polynomials approach for solving auto-convolution VIEs and the initial value problem of auto-convolution VIDEs

2025-02-12

E. Aourir , H. Laeli Dastjerdi , Mustapha Oudani +2 more

Existence of solutions of multi-order fractional differential equations

2025-02-01

Faycal Bouchelaghem , Hamid Boulares , Abdelouaheb Ardjouni +4 more

Numerical solution of two dimensional time-fractional telegraph equation using Chebyshev spectral collocation method

2025-02-01

Kamran Kamran , Farman Ali Shah , Kamal Shah +1 more

On Generalized Fractal-Fractional Derivative and Integral Operators Associated with Generalized Mittag-Leffler Function

2025-01-23

Hasib Khan , Gauhar Rahman , Muhammad Samraiz +2 more

In recent years, the Atangana-Baleanu (AB) fractal-fractional derivatives are widely used in many fields. In 2017, Atangana defined such operators by utilizing one parameter Mittag-Leffler function (M-L) function. Such operators … In recent years, the Atangana-Baleanu (AB) fractal-fractional derivatives are widely used in many fields. In 2017, Atangana defined such operators by utilizing one parameter Mittag-Leffler function (M-L) function. Such operators have not yet been studied for three parameters M-L function. In this paper, we discuss further modifications of Caputo Fabrizio (CF), AB and generalized Hattaf fractal-fractional (GHF) operators. We used the modified three parameters M-L function to define the generalized fractal-fractional (GFF) differential and integral operators. We study an innovative class of new generalized weighted differential and integral operators. We define the generalized fractal-fractional (GFF) differential and integral operators with generalized Mittag-Leffler (M-L) kernels, which are used to simulate the complex dynamics of several natural and physical phenomena in a variety of scientific and engineering domains. There are a few established features of the newly defined operators. An example of an application for this new class of GFF integral is presented. Also, we discussed the graphical comparison of this new GFF operator with the existing GHF, AB and CF derivatives. Our case is the more general case compared with the existing fractal-fractional operators. We have presented some novel results for the new operators both analytically and graphically. Also, we discussed some special cases by giving specific value to the parameter θ . All the classical operators are restored by applying certain conditions on parameters.

Exploring the dynamics of Gumboro-Salmonella co-infection with fractal fractional analysis

2025-01-18

Israr Ahmad , Zeeshan Ali , Babar Khan +2 more

Advancements in integral inequalities of Ostrowski type via modified Atangana-Baleanu fractional integral operator

2025-01-01

Gauhar Rahman , Muhammad Samraiz , Kamal Shah +2 more

Corrections to “Development p, q, r−Spherical Fuzzy Einstein Aggregation Operators: Application in Decision-Making in Logo Design”

2025-01-01

Lin Kang , Salma Khan , Muhammad Rahim +2 more

Haar wavelet collocation method for existence and numerical solutions of fourth-order integro-differential equations with bounded coefficients

2025-01-01

Rohul Amin , Muhammad Azher Nawaz , Kamal Shah +1 more

Abstract In this article, Haar wavelet collocation method is applied for the solution of fourth-order integro-differential equations. Also, a fixed point approach is used to investigate the existence theory of … Abstract In this article, Haar wavelet collocation method is applied for the solution of fourth-order integro-differential equations. Also, a fixed point approach is used to investigate the existence theory of solution to the considered problem. The fourth-order derivative is approximated using Haar function. In addition, third-, second-, and first-order derivatives together with unknown functions are obtained by the process of successive integrations. On applying the Haar collocation method, the suggested problem of IDEs is transformed to a system of algebraic equations. The Gauss elimination scheme is used for the solution of linear algebraic equations. The precision, effectiveness, and convergence of the Haar approach are checked on some test problems. Different collocation and Gauss points are used to determine the absolute and root mean square errors. To demonstrate the applicability of the proposed method, an experimental rate of convergence is calculated, which is almost equal to 2. The method is accurate, easily applicable, and efficient.

Investigating a Nonlinear Fractional Evolution Control Model Using W-Piecewise Hybrid Derivatives: An Application of a Breast Cancer Model

2024-12-13

Hicham Saber , Mohammed A. ‬Almalahi , Hussien Albala +4 more

Many real-world phenomena exhibit multi-step behavior, demanding mathematical models capable of capturing complex interactions between distinct processes. While fractional-order models have been successfully applied to various systems, their inherent smoothness … Many real-world phenomena exhibit multi-step behavior, demanding mathematical models capable of capturing complex interactions between distinct processes. While fractional-order models have been successfully applied to various systems, their inherent smoothness often limits their ability to accurately represent systems with discontinuous changes or abrupt transitions. This paper introduces a novel framework for analyzing nonlinear fractional evolution control systems using piecewise hybrid derivatives with respect to a nondecreasing function W(ι). Building upon the theoretical foundations of piecewise hybrid derivatives, we establish sufficient conditions for the existence, uniqueness, and Hyers–Ulam stability of solutions, leveraging topological degree theory and functional analysis. Our results significantly improve upon existing theoretical understanding by providing less restrictive conditions for stability compared with standard fixed-point theorems. Furthermore, we demonstrate the applicability of our framework through a simulation of breast cancer disease dynamics, illustrating the impact of piecewise hybrid derivatives on the model’s behavior and highlighting advantages over traditional modeling approaches that fail to capture the multi-step nature of the disease. This research provides robust modeling and analysis tools for systems exhibiting multi-step behavior across diverse fields, including engineering, physics, and biology.

On modified Mittag–Leffler coupled hybrid fractional system constrained by Dhage hybrid fixed point in Banach algebra

2024-12-04

Mohammed A. ‬Almalahi , Khaled Aldowah , Faez A. Alqarni +3 more

This research investigates the dynamics of nonlinear coupled hybrid systems using a modified Mittag–Leffler fractional derivative. The primary objective is to establish criteria for the existence and uniqueness of solutions … This research investigates the dynamics of nonlinear coupled hybrid systems using a modified Mittag–Leffler fractional derivative. The primary objective is to establish criteria for the existence and uniqueness of solutions through the implementation of Dhage's hybrid fixed-point theorem. The study further analyzes the stability of the proposed model. To demonstrate the practical application of this framework, we utilize a modified Mittag–Leffler operator to model the transmission of the Ebola virus, known for its complex and diverse dynamics. The analysis is conducted using a combination of theoretical and numerical methods, including transforming the system of equations into an equivalent integral form, applying the fixed-point theorem, and developing a numerical scheme based on Lagrange's interpolation for simulating the Ebola virus model. This study aims to enhance our understanding of Ebola virus dynamics and provide valuable insights for developing effective control strategies.

Using Extended Direct Algebraic Method to Investigate Families of Solitary Wave Solutions for the Space-Time Fractional Modified Benjamin Bona Mahony Equation

2024-11-25

Muhammad Bilal , Javed Iqbal , Ikram Ullah +2 more

Abstract This work studies the space-time fractional modified Benjamin-Bona-Mahony equation, a mathematical&#xD;model of nonlinear wave propagation in various physical systems, for solitary wave solutions. Among the&#xD;precise solutions we produce with … Abstract This work studies the space-time fractional modified Benjamin-Bona-Mahony equation, a mathematical&#xD;model of nonlinear wave propagation in various physical systems, for solitary wave solutions. Among the&#xD;precise solutions we produce with the Extended Direct Algebraic method are solitary waves and periodic&#xD;wave patterns. These solutions reveal information on soliton interactions and propagation processes, offering&#xD;insight into the dynamics of the problem. Characterizing the answers is made easier with the use of graphic&#xD;representations. Our work bridges the gap between chemical reaction-diffusion mechanisms and biological&#xD;mathematics to improve comprehension of complicated events in interdisciplinary study.

An innovative method for solving the nonlinear fractional diffusion reaction equation with quadratic nonlinearity analysis

2024-11-20

Alamgir Khan , Ikram Ullah , Javed Iqbal +2 more

Abstract This study generates and investigates spreading solitons in the fractional DR quadratic equation (FDE) with frac-&#xD;tional derivatives using the Extended Direct Algebraic Method (EDAM). In population growth, mathematical biology,&#xD;and … Abstract This study generates and investigates spreading solitons in the fractional DR quadratic equation (FDE) with frac-&#xD;tional derivatives using the Extended Direct Algebraic Method (EDAM). In population growth, mathematical biology,&#xD;and reaction-diffusion mechanisms, the FDE is crucial. Applying the series form solution to the NODE from the FDE&#xD;conversion into a recommended EDAM yields many traveling soliton solutions. To characterize and explore soliton&#xD;structure propagation, we draw shock and kink soliton solutions. Through reaction-diffusion mechanics and mathemat-&#xD;ical biology, we may explain complex processes in many academic subjects.

An inverse nodal problem of a conformable Sturm-Liouville problem with restrained constant delay

2024-11-12

Auwalu Sa’idu , Hikmet Koyunbakan , Kamal Shah +1 more

This paper presents a new technique: a conformable derivative for the inverse problem of a Sturm-Liouville problem with restrained constant delay. Solutions to the Sturm-Liouville problem often involve eigenfunctions and … This paper presents a new technique: a conformable derivative for the inverse problem of a Sturm-Liouville problem with restrained constant delay. Solutions to the Sturm-Liouville problem often involve eigenfunctions and eigenvalues, which have important applications in physics, engineering, and other fields. The presence of a constant delay introduces unique challenges in formulating and solving this problem. In this case, we derived the asymptotic formulas for the eigenvalues with their corresponding eigenfunctions and demonstrated the existence of the solution. Additionally, we identified the nodal points used to generate the problem's potential function. Finally, we applied the Lipschitz stability approach and demonstrated the stability of the solution to the problem.

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Presentation of the efficient scheme for solving fractional order telegraph problems

2024-11-05

Wasim Sajjad Hussain , Sajjad Ali , Nahid Fatima +2 more

Dynamical behavior of whooping cough SVEIQRP model via system of fractal fractional differential equations

2024-11-01

Razia Begum , Sajjad Ali , Nahid Fatima +2 more

On computational analysis via fibonacci wavelet method for investigating some physical problems

2024-10-03

Shahid Ahmed , Shah Jahan , Kamal Shah +1 more

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.

Exploring the dynamical behaviour of optical solitons in integrable kairat-II and kairat-X equations

2024-08-12

Yeliang Xiao , Shoaib Barak , Manel Hleili +1 more

Abstract The current research focusses on the establishment of an analytical approach known as the Riccati Modified Extended Simple equation Method (RMESEM) for the development and assessment of optical soliton … Abstract The current research focusses on the establishment of an analytical approach known as the Riccati Modified Extended Simple equation Method (RMESEM) for the development and assessment of optical soliton solutions in two important Kairat equations. These models are known as Kairat-X equation (K-XE) and the Kairat-II equation (K-IIE), which describe the trajectory of optical pulses in optical fibres. Using RMESEM, the soliton solutions in five families–the periodic, rational, hyperbolic, rational-hyperbolic, and exponential functional families–are achieved for the targeted models. A set of 3D, 2D, and contour visualisations are presented to visually illustrate the dynamics of some produced optical soliton solutions which demonstrates that the due to the axial-periodic perturbation, the optical soliton solutions exhibit fractal phenomena in the realm of K-IIE whereas in the setting of K-XE the optical solitons adopt the form of kink solitons such as solitary kink, lump-type kink, dromion and periodic kink soliton structures. Moreover, our suggested RMESEM illustrates its usefulness by building a multitude of optical soliton solutions, providing valuable insights into the dynamics of the targeted models and indicating potential uses in addressing other nonlinear models.

Mathematical analysis of fractional order co-abuse infection model using power law type kernel

2024-08-10

Izhar Ul Haq , Amjad Ali , Aman Ullah +2 more

In this study, we present a mathematical model designed to illustrate the simultaneous occurrence of smoking and heroin co-abuse infections.To explore non-negative solutions and identify a stable equilibrium point, as … In this study, we present a mathematical model designed to illustrate the simultaneous occurrence of smoking and heroin co-abuse infections.To explore non-negative solutions and identify a stable equilibrium point, as well as the fundamental reproductive number, we enhance the model by integrating Caputo fractional-order (FO) derivative operators.Employing functional analysis concepts, we derive several results pertaining to the existence of a unique solution.Additionally, we utilize the Ulam-Hyres (UH) notion to establish the stability of the model solutions.To offer further insights, we present numerical results for the fractional-order system using an Euler-type numerical technique.These results are visually represented in graphs, illustrating the diverse responses of the model under different parameter values.

Pioneering the plethora of soliton for the (3+1)-dimensional fractional heisenberg ferromagnetic spin chain equation

2024-08-02

Ikram Ullah , Kamal Shah , Thabet Abdeljawad +1 more

Abstract In this scholarly article, we investigate the complex structured (3+1)-dimensional Fractional Heisenberg Ferromagnetic Spin Chain equation (FHFSCE) with conformable fractional derivatives. We develop a diverse glut of soliton solutions … Abstract In this scholarly article, we investigate the complex structured (3+1)-dimensional Fractional Heisenberg Ferromagnetic Spin Chain equation (FHFSCE) with conformable fractional derivatives. We develop a diverse glut of soliton solutions using an improved version of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mfenced close=")" open="("><mml:mrow><mml:mstyle displaystyle="false"><mml:mfrac><mml:mrow><mml:mi>G</mml:mi><mml:mo accent="false">′</mml:mo></mml:mrow><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:mfenced></mml:math> -expansion method, namely the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mfenced close=")" open="("><mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mstyle displaystyle="false"><mml:mfrac><mml:mrow><mml:mi>G</mml:mi><mml:mo accent="false">′</mml:mo></mml:mrow><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:mfenced></mml:math> -expansion method. The constraints for the existence of these solutions are painstakingly explained. Our findings are clearly communicated through a number of 3D and 2D graphical representations displaying periodic, multiple periodic, kink, and shock soliton solutions. These soliton solutions are expressed in several mathematical function forms, such as hyperbolic, trigonometric, and rational functions. Our findings support the suggested method’s efficacy as a powerful symbolic algorithm for discovering innovative soliton solutions within nonlinear evolution systems.

On analysis of a system of non-homogenous boundary value problems using hausdorff derivative with exponential kernel

2024-07-31

Shafi Ullah , Kamal Shah , Muhammad Sarwar +3 more

Abstract In recent years, the fractals (Hausdorff) derivatives with fractional order under various types kernel have gained attention from researchers. The aforesaid area has many applications in the description of … Abstract In recent years, the fractals (Hausdorff) derivatives with fractional order under various types kernel have gained attention from researchers. The aforesaid area has many applications in the description of intricate and irregular geometry of various processes. Numerous studies utilizing the fractional derivatives (HFDs) for initial value problems have been carried out. But the boundary value problems using the said concepts have been very rarely studied. Thus, a coupled system with non-homogenous boundary conditions (BCs) is examined in this study by using fractals fractional derivative in Caputo Fabrizio sense. To establish the required conditions for the existence and uniqueness of solution to the considered problem, we apply the Banach and Krasnoselskii’s fixed point theorems. Furthermore, some results related to Hyers-Ulam (H-U) stability have also deduced. We have included two pertinent examples to verify our results.

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Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method

2024-07-24

Muhammad Bilal , Javed Iqbal , Kamal Shah +3 more

The study of soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) has gained prominence recently because of its ability to realistically recreate complex physical processes. Numerous mathematical techniques have … The study of soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) has gained prominence recently because of its ability to realistically recreate complex physical processes. Numerous mathematical techniques have been devised to handle the problem of NFPDEs where soliton solutions are difficult to obtain. Due to their accuracy in reproducing complex physical phenomena, soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) have recently attracted interest. Several mathematical techniques have been devised to tackle the difficult task of solving non-finite partial differential equations (NFPDEs) soliton. Studies of soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) have garnered increased attention recently due to its capacity to accurately represent complex physical processes. Due to the difficulty of obtaining soliton solutions, NFPDEs can be solved using a wide variety of mathematical methods. In this way, it facilitates the extraction of the recently found abundance of optical soliton solutions. To further understanding of the results, the study also includes contour and three-dimensional images that visually depict particular optical soliton solutions for particular parameter selections, suggesting the existence of different soliton structures in the nonlinear fractional Kundu–Eckhaus equation (NFKEE) region. It is shown that the proposed technique is quite powerful and effective in solving several nonlinear FDEs.

Theoretical and computational results for implicit non-singular hybrid fractional differential equation subject to multi-terms non-local initial conditions

2024-07-15

Shafiullah , Kamal Shah , Muhammad Sarwar +1 more

This research work is devoted to investigate a class of hybrid fractional differential equations with n + 1 terms initial conditions.The aforesaid problem is considered under the Atangana-Baleanue-Caputo fractional order … This research work is devoted to investigate a class of hybrid fractional differential equations with n + 1 terms initial conditions.The aforesaid problem is considered under the Atangana-Baleanue-Caputo fractional order derivative.Here it is remarkable that hybrid differential equations with linear perturbations have significant applications in modeling various dynamical problems.Sufficient conditions are established for the existence and uniqueness of solution to the problem under investigation by using the Banach and Krasnoselsikii's fixed point theorems.Since stability theory plays important role in establishing various numerical and optimizations results, therefore, Hyers-Ulam type stability results are deduced for the considered problems using the tools of nonlinear functional analysis.Additionally, a numerical method based on Euler procedure is established to study some approbation results for the proposed problem.By a pertinent example, we demonstrate our results.Also some graphical illustrations for different fractional orders are given.

Analysis of nonlinear Burgers equation with time fractional Atangana-Baleanu-Caputo derivative

2024-07-01

Abdul Ghafoor , Muhammad Fiaz , Kamal Shah +1 more

This paper demonstrates, a numerical method to solve the one and two dimensional Burgers' equation involving time fractional Atangana-Baleanu Caputo (ABC) derivative with a non-singular kernel. The numerical stratagem consists … This paper demonstrates, a numerical method to solve the one and two dimensional Burgers' equation involving time fractional Atangana-Baleanu Caputo (ABC) derivative with a non-singular kernel. The numerical stratagem consists of a quadrature rule for time fractional (ABC) derivative along with Haar wavelet (HW) approximations of one and two dimensional problems. The key feature of the scheme is to reduce fractional problems to the set of linear equations via collocation procedure. Solving the system gives the approximate solution of the given problem. To verify the effectiveness of the developed method five numerical examples are considered. Besides this, the obtained simulations are compared with some published work and identified that proposed technique is better. Moreover, computationally the convergence rate in spatiotemporal directions is presented which shows order two convergence. The stability of the proposed scheme is also described via Lax-Richtmyer criterion. From simulations it is obvious that the scheme is quite useful for the time fractional problems.

Dust acoustic nonlinearity of nonlinear mode in plasma to compute temporal and spatial results

2024-06-22

Aziz Khan , Muhammad Sinan , Sumera Bibi +4 more

Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For … Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For the require results, we use reduction perturbation method (RPM) to find the analytical solution of the dust acoustic shock waves. Further, using the same technique we find the solitary wave potential and compared the solutions obtained with another very useful technique known as Homotopy perturbation method (HPM). The comparison of results for both approaches are more precise and agreed with the exact solution of the problem. Finally, we present graphical representation for different fractional order for both temporal and spatial approximate solution.

Study of a class of fractional-order evolution hybrid differential equations using a modified Mittag-Leffler-type derivative

2024-06-20

Kamal Shah , Thabet Abdeljawad , Bahaaeldin Abdalla +1 more

Abstract This work is devoted to using topological degree theory to establish a mathematical analysis for a class of fractional-order evolution hybrid differential equations using a modified Mittag–Leffler-type derivative. In … Abstract This work is devoted to using topological degree theory to establish a mathematical analysis for a class of fractional-order evolution hybrid differential equations using a modified Mittag–Leffler-type derivative. In addition, two kinds of Ulam–Hyers (U–H) stability results are deduced for the mentioned problem. A pertinent example is given to verify the results.

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A comprehensive mathematical analysis of fractal–fractional order nonlinear re-infection model

2024-06-19

Eiman , Kamal Shah , Muhammad Sarwar +1 more

Correction to: Analytical study of a modified-ABC fractional order breast cancer model

2024-06-18

Khaled Aldwoah , Mohammed A. ‬Almalahi , Manel Hleili +3 more

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On the numerical solution of second order delay differential equations via a novel approach

2024-06-14

Salma Aljawi , Kamran Kamran , Sultan Hussain +2 more

Delay differential equations belong to an important class of differential equations in which the evolution of the state depends on the previous time.This work proposes a novel approach for the … Delay differential equations belong to an important class of differential equations in which the evolution of the state depends on the previous time.This work proposes a novel approach for the numerical solution of delay differential equations of second order.The suggested numerical scheme is based on Laplace transform (LT) technique.In the suggested technique, first, the given equation is transformed using the LT method to an algebraic expression.The expression is then solved for the unknown transformed function and finally the well-known Weeks method is utilized to convert the solution back to time domain.Functional analysis was used to examine the existence and uniqueness of the considered equations and to generate sufficient requirements for Ulam-Hyers (UH) type stability.Furthermore, we consider different numerical example from literature to validate our method.

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Solution of the foam-drainage equation with cubic B-spline hybrid approach

2024-06-12

Alina Yousafzai , Sirajul Haq , Abdul Ghafoor +2 more

Abstract This work presents a robust and efficient numerical stratagem for the study of integer and fractional order non-linear Foam-Drainage (FD) model. The scheme first uses, usual forward difference and … Abstract This work presents a robust and efficient numerical stratagem for the study of integer and fractional order non-linear Foam-Drainage (FD) model. The scheme first uses, usual forward difference and the L 1 formula, in integer and fractional cases, respectively. Then, the collocation approach together with cubic B-splines (CBS) basis are employed to estimate the unknown solution and its derivatives. With the help of these discretizations and Quasi-linearization, solving non-linear FD model transforms to the system of linear algebraic equations. The solution of the linear system approximates the CBS coefficients which further leads to the numerical solutions. Moreover, by Von Neumann stability it is proved that the proposed scheme is unconditionally stable. To evaluate the performance and accuracy of the technique, absolute error (AE), L 2 , and L ∞ norms are presented. The obtained outcomes are also matched with some existing results in literature. It is noted from simulations that the proposed method gives quite accurate solutions.

Fuzzy fixed point approach to study the existence of solution for Volterra type integral equations using fuzzy Sehgal contraction

2024-06-06

Muhammad Tariq Zahid , Fahim Ud Din , Kamal Shah +1 more

In this manuscript, we present a novel concept known as the fuzzy Sehgal contraction, specifically designed for self-mappings defined in the context of a fuzzy metric space. Our primary objective … In this manuscript, we present a novel concept known as the fuzzy Sehgal contraction, specifically designed for self-mappings defined in the context of a fuzzy metric space. Our primary objective is to explore the existence and uniqueness of fixed points for self-mappings in fuzzy metric space. To support our conclusions, we present a detailed illustrative case that demonstrates the superiority of the convergence obtained with our suggested method to those currently recorded in the literature. Moreover, we provide graphical depictions of the convergence behavior, which makes our study more understandable and transparent. Additionally, we extend the application of our results to address the existence and uniqueness of solutions for Volterra integral equations.

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Cosine similarity and distance measures for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>−</mml:mo></mml:mrow></mml:math> quasirung orthopair fuzzy sets: Applications in investment decision-making

2024-05-31

Muhammad Rahim , Shougi Suliman Abosuliman , Roobaea Alroobaea +2 more

Similarity measures and distance measures are used in a variety of domains, such as data clustering, image processing, retrieval of information, and recognizing patterns, in order to measure the degree … Similarity measures and distance measures are used in a variety of domains, such as data clustering, image processing, retrieval of information, and recognizing patterns, in order to measure the degree of similarity or divergence between elements or datasets. quasirung orthopair fuzzy ( QOF) sets are a novel improvement in fuzzy set theory that aims to properly manage data uncertainties. Unfortunately, there is a lack of research on similarity and distance measure between QOF sets. In this paper, we investigate different cosine similarity and distance measures between to quasirung orthopair fuzzy sets ( ROFSs). Firstly, the cosine similarity measure and the Euclidean distance measure for QOFSs are defined, followed by an exploration of their respective properties. Given that the cosine measure does not satisfy the similarity measure axiom, a method is presented for constructing alternative similarity measures for QOFSs. The structure is based on the suggested cosine similarity and Euclidean distance measures, which ensure adherence to the similarity measure axiom. Furthermore, we develop a cosine distance measure for QOFSs that connects similarity and distance measurements. We then apply this technique to decision-making, taking into account both geometric and algebraic perspectives. Finally, we present a practical example that demonstrates the proposed justification and efficacy of the proposed method, and we conclude with a comparison to existing approaches.

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Coauthor	Papers Together
Thabet Abdeljawad	189
Rahmat Ali Khan	54
Bahaaeldin Abdalla	31
Muhammad Sarwar	30
Aman Ullah	28
Mohammed S. ‬Abdo	26
Aziz Khan	25
Fahd Jarad	25
Muhammad Arfan	22
Hussam Alrabaiah	22
Ghaus ur Rahman	21
Rohul Amin	21
Manar A. Alqudah	19
Zeeshan Ali	18
Ibrahim Mahariq	17
Arshad Ali	16
Mohammed A. ‬Almalahi	16
Yongjin Li	15
Sher Muhammad	15
Amjad Ali	14
Sajjad Ali	14
Poom Kumam	13
Eiman	13
Israr Ahmad	13
Amjad Ali	12
Zareen A. Khan	12
Nabil Mlaiki	11
Khaled Aldwoah	11
Hammad Khalil	11
Dumitru Băleanu	10
Kamran Kamran	10
Shabir Ahmad	10
Samia Bushnaq	10
Fazal Haq	9
Muhammad Arif	9
Manel Hleili	9
Khursheed J‎. ‎Ansari	8
Ali Ahmadian	8
Mdi Begum Jeelani	8
Faranak Rabiei	8
Qasem M. Al‐Mdallal	8
Gauhar Ali	8
Hasib Khan	8
Akbar Zada	7
Rahim Ud Din	7
Satish K. Panchal	7
Muhammad Sinan	7
Soheil Salahshour	7
Salman Zeb	7
Rozi Gul	7

Applications of Fractional Calculus in Physics

2000-03-01

R. Hilfer

Theory and Applications of Fractional Differential Equations

2006-01-01

Anatoly A. Kilbas , H. M. Srivastava , Juan J. Trujillo

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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On the Stability of the Linear Functional Equation

1941-04-15

D. H. Hyers

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Applications of Fractional Calculus in Physics

2000-01-01

R. Hilfer

An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time … An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.

Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids

1997-01-01

Yury A. Rossikhin , Marina V. Shitikova

The aim of this review article is to collect together separated results of research in the application of fractional derivatives and other fractional operators to problems connected with vibrations and … The aim of this review article is to collect together separated results of research in the application of fractional derivatives and other fractional operators to problems connected with vibrations and waves in solids having hereditarily elastic properties, to make critical evaluations, and thereby to help mechanical engineers who use fractional derivative models of solids in their work. Since the fractional derivatives used in the simplest viscoelastic models (Kelvin-Voigt, Maxwell, and standard linear solid) are equivalent to the weakly singular kernels of the hereditary theory of elasticity, then the papers wherein the hereditary operators with weakly singular kernels are harnessed in dynamic problems are also included in the review. Merits and demerits of the simplest fractional calculus viscoelastic models, which manifest themselves during application of such models in the problems of forced and damped vibrations of linear and nonlinear hereditarily elastic bodies, propagation of stationary and transient waves in such bodies, as well as in other dynamic problems, are demonstrated with numerous examples. As this takes place, a comparison between the results obtained and the results found for the similar problems using viscoelastic models with integer derivatives is carried out. The methods of Laplace, Fourier and other integral transforms, the approximate methods based on the perturbation technique, as well as numerical methods are used as the methods of solution of the enumerated problems. This review article includes 174 references.

On the stability of the linear mapping in Banach spaces

1978-11-01

Themistocles M. Rassias

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1 comma upper E 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1 comma upper E 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{E_1},{E_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be two Banach spaces, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper E 1 right-arrow upper E 2"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f:{E_1} \to {E_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a mapping, that is “approximately linear". S. M. Ulam posed the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam’s problem.

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Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system

2017-04-25

Abdon Atangana

Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations

2015-06-23

Kamal Shah , Hammad Khalil , Rahmat Ali Khan

Fractional Calculus in Bioengineering

2007-02-12

Bruce J. West

Existence results for boundary value problems with non-linear fractional differential equations

2008-07-01

Mouffak Benchohra , John R. Graef , Samira Hamani

In this article, the authors establish sufficient conditions for the existence of solutions to a class of boundary value problem for fractional differential equations involving the Caputo fractional derivative of … In this article, the authors establish sufficient conditions for the existence of solutions to a class of boundary value problem for fractional differential equations involving the Caputo fractional derivative of order α ∈ (1, 2] and non-linear integral conditions.

Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions

2009-09-10

Bashir Ahmad , Juan J. Nieto

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.

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

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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Analytical Solutions of Fractional Order Diffusion Equations by Natural Transform Method

2016-10-27

Kamal Shah , Hammad Khalil , Rahmat Ali Khan

Hyers–Ulam stability of linear differential equations of first order, II

2005-12-22

‎Soon-Mo Jung

Study in Fractional Differential Equations by Means of Topological Degree Methods

2012-01-06

JinRong Wang , Yong Zhou , Wei Wei

In this article, we study some class of fractional differential equations involving the Caputo fractional derivative. By using a fixed point theorem on topological degree for condensing maps via a … In this article, we study some class of fractional differential equations involving the Caputo fractional derivative. By using a fixed point theorem on topological degree for condensing maps via a priori estimate method, some sufficient conditions for the existence of solutions are presented. Uniqueness and data dependence results of solutions are also deduced.

Recent history of fractional calculus

2010-06-01

J. A. Tenreiro Machado , Virginia Kiryakova , Francesco Mainardi

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Fractional differential equations in electrochemistry

2009-02-24

Keith B. Oldham

Ulam stability and data dependence for fractional differential equations with Caputo derivative

2011-01-01

JinRong Wang , Linli Lv , Yong Zhou

In this paper, Ulam stability and data dependence for fractional differential equations with Caputo fractional derivative of orderare studied. We present four types of Ulam stability results for the fractional … In this paper, Ulam stability and data dependence for fractional differential equations with Caputo fractional derivative of orderare studied. We present four types of Ulam stability results for the fractional differential equation in the case of 0 < � < 1 and b = +1 by virtue of the Henry-Gronwall inequality. Meanwhile, we give an interesting data dependence results for the fractional differential equation in the case of 1 < � < 2 and b < +1 by virtue of a generalized Henry-Gronwall inequality with mixed integral term. Finally, examples are given to illustrate our theory results.

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On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative

2020-05-08

Mohammed S. ‬Abdo , Kamal Shah , Hanan A. Wahash +1 more

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Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations

2010-05-03

Mujeeb ur Rehman , Rahmat Ali Khan

A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions

2008-11-10

Ravi P. Agarwal , Mouffak Benchohra , Samira Hamani

New concept in calculus: Piecewise differential and integral operators

2021-03-20

Abdon Atangana , Seda İğret Araz

A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action

2020-03-04

Qianying Lin , Shi Zhao , Daozhou Gao +7 more

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A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions

2014-06-27

Rasiel Toledo-Hernandez , Vicente Rico-Ramı́rez , Gustavo A. Iglesias‐Silva +1 more

Boundary value problem for a coupled system of nonlinear fractional differential equations

2008-04-02

Xinwei Su

Novel techniques in parameter estimation for fractional dynamical models arising from biological systems

2011-04-15

F. Liu , Kevin Burrage

Numerical solution of fractional order smoking model via laplace Adomian decomposition method

2017-03-14

Fazal Haq , Kamal Shah , Ghaus ur Rahman +1 more

Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions

2015-01-01

Kamal Shah , Rahmat Ali Khan

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

1999-01-01

Solution of the epidemic model by Adomian decomposition method

2005-06-29

Jafar Biazar

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

2018-10-10

Fahd Jarad , Thabet Abdeljawad , Zakia Hammouch

On Ulam’s Stability for a Coupled Systems of Nonlinear Implicit Fractional Differential Equations

2018-04-30

Zeeshan Ali , Akbar Zada , Kamal Shah

On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative

2020-03-14

Kamal Shah , Fahd Jarad , Thabet Abdeljawad

Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order

2016-03-09

Abdon Atangana , İlknur Koca

A fixed-point theorem of Krasnoselskii

1998-01-01

T. A. Burton

North-Holland Mathematics Studies 202

2005-01-01

Nonlinear Functional Analysis

1985-01-01

Klaus Deimling

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.

Existence of solutions of nonlinear fractional pantograph equations

2013-05-01

K. Balachandran , S. Kiruthika , Juan J. Trujillo

Fractional derivatives with no-index law property: Application to chaos and statistics

2018-08-30

Abdon Atangana , J.F. Gómez‐Aguilar

Coupled fixed point theorems and applications to periodic boundary value problems

2013-01-01

Cristina Urs

In this paper we present some existence, uniqueness and Hyers-Ulam stability results for the coupled fixed point of a pair of contractive type operators on complete metric spaces.The approach is … In this paper we present some existence, uniqueness and Hyers-Ulam stability results for the coupled fixed point of a pair of contractive type operators on complete metric spaces.The approach is based on a Perov type fixed point theorem for contractions.Some applications to integral equations and to boundary value problems are also given.

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

2017-10-01

Toufik Mekkaoui , Abdon Atangana

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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

Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet

2016-07-28

Imran Aziz , Rohul Amin