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Mathematics › Modeling and Simulation

Fractional Differential Equations Solutions

Works Count: 60998

Description

This cluster of papers focuses on the modeling, analysis, and applications of anomalous diffusion using fractional calculus, nonlinear equations, and the homotopy analysis method. It explores phenomena such as random walk, fractional derivatives, time-fractional diffusion equations, single particle tracking in biological cells, and diffusion in crowded environments.

Keywords

Anomalous Diffusion; Fractional Calculus; Nonlinear Equations; Homotopy Analysis Method; Random Walk; Fractional Derivatives; Time-Fractional Diffusion Equation; Single Particle Tracking; Biological Cells; Crowded Environments

Nonlinear Functional Analysis and its Applications

1990-01-01
Eberhard Zeidler

Theory and Applications of Fractional Differential Equations

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

The Fractional Calculus - Theory and Applications of Differentiation and Integration to Arbitrary Order

1974-01-01
Keith B. Oldham , Jerome Spanier

Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers

1999-01-01
Igor Podlubný
Dynamic systems of an arbitrary real order (fractional-order systems) are considered. The concept of a fractional-order PI/sup /spl lambda//D/sup /spl mu//-controller, involving fractional-order integrator and fractional-order differentiator, is proposed. The … Dynamic systems of an arbitrary real order (fractional-order systems) are considered. The concept of a fractional-order PI/sup /spl lambda//D/sup /spl mu//-controller, involving fractional-order integrator and fractional-order differentiator, is proposed. The Laplace transform formula for a new function of the Mittag-Leffler-type made it possible to obtain explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional-order system with fractional-order controller for both open- and closed-loops. An example demonstrating the use of the obtained formulas and the advantages of the proposed PI/sup /spl lambda//D/sup /spl mu//-controllers is given.

SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS

2006-04-19
Ji‐Huan He
This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate … This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this field in China so the references, therefore, are not exhaustive.

Approximate analytical solution for seepage flow with fractional derivatives in porous media

1998-12-01
Ji‐Huan He

Fractional Schrödinger equation

2002-11-18
Nick Laskin
Some properties of the fractional Schrödinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical … Some properties of the fractional Schrödinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrödinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schrödinger equations.
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Analysis of Fractional Differential Equations

2002-01-01
Kai Diethelm , Neville J. Ford

Beyond Perturbation: Introduction to the Homotopy Analysis Method

2004-09-01
Shijun Liao , S. A. Sherif
9R2. Beyond Perturbation: Introduction to the Homotopy Analysis Method. - Edited by Shijun Liao (Shanghai Jiao Tong University, Shanghai, China). Chapman and Hall/CRC Press, Boca Raton FL. 2004. 322 pp. … 9R2. Beyond Perturbation: Introduction to the Homotopy Analysis Method. - Edited by Shijun Liao (Shanghai Jiao Tong University, Shanghai, China). Chapman and Hall/CRC Press, Boca Raton FL. 2004. 322 pp. ISBN 1-58488-407-X.Reviewed by SA Sherif (Dept of Mech and Aerospace Eng, Univ of Florida, 232 MAE Bldg B, PO Box 116300, Gainesville FL 32611-6300).This book deals with a very interesting mathematical technique that is rather powerful. While perturbation methods work nicely for slightly nonlinear problems, the homotopy analysis technique addresses nonlinear problems in a more general manner. Through this method, the author demonstrates that a nonlinear problem that normally has a unique solution can have an infinite number of different solution expressions whose convergence region and rate are dependent on an auxiliary parameter. The method provides for ways to control and adjust the convergence region. This makes the method particularly suited for problems with strong nonlinearity. The book is comprised of two parts. Part I contains Chapters 1 through 5, while Part II contains Chapters 6 through 18. The first part covers the basic ideas and concepts of the method, while the second part focuses on applications of the method to different situations. In addition to introducing the method in Part I, the author discusses the relation of the method to other analytical methods as well as the advantages and limitations of the method. Applications discussed in Part II are varied in scope covering areas such as simple bifurcation of nonlinear problems, nonlinear eigenvalue problems, the Thomas-Fermi atom model, free oscillation systems with both odd and quadratic nonlinearities, Blasius' viscous flow, boundary layer flows with exponential and algebraic properties, von Karman's swirling viscous flow, and nonlinear progressive waves in deep water. The book should serve as an excellent reference to researchers, engineers, and interested individuals in helping them tackle nonlinear problems in an analytical fashion. It has a good subject index and an outstanding list of bibliography with 136 references cited. The book is very well written and is relatively easy to follow to the mathematically literate person. I highly recommend that it be acquired by interested individuals and libraries throughout.
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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

2004-07-22
Ralf Metzler , J. Klafter
Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research … Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.
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Variational iteration method – a kind of non-linear analytical technique: some examples

1999-07-01
Ji‐Huan He

Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon

1981-11-09
Thomas A. Witten , Leonard M. Sander
A model for random aggregates is studied by computer simulation. The model is applicable to a metal-particle aggregation process whose correlations have been measured previously. Density correlations within the model … A model for random aggregates is studied by computer simulation. The model is applicable to a metal-particle aggregation process whose correlations have been measured previously. Density correlations within the model aggregates fall off with distance with a fractional power law, like those of the metal aggregates. The radius of gyration of the model aggregates has power-law behavior. The model is a limit of a model of dendritic growth.

Homotopy perturbation method: a new nonlinear analytical technique

2002-10-07
Ji‐Huan He

Mittag–Leffler stability of fractional order nonlinear dynamic systems

2009-05-14
Yan Li , YangQuan Chen , Igor Podlubný

Application of homotopy perturbation method to nonlinear wave equations

2005-05-18
Ji‐Huan He

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

2014-01-01
Ralf Metzler , Jae‐Hyung Jeon , Andrey G. Cherstvy +1 more
This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data. This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.
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Mean shift, mode seeking, and clustering

1995-01-01
Yizong Cheng
Mean shift, a simple interactive procedure that shifts each data point to the average of data points in its neighborhood is generalized and analyzed in the paper. This generalization makes … Mean shift, a simple interactive procedure that shifts each data point to the average of data points in its neighborhood is generalized and analyzed in the paper. This generalization makes some k-means like clustering algorithms its special cases. It is shown that mean shift is a mode-seeking process on the surface constructed with a "shadow" kernal. For Gaussian kernels, mean shift is a gradient mapping. Convergence is studied for mean shift iterations. Cluster analysis if treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data. Applications in clustering and Hough transform are demonstrated. Mean shift is also considered as an evolutionary strategy that performs multistart global optimization.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>
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User’s guide to viscosity solutions of second order partial differential equations

1992-01-01
Michael G. Crandall , Hitoshi Ishii , Pierre-Louis Lions
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about … The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.
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A family of variable-metric methods derived by variational means

1970-01-01
Donald Goldfarb
A new rank-two variable-metric method is derived using Greenstadt’s variational approach [<italic>Math. Comp.</italic>, this issue]. Like the Davidon-Fletcher-Powell (DFP) variable-metric method, the new method preserves the positive-definiteness of the approximating … A new rank-two variable-metric method is derived using Greenstadt’s variational approach [<italic>Math. Comp.</italic>, this issue]. Like the Davidon-Fletcher-Powell (DFP) variable-metric method, the new method preserves the positive-definiteness of the approximating matrix. Together with Greenstadt’s method, the new method gives rise to a one-parameter family of variable-metric methods that includes the DFP and rank-one methods as special cases. It is equivalent to Broyden’s one-parameter family [<italic>Math. Comp.</italic>, v. 21, 1967, pp. 368–381]. Choices for the inverse of the weighting matrix in the variational approach are given that lead to the derivation of the DFP and rank-one methods directly.
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Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability

2009-08-29
Yan Li , YangQuan Chen , Igor Podlubný

A new definition of fractional derivative

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

Chaos, fractional kinetics, and anomalous transport

2002-12-01
G. M. Zaslavsky

On conformable fractional calculus

2014-10-29
Thabet Abdeljawad

A coupling method of a homotopy technique and a perturbation technique for non-linear problems

2000-01-01
Ji‐Huan He

Application of a fractional advection‐dispersion equation

2000-02-01
David A. Benson , Stephen W. Wheatcraft , Mark M. Meerschaert
A transport equation that uses fractional‐order dispersion derivatives has fundamental solutions that are Lévy's α‐stable densities. These densities represent plumes that spread proportional to time 1/α , have heavy tails, … A transport equation that uses fractional‐order dispersion derivatives has fundamental solutions that are Lévy's α‐stable densities. These densities represent plumes that spread proportional to time 1/α , have heavy tails, and incorporate any degree of skewness. The equation is parsimonious since the dispersion parameter is not a function of time or distance. The scaling behavior of plumes that undergo Lévy motion is accounted for by the fractional derivative. A laboratory tracer test is described by a dispersion term of order 1.55, while the Cape Cod bromide plume is modeled by an equation of order 1.65 to 1.8.
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Finite difference approximations for fractional advection–dispersion flow equations

2004-03-17
Mark M. Meerschaert , Charles Tadjeran

The random walk's guide to anomalous diffusion: a fractional dynamics approach

2000-12-01
Ralf Metzler , J. Klafter

Exp-function method for nonlinear wave equations

2006-05-03
Ji‐Huan He , Xuhong Wu

Applications of Fractional Calculus to the Theory of Viscoelasticity

1984-06-01
R. C. Koeller
The connection between the fractional calculus and the theory of Abel’s integral equation is shown for materials with memory. Expressions for creep and relaxation functions, in terms of the Mittag-Leffler … The connection between the fractional calculus and the theory of Abel’s integral equation is shown for materials with memory. Expressions for creep and relaxation functions, in terms of the Mittag-Leffler function that depends on the fractional derivative parameter β, are obtained. These creep and relaxation functions allow for significant creep or relaxation to occur over many decade intervals when the memory parameter, β, is in the range of 0.05–0.35. It is shown that the fractional calculus constitutive equation allows for a continuous transition from the solid state to the fluid state when the memory parameter varies from zero to one.

Fractional calculus - A different approach to the analysis of viscoelastically damped structures

1983-05-01
Ronald L. Bagley , Peter J. Torvik
Fractional calculus is used to construct stress-strain relationships for viscoelastic materials. These relationships are used in the finite element analysis of viscoelastically damped structures and closed-form solutions to the equations … Fractional calculus is used to construct stress-strain relationships for viscoelastic materials. These relationships are used in the finite element analysis of viscoelastically damped structures and closed-form solutions to the equations of motion are found. The attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions.

Finite difference/spectral approximations for the time-fractional diffusion equation

2007-02-15
Yumin Lin , Chuanju Xu

Homotopy perturbation technique

1999-08-01
Ji‐Huan He

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

1999-01-01

Advances in Fractional Calculus

2007-01-01
Jocelyn Sabatier , Om P. Agrawal , J. A. Tenreiro Machado

A new collection of real world applications of fractional calculus in science and engineering

2018-04-24
HongGuang Sun , Yong Zhang , Dumitru Băleanu +2 more

A Note on the Multiplying Factors for Various χ2 Approximations

1954-07-01
M. S. Bartlett
Summary A list is given, for reference, of various approximate tests, based on the asymptotic approximations for likelihood ratios, but with adjusted multiplying factors. Summary A list is given, for reference, of various approximate tests, based on the asymptotic approximations for likelihood ratios, but with adjusted multiplying factors.

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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Nonlinear Functional Analysis and its Applications

1986-01-01
Eberhard Zeidler

The Analysis of Fractional Differential Equations

2010-01-01
Kai Diethelm

Applications of Fractional Calculus in Physics

2000-03-01
R. Hilfer

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.

Stability results for fractional differential equations with applications to control processing

1996-01-01
Denis Matignon
In this paper, stability results of main concern for control theory are given for finite-dimensional linear fractional differential systems. For fractional differential systems in state-space form, both internal and external … In this paper, stability results of main concern for control theory are given for finite-dimensional linear fractional differential systems. For fractional differential systems in state-space form, both internal and external stabilities are investigated. For fractional differential systems in polynomial representation, external stability is thoroughly examined. Our main qualitative result is that stabilities are guaranteed iff the roots of some polynomial lie outside the closed angular sector |arg(σ)| ≤ απ/2, thus generalizing in a stupendous way the well-known results for the integer case α = 1.

Elegant Rational Contractive Conditions with Applications to Implicit Functional Integral Equations

2025-06-25
Syed Irtaza Hassnain , Wutiphol Sintunavarat | Bangmod International Journal of Mathematical and Computational Science
This paper aims to introduce several new classes of contraction mappings inspired by convex and rational contraction mappings. We establish the existence and uniqueness of fixed points for each newly … This paper aims to introduce several new classes of contraction mappings inspired by convex and rational contraction mappings. We establish the existence and uniqueness of fixed points for each newly proposed contraction mapping in metric spaces. To validate our theoretical findings, we provide several illustrative examples that demonstrate cases where well-known fixed point results, such as the Banach contraction principle, the Kannan fixed point theorem, the Chatterjea fixed point theorem, the Jaggi fixed point theorem, and the Istratescu fixed point theorem, are not applicable. As an application, we employ our theoretical fixed point results to investigate the existence and uniqueness of solutions to nonlinear implicit integral equations.

H1-norm error analysis of an ADI compact finite difference method for a two-dimensional time-fractional reaction-diffusion equation with variable coefficients

2025-06-25
Pradip Roul , Sameer N. Khandagale , Jianxiong Cao | Computers & Mathematics with Applications

Approximate Analytical Solution of an Extended COVID-19 Model Incorporating Two-Dose Vaccination and Physical Control Measures

2025-06-24
Stanley Chinwekele Nwasuka , U.A. Osisiogu , E. Ebele +1 more | World Journal of Advanced Research and Reviews
The COVID-19 pandemic, which severely disrupted the global economy, remains a vital area of study for effective preparedness against future epidemics. The emergence of different variants has led to successive … The COVID-19 pandemic, which severely disrupted the global economy, remains a vital area of study for effective preparedness against future epidemics. The emergence of different variants has led to successive waves of the disease, prompting the development of numerous mathematical models. This study investigates an extended COVID-19 model that incorporates both first and second doses of vaccination as control strategies, alongside previously established physical preventive measures. The model is demonstrated to be mathematically and epidemiologically well-posed, with the existence and uniqueness of solutions to the state system established prior to analysis. Using the Next-Generation Matrix method, the control reproduction number was derived. Analytical results indicate that the disease-free equilibrium is locally and globally asymptotically stable when the control reproduction number, RcR_cRc​, is less than one, and unstable when it exceeds one. Sensitivity analysis was conducted to determine the influence of key parameters on Rc​. Findings highlight that improving compliance with hand sanitizing, social distancing, mask usage, testing, isolation, and vaccination significantly aids disease control. Conversely, reducing the rate of contact with exposed individuals, infectiousness development, and transmission probabilities also contributes to containment. Numerical simulations further illustrate the impact of these control measures, emphasizing the effectiveness of vaccination and adherence to physical protocols. The study recommends promoting vaccination and reinforcing compliance with physical preventive measures to mitigate the spread of COVID-19.

Fractional order Lorenz dynamics: investigating existence and uniqueness via basic contraction

2025-06-24
Haroon Ahmad , Fahim Ud Din , Mudasir Younıs | Journal of Applied Mathematics and Computing

A second-order difference scheme for time-fractional Burgers-Fisher equation with initial singularity

2025-06-24
Zhibo Wang , Wenguang Yang , Feng Li | International Journal of Computer Mathematics

Fractional Wave Potentials and Capacities with Some Applications

2025-06-24
Zhen Shuang , J. Xiao , C. Yue | Journal of Geometric Analysis

Mathematical modelling of child mortality with Caputo–Fabrizio fractional derivative

2025-06-24
Patience Pokuaa Gambrah , Herman Matondo Mananga , Gratien Twagirumukiza +3 more | Open Journal of Mathematical Sciences
In this article, we proposed a fractional-order mathematical model of Child mortality. We analyzed the existence of a unique solution for our model using the fixed point theory and Picard–Lindelöf … In this article, we proposed a fractional-order mathematical model of Child mortality. We analyzed the existence of a unique solution for our model using the fixed point theory and Picard–Lindelöf technique. We propose a Caputo operator for modeling child mortality in a given population of 1000 susceptible under five children. Our stability analysis was based on the fixed point theory, which was used to prove that our Picard iteration was stable. Using the Julia software and some real world values for our parameters, we numerically simulated the system through graphs. Our findings were that, reducing child mortality rates alone is insufficient to significantly improve survival rates for children under five. To make a real impact, a holistic approach is necessary, including access to healthcare, proper nutrition, vaccination programs, hygiene practices, clean water sources and comprehensive public health campaigns can greatly enhance the survival rates of children under five.

Convergence analysis of a Kantorovich approximation technique for solving fractional Volterra integral equations

2025-06-24
Faruk Özger , Sinan Deni̇z , Cheima Khennaoui +2 more | Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas

Adsorption–Desorption at Anomalous Diffusion: Fractional Calculus Approach

2025-06-24
Ivan Bazhlekov , Emilia Bazhlekova | Fractal and Fractional
A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional … A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of the flux bulk–interface. The general case, when the surfactant is soluble in both phases, is considered under the assumption that the adsorption–desorption process is diffusion-controlled. Some of the most popular kinetic models of Henry, Langmuir, and Volmer are considered. Applying the Laplace transform, the partial differential model is transformed into a single multi-term time-fractional nonlinear ordinary differential equation for the surfactant concentration on the interface. Based on existing analytical solutions of linear time-fractional differential equations, the exact solution in the case of the Henry model is derived in terms of multinomial Mittag–Leffler functions, and its asymptotic behavior is studied. Further, the fractional differential model in the general nonlinear case is rewritten as an integral equation, which is a generalization of the well-known Ward–Tordai equation. For computer simulations, based on the obtained integral equation, a predictor–corrector numerical technique is developed. Numerical results are presented and analyzed.

Theoretical study and numerical analysis using step spectral collocation method for stochastic M-fractional differential models of simplified Brownian motion within constant delays process

2025-06-24
Haneen Badawi , Omar Abu Arqub , Nabil Shawagfeh | Journal of Applied Mathematics and Computing

Numerical solutions of the nonlinear Fisher’s equation using a one-level meshless method

2025-06-24
Yingzi Jiang , Fuzhang Wang , Zhongyang Sun | Frontiers in Physics
This study introduces a relatively new numerical technique for solving one-dimensional Fisher’s equation. The proposed numerical technique is a simple direct meshless method, which is based on the collocation scheme. … This study introduces a relatively new numerical technique for solving one-dimensional Fisher’s equation. The proposed numerical technique is a simple direct meshless method, which is based on the collocation scheme. To circumvent the traditional two-level numerical procedure, the space-time radial basis function is considered. Under such circumstances, the time-dependent one-dimensional nonlinear Fisher’s equation can be solved by a one-level numerical procedure. Several numerical results are investigated to show advantages of the proposed meshless method.

Optimal control of conformable fractional neutral stochastic integrodifferential systems with infinite delay

2025-06-23
Dimplekumar Chalishajar , Ramkumar Kasinathan , K. Ravikumar +2 more | Optimization

The numerical solution of a Fredholm integral equation of the second kind using the Galerkin method based on optimal interpolation

2025-06-23
Samandar Babaev , Abdullo Hayotov , Asliddin Boltaev +2 more | Results in Applied Mathematics

A useful extension of Mittag-Leffler type fractional derivatives with respect to functions: An application to a fractional Lotka-Volterra system

2025-06-23
Zaid Odibat | Journal of Pseudo-Differential Operators and Applications

Exploring SEIR Influenza Epidemic Model via Fuzzy ABC Fractional Derivatives with Crowley–Martin Incidence Rate

2025-06-23
F. Gassem , Ashraf Ahmed Qurtam , Mohammed A. ‬Almalahi +4 more | Fractal and Fractional
Despite initial changes in respiratory illness epidemiology due to SARS-CoV-2, influenza activity has returned to pre-pandemic levels, highlighting its ongoing challenges. This paper investigates an influenza epidemic model using a … Despite initial changes in respiratory illness epidemiology due to SARS-CoV-2, influenza activity has returned to pre-pandemic levels, highlighting its ongoing challenges. This paper investigates an influenza epidemic model using a Susceptible-Exposed-Infected-Recovered (SEIR) framework, extended with fuzzy Atangana–Baleanu–Caputo (ABC) fractional derivatives to incorporate uncertainty (via fuzzy numbers for state variables) and memory effects (via the ABC fractional derivative for non-local dynamics). We establish the theoretical foundation by defining the fuzzy ABC derivatives and integrals based on the generalized Hukuhara difference. The existence and uniqueness of the solutions for the fuzzy fractional SEIR model are rigorously proven using fixed-point theorems. Furthermore, we analyze the system’s disease-free and endemic equilibrium points under the fractional framework. A numerical scheme based on the fractional Adams–Bashforth method is used to approximate the fuzzy solutions, providing interval-valued results for different uncertainty levels. The study demonstrates the utility of fuzzy fractional calculus in providing a more flexible and potentially realistic approach to modeling epidemic dynamics under uncertainty.

حل المعادلات التفاضلية العادية الخطية ذات المعاملات المتغيرة باستخدام تحويل تكاملي جديد

2025-06-23
غادة اشتيوي | Journal of Pure & Applied Sciences
يهدف هذا البحث إلى تطبيق تحويل تكاملي جديد لحل المعادلات التفاضلية العادية الخطية ذات المعاملات المتغيرة، حيث يعد هذا النوع من المعادلات من النماذج الرياضية المعقدة التي تواجه العديد من … يهدف هذا البحث إلى تطبيق تحويل تكاملي جديد لحل المعادلات التفاضلية العادية الخطية ذات المعاملات المتغيرة، حيث يعد هذا النوع من المعادلات من النماذج الرياضية المعقدة التي تواجه العديد من التحديات في مجال الرياضيات التطبيقية. في هذا الإطار، تم اقتراح تحويل تكاملي جديد يعتمد على دالة نواة تعمل على تبسيط التعامل مع معاملات غير ثابتة. وقد تم تعريف هذا التحويل رياضيًا بالإضافة إلى تقديم الشروط اللازمة لوجوده، ثم استُعرضت خصائصه الأساسية، بما في ذلك تطبيقه على بعض الدوال الأساسية مثل كثيرات الحدود والدوال الأسية، كما تم ايضاح التحويل العكسي الذي يُمكن من استرجاع الدالة الأصلية بعد إجراء عملية التحويل، بالإضافة إلى اشتقاق قواعد تحويل المشتقات من الرتبتين الأولى والثانية، مع الإشارة إلى إمكانية تعميم هذا التحويل على مشتقات من رتب أعلى. تمت دراسة كفاءة هذا التحويل من خلال تطبيقه على مجموعة من مسائل القيمة الابتدائية للمعادلات التفاضلية، وقد أظهرت النتائج أن التحويل التكاملي الجديد يُتيح الوصول إلى حلول تحليلية دقيقة دون الحاجة إلى إجراء عمليات رياضية معقدة أو اللجوء إلى طرق عددية تقريبية. ومن خلال المقارنة بين الحلول المتحصّل عليها باستخدام هذا التحويل وتلك التي تنتج عن استخدام تحويل لابلاس، وُجد أن التحويل الجديد قد يُظهر تفوقاً في بعض الحالات من حيث الدقة والبساطة. من خلال هذه الدراسة، يتضح أن التحويل التكاملي الجديد يمكن أن يشكل أداة رياضية قوية لتحليل المعادلات ذات المعاملات المتغيرة، ويمهد الطريق أمام استخدامه في حل أنظمة المعادلات التفاضلية المعقدة والمعادلات التفاضلية الجزئية. لذا، فإن هذا التحويل يمثل مساهمة مبتكرة في التحويلات الرياضية وتطبيقاتها المستقبلية وتوسيع آفاق استخدام التحويلات التكامليّة كأدوات فعّالة في حل المعادلات التفاضلية. علاوة على ذلك، تم كتابة مجموعة من الأكواد البرمجية باستخدام برنامج MATLAB، تتيح حساب التحويل التكاملي الجديد تلقائيًا للدوال من النوع ‎t^n f''(x),t^n f^' (x),t^n f(t)، حيث ‎n عدد صحيح موجب. وقد ساهمت هذه الأدوات البرمجية في تسريع عملية الحل وتقليل الجهد الحسابي، مما يعزز إمكانية استخدام هذا التحويل في التطبيقات العملية والهندسية.

Efficient discretization of fractional SPDEs via Galerkin and exponential Euler methods

2025-06-23
Minoo Kamrani | Computers & Mathematics with Applications

A fitted scheme for the nonlinear time fractional Gray-Scott model with nonsmooth solutions

2025-06-22
Xiaolong Deng , Caixia Ou , Zhibo Wang +1 more | Journal of Applied Mathematics and Computing

Oscillation of non-instantaneous impulsive hybrid-fractional delay differential inclusions with Hilfer-Katugampola derivative

2025-06-22
Assia Boudjerida , Djamila Seba , Naima Meskine +1 more | Journal of Applied Mathematics and Computing

On a new definition of fractional derivative: a comparative study of singular kernel fractional derivative operators

2025-06-22
Zaid Odibat | Journal of Applied Mathematics and Computing

Fractional-based nonlinear mechanical system modeling with FELEs: numerical analysis of oscillatory and nonoscillatory behavior of the inverted pendulum system

2025-06-21
Esra Demir , İbrahim Özkol | Archive of Applied Mechanics
Abstract This paper examines the behavior of the inverted pendulum, a notably nonlinear system, in fractional dimensions using different fractional derivative types and order. The inverted pendulum, a two-degree-of-freedom system, … Abstract This paper examines the behavior of the inverted pendulum, a notably nonlinear system, in fractional dimensions using different fractional derivative types and order. The inverted pendulum, a two-degree-of-freedom system, exhibits both linear behavior due to the cart’s motion in the horizontal plane and oscillatory behavior due to the pendulum’s angular motion. Initially, the system’s equations of motion have been derived using the classical Euler–Lagrange equation (CELE), thereby obtaining the classical integer-order model. Subsequently, the fractional model has been developed using the fractional Euler–Lagrange equation (FELE) with the Riemann-Liouville and the Caputo–Fabrizio fractional derivatives. The results of the models obtained were shown in the simulation platform and presented comparatively. In this paper, the impact of fractional-order modeling on both oscillatory and nonoscillatory motions of mechanical systems is analyzed. This is achieved by introducing the inverted pendulum model and employing two different types of fractional-order derivatives.

Modified Numerical Method for Solving Fractional Order Infectious Disease Model

2025-06-21
| Deleted Journal

Fractional-order modeling of tuberculosis and diabetes mellitus co-existence dynamics

2025-06-21
Mohammed S. ‬Abdo , Najla Alghamdi , Hadeel Z. Alzumi +1 more | Computers in Biology and Medicine

Novel and Reliable Techniques for Two‐Dimensional Stochastic Itô–Volterra Fredholm Integral Equations

2025-06-21
S. Saha Ray | Mathematical Methods in the Applied Sciences
ABSTRACT This article proposes two novel numerical techniques for solving two‐dimensional stochastic Itô–Volterra Fredholm integral equations. Many two‐dimensional stochastic integral equations present significant challenges for analytical solutions. Consequently, possessing an … ABSTRACT This article proposes two novel numerical techniques for solving two‐dimensional stochastic Itô–Volterra Fredholm integral equations. Many two‐dimensional stochastic integral equations present significant challenges for analytical solutions. Consequently, possessing an efficient method to get very precise numerical solutions for these stochastic integral equations is of paramount importance. In this article, two novel numerical techniques based on two‐dimensional wavelets have been successfully introduced to acquire the numerical solutions of the two‐dimensional stochastic Itô–Volterra Fredholm integral equations. This efficient and noteworthy method transforms the stochastic integral equation into a system of algebraic equations, which is then solved using an appropriate numerical technique. Moreover, convergence analysis and the error analysis have also been well established successfully. At the end, some test problems have been presented to illustrate the accuracy, efficiency, simplicity, and plausibility of the proposed techniques.

Analytical Solutions to Fractional Nonlinear Dispersive Models using Riccati-type Equations

2025-06-20
Hamı Gündoğdu | WSEAS TRANSACTIONS ON SYSTEMS
In this study, two important fractional wave equations that describe nonlinear dispersion behaviors in complex media, namely the fractional Schrödinger equation and the complex modified Korteweg–de Vries equation, are considered. … In this study, two important fractional wave equations that describe nonlinear dispersion behaviors in complex media, namely the fractional Schrödinger equation and the complex modified Korteweg–de Vries equation, are considered. In order to model the memory effects in wave propagation more realistically, the Caputo-type fractional derivative is used. Analytical solutions are obtained by the sub-equation method based on the Riccati equation. These solutions are expressed in the form of hyperbolic, trigonometric, and rational functions. In order to reveal the effect of fractional order on wave dynamics, the real and imaginary components of the solutions are visualized in three dimensions for various parameter values. The obtained results improve the analytical understanding of fractional nonlinear systems and make a significant contribution to the current research in the field of complex wave theory.

Spatiotemporal pattern evolution on a spatial fractional Rosenzweig-MacArthur model

2025-06-20
Yong Ye , Jiaying Zhou | Ricerche di Matematica

An efficient collocation method based on Legendre and Romanovski polynomials for solving Riesz distributed fractional differential equations

2025-06-20
Ammar Lachin , M. A. Abdelkawy , Saratha Sathasivam | Alexandria Engineering Journal

Time-space finite element analysis for time-space fractional parabolic problems involving spectral fractional Laplacian

2025-06-20
Yi Yang , Jin Huang , Hongyan Liu +1 more | Journal of Applied Mathematics and Computing

A Combined Separation of Variables and Fractional Power Series Approach for Selected Boundary Value Problems

2025-06-20
Gabriel Antonio Felipe , Carlos Alberto Valentim , Sérgio Adriani David | Dynamics
Fractional modeling has emerged as an important resource for describing complex phenomena and systems exhibiting non-local behavior or memory effects, finding increasing application in several areas in physics and engineering. … Fractional modeling has emerged as an important resource for describing complex phenomena and systems exhibiting non-local behavior or memory effects, finding increasing application in several areas in physics and engineering. This study presents the analytical derivation of equations pertinent to the modeling of different systems, with a focus on heat conduction. Two specific boundary value problems are addressed: a Helmholtz equation modified with a fractional derivative term, and a fractional formulation of the Laplace equation applied to steady-state heat conduction in circular geometry. The methodology combines the separation of variables technique with fractional power series expansions, primarily utilizing the Caputo fractional derivative. An important aspect of this paper is its instructional emphasis, wherein the mathematical derivations are presented with detail and clarity. This didactic approach is intended to make the analytical methodology transparent and more understandable, thereby facilitating greater comprehension of the application of these established methods to non-integer-order systems. The final goal is not only to provide a different approach of solving these physical models analytically, but to provide a clear, guided pathway for those engaging in the treatment of fractional differential equations.

Advancing Fractional Boundary Value Problems Solutions Using Shifted Vieta-Lucas Polynomials and Spectral Approaches

2025-06-20
Zulfiqar Ahmad Noor , Mohie M. Alqezweeni , Muhammad Bilal Riaz +1 more | Fractals

On Computation of Prefactor of Free Boundary in One Dimensional One-Phase Fractional Stefan Problems

2025-06-20
N. Caruso , Sabrina Roscani , Lucas Venturato +1 more | Fractal and Fractional
We consider the melting of a one-dimensional domain (x≥0), initially at the melting temperature u=0, by fixing the boundary temperature to a value u(0,t)=U0&gt;0—the so called Stefan melting problem. The … We consider the melting of a one-dimensional domain (x≥0), initially at the melting temperature u=0, by fixing the boundary temperature to a value u(0,t)=U0&gt;0—the so called Stefan melting problem. The governing transient heat-conduction equation involves a time derivative and the spatial derivative of the temperature gradient. In the general case the order of the time derivative and the gradient can take values in the range (0,1]. In these problems it is known that the advance of the melt front s(t) can be uniquely determined by a specified prefactor multiplying a power of time related to the order of the fractional derivatives in the governing equation. For given fractional orders the value of the prefactor is the unique solution to a transcendental equation formed in terms of special functions. Here, our main purpose is to provide efficient numerical schemes with low computational complexity to compute these prefactors. The values of the prefactors are obtained through a dimensionalization that allows the recovery of the solution for the quasi-stationary case when the Stefan number approaches zero. The mathematical analysis of this convergence is given and provides consistency to the numerical results obtained.

Theoretical and numerical investigation of long-time behaviors for time-fractional Fisher equations

2025-06-20
Ahmed S. Rahby , Zhanwen Yang | Journal of Applied Mathematics and Computing

A Virtual Element Method for a (21)-Dimensional Wave Equation with Time-Fractional Dissipation on Polygonal Meshes+

2025-06-20
Zaffar Mehdi Dar , M. Chandru , Higinio Ramos | Fractal and Fractional
We propose a novel space-time discretization method for a time-fractional dissipative wave equation. The approach employs a structured framework in which a fully discrete formulation is produced by combining virtual … We propose a novel space-time discretization method for a time-fractional dissipative wave equation. The approach employs a structured framework in which a fully discrete formulation is produced by combining virtual elements for spatial discretization and the Newmark predictor–corrector method for the temporal domain. The virtual element technique is regarded as a generalization of the finite element method for polygonal and polyhedral meshes within the Galerkin approximation framework. To discretize the time-fractional dissipation term, we utilize the Grünwald-Letnikov approximation in conjunction with the predictor–corrector scheme. The existence and uniqueness of the discrete solution are theoretically proved, together with the optimal convergence order achieved and an error analysis associated with the H1-seminorm and the L2-norm. Numerical experiments are presented to support the theoretical findings and demonstrate the effectiveness of the proposed method with both convex and non-convex polygonal meshes.

An Efficient Multi Projection Methods for Systems of Fredholm Integral Equations with Mixed Weakly Singular Kernels: A Superconvergence Approach

2025-06-19
Krishna Murari Malav , Kapil Kant , Joydip Dhar +1 more | Applied Numerical Mathematics

Novel solutions structure for fractional Schrödinger equations (FSE) via a new scheme

2025-06-19
Loubna Ouahid | Modern Physics Letters B
In this paper, we employ a recently developed novel scheme, namely, the extended mapping approach, for constructing new periodic soliton solutions, including rational, hyperbolic and trigonometric function solutions for nonlinear … In this paper, we employ a recently developed novel scheme, namely, the extended mapping approach, for constructing new periodic soliton solutions, including rational, hyperbolic and trigonometric function solutions for nonlinear systems of fractional Schrödinger equations (FSE) involving conformable (CF) fractional derivatives. We examine the influence of specific factors on the solutions of these governing equations and use 3D and 2D graphs to illustrate the dynamic wave patterns, a method that has not been probed in previous studies. The results of this study may assist researchers in gaining a deeper understanding of the dynamic behaviors in plasma physics, optical fibers, quantum electronics and nonlinear sciences. This work emphasizes the dynamic nature of the FSE by presenting various sets of soliton solutions and their intricate interconnections. These findings are crucial not only for comprehending the dynamics of FSE but also for their applications in nonlinear fractional partial differential equations.

Subdifferentially Polynomially Bounded Functions and Gaussian Smoothing–Based Zeroth-Order Optimization

2025-06-19
Ming Lei , Ting Kei Pong , Shuqin Sun +1 more | SIAM Journal on Optimization
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Two-Scale Approach for the Analysis of Time-Fractional Diffusion Problems in Heterogeneous Media

2025-06-19
Jingmei Peng , Huaping Xiao , Asim Suleman A. Alwabel +2 more | Fractals

Solvability of Fuzzy Partially Differentiable Models for Caputo–Hadamard-Type Goursat Problems Involving Generalized Hukuhara Difference

2025-06-19
Si‐Jian Lin , Heng-you Lan , Jihong Li | Fractal and Fractional
In this paper, we investigate a class of fuzzy partially differentiable models for Caputo–Hadamard-type Goursat problems with generalized Hukuhara difference, which have been widely recognized as having a significant role … In this paper, we investigate a class of fuzzy partially differentiable models for Caputo–Hadamard-type Goursat problems with generalized Hukuhara difference, which have been widely recognized as having a significant role in simulating and analyzing various kinds of processes in engineering and physical sciences. By transforming the fuzzy partially differentiable models into equivalent integral equations and employing classical Banach and Schauder fixed-point theorems, we establish the existence and uniqueness of solutions for the fuzzy partially differentiable models. Furthermore, in order to overcome the complexity of obtaining exact solutions of systems involving Caputo–Hadamard fractional derivatives, we explore numerical approximations based on trapezoidal and Simpson’s rules and propose three numerical examples to visually illustrate the main results presented in this paper.

A Novel Approach to Solving Fractional Navier–Stokes Equations Using the Elzaki Transform

2025-06-19
Tarig M. Elzaki , Eltaib M. Abd Elmohmoud | Fractal and Fractional
A clear method is provided to explain a new approach for solving systems of fractional Navier–Stokes equations (SFNSEs) using initial conditions (ICs) that rely on the Elzaki transform (ET). A … A clear method is provided to explain a new approach for solving systems of fractional Navier–Stokes equations (SFNSEs) using initial conditions (ICs) that rely on the Elzaki transform (ET). A few steps show the technique’s validity and utility for handling SFNSE solutions. For fractional derivatives, the Caputo sense is used. This method does not need discretization or limiting assumptions and may be used to solve both linear and nonlinear SFNSEs. By eliminating round-off mistakes, the technique reduces the need for numerical calculations. Using examples, the new technique’s accuracy and efficacy are illustrated.

REGULARIZED ITERATIVE FDEM: A NUMERICAL APPROACH FOR NONLINEAR CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS

2025-06-19
Ramzi B. Albadarneh , Iqbal M. Batiha , Radwan M. Batyha +1 more | International Journal of Apllied Mathematics

UNIQUENESS AND CONVERGENCE ANALYSIS OF THE FRACTIONAL VOLTERRA-FREDHOLM MODEL

2025-06-19
E.A. Hussein , S.A.M. Jameel | International Journal of Apllied Mathematics

Analytical and Numerical Treatment of Evolutionary Time-Fractional Partial Integro-Differential Equations with Singular Memory Kernels

2025-06-19
Kamel Al‐Khaled , Isam Al‐Darabsah , Amer Darweesh +1 more | Fractal and Fractional
Evolution equations with fractional-time derivatives and singular memory kernels are used for modeling phenomena exhibiting hereditary properties, as they effectively incorporate memory effects into their formulation. Time-fractional partial integro-differential equations … Evolution equations with fractional-time derivatives and singular memory kernels are used for modeling phenomena exhibiting hereditary properties, as they effectively incorporate memory effects into their formulation. Time-fractional partial integro-differential equations (FPIDEs) represent a significant class of such evolution equations and are widely used in diverse scientific and engineering fields. In this study, we use the sinc-collocation and iterative Laplace transform methods to solve a specific FPIDE with a weakly singular kernel. Specifically, the sinc-collocation method is applied to discretize the spatial domain, while a combination of numerical techniques is utilized for temporal discretization. Then, we prove the convergence analytically. To compare the two methods, we provide two examples. We notice that both the sinc-collocation and iterative Laplace transform methods provide good approximations. Moreover, we find that the accuracy of the methods is influenced by fractional order α∈(0,1) and the memory-kernel parameter β∈(0,1). We observe that the error decreases as β increases, where the kernel becomes milder, which extends the single-value study of β=1/2 in the literature.

On special ruled surface pairs in fractional calculus

2025-06-19
Zehra Nur Koçak , Emel Karaca | Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics
In this paper, we extend the theory of special fractional curve pairs (i.e., F-Bertrand, FMannheim, and F-involute-evolute curve pairs) to fractional ruled surfaces with the perspective of fractional calculus. Next, … In this paper, we extend the theory of special fractional curve pairs (i.e., F-Bertrand, FMannheim, and F-involute-evolute curve pairs) to fractional ruled surfaces with the perspective of fractional calculus. Next, we characterize two fractional ruled surfaces, offset in the senses of F-Bertrand, F-Mannheim, and F-involute-evolute. Moreover, considering the chain rules in fractional calculus, some significant theorems are proved, and the developability conditions are examined by calculating the distribution parameters. Finally, we give examples to verify the results.

On the Existence of (p,q)-Solutions for the Post-Quantum Langevin Equation: A Fixed-Point-Based Approach

2025-06-19
Mohammed Jasim Mohammed , Ali Ghafarpanah , Sina Etemad +2 more | Axioms
The two-parameter (p,q)-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we … The two-parameter (p,q)-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we present a new construction of the Langevin equation using two-parameter (p,q)-Caputo derivatives. For this new Langevin equation, equivalently, we obtain the solution structure as a post-quantum integral equation and then conduct an existence analysis via a fixed-point-based approach. The use of theorems such as the Krasnoselskii and Leray–Schauder fixed-point theorems will guarantee the existence of solutions to this equation, whose uniqueness is later proven by Banach’s contraction principle. Finally, we provide three examples in different structures and validate the results numerically.

Proposal for the Application of Fractional Operators in Polynomial Regression Models to Enhance the Determination Coefficient R2 on Unseen Data

2025-06-19
A. Torres-Hernandez , Rafael Ramírez , F. Brambila-Paz | Fractal and Fractional
Since polynomial regression models are generally quite reliable for data that can be handled using a linear system, it is important to note that in some cases, they may suffer … Since polynomial regression models are generally quite reliable for data that can be handled using a linear system, it is important to note that in some cases, they may suffer from overfitting during the training phase. This can lead to negative values of the coefficient of determination R2 when applied to unseen data. To address this issue, this work proposes the partial implementation of fractional operators in polynomial regression models to construct a fractional regression model. The aim of this approach is to mitigate overfitting, which could potentially improve the R2 value for unseen data compared to the conventional polynomial model, under the assumption that this could lead to predictive models with better performance. The methodology for constructing these fractional regression models is presented along with examples applicable to both Riemann–Liouville and Caputo fractional operators, where some results show that regions with initially negative or near-zero R2 values exhibit remarkable improvements after the application of the fractional operator, with absolute relative increases exceeding 800% on unseen data. Finally, the importance of employing sets in the construction of the fractional regression model within this methodological framework is emphasized, since from a theoretical standpoint, one could construct an uncountable family of fractional operators derived from the Riemann–Liouville and Caputo definitions that, although differing in their formulation, would yield the same regression results as those shown in the examples presented in this work.

An enhanced Artificial Neural Network approach for solving nonlinear fractional-order differential equations

2025-06-18
Nikhil Sharma , Sunil Joshi , Pranay Goswami | Partial Differential Equations in Applied Mathematics

(Non)Commutativity and associativity of general fractional derivatives with different Sonin kernels

2025-06-18
Vasily E. Tarasov | Communications in Nonlinear Science and Numerical Simulation

On the solutions of fractional differential equations with modified Mittag-Leffler kernel and Dirac Delta function: Analytical results and numerical simulations

2025-06-18
Mohammed Al-Refai , Dumitru Băleanu , A‎. ‎K‎. Alomari | PLoS ONE
In this paper we define, for the first time, the modified fractional derivative with Mittage-Leffler kernel of Riemann–Liouville (R-L) type of arbitrary order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="M1"><mml:mrow><mml:mi>δ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> delta. We derive … In this paper we define, for the first time, the modified fractional derivative with Mittage-Leffler kernel of Riemann–Liouville (R-L) type of arbitrary order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="M1"><mml:mrow><mml:mi>δ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> delta. We derive the infinite series representations for the modified derivatives of R-L and Caputo types and present a relationship between them. We also investigate the modified derivatives for the Dirac delta functions, and study related fractional differential equations. Explicit solutions were presented for linear fractional differential equations with constant coefficients via the Laplace transform. A fractional model with the modified derivative is considered and numerical simulations were presented.

Conservation laws and discrete counterparts for the time-fractional generalized nonlinear Schrödinger equation

2025-06-18
Wei Yao , Pin Lyu , Seakweng Vong | Applied Mathematics Letters