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20 pages, 1370 KB

Open AccessArticle

Triply Coupled Systems of Differential Equations with Time-Dependent Delay and Application to Three-Species Food-Chain Dynamics

by F. Gassem, L. M. Abdalgadir, Arshad Ali, Alwaleed Kamel, Alawia Adam, Khaled Aldwoah and M. M. Rashed

Fractal Fract. 2025, 9(10), 651; https://doi.org/10.3390/fractalfract9100651 - 8 Oct 2025

Abstract

We introduce a class of triply coupled systems of differential equations with fractal–fractional Caputo derivatives and time-dependent delays. This framework captures long-memory effects and complex structural patterns while allowing delays to evolve over time, offering greater realism than constant-delay models. The existence and [...] Read more.

We introduce a class of triply coupled systems of differential equations with fractal–fractional Caputo derivatives and time-dependent delays. This framework captures long-memory effects and complex structural patterns while allowing delays to evolve over time, offering greater realism than constant-delay models. The existence and uniqueness of solutions are established using fixed point theory, and Hyers–Ulam stability is analyzed. A numerical scheme based on the Adams–Bashforth method is implemented to approximate solutions. The approach is illustrated through a numerical example and applied to a three-species food-chain model, comparing scenarios with and without time-dependent delays to demonstrate their impact on system dynamics. Full article

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29 pages, 971 KB

Open AccessArticle

Fractional-Order Numerical Scheme with Symmetric Structure for Fractional Differential Equations with Step-Size Control

by Mudassir Shams and Mufutau Ajani Rufai

Symmetry 2025, 17(10), 1685; https://doi.org/10.3390/sym17101685 - 8 Oct 2025

Abstract

This research paper uses two-stage explicit fractional numerical schemes to solve fractional-order initial value problems of ODEs. The proposed methods exhibit structural symmetry in their formulation, contributing to enhanced numerical stability and balanced error behavior across computational steps. The schemes utilize constant and [...] Read more.

This research paper uses two-stage explicit fractional numerical schemes to solve fractional-order initial value problems of ODEs. The proposed methods exhibit structural symmetry in their formulation, contributing to enhanced numerical stability and balanced error behavior across computational steps. The schemes utilize constant and variable step sizes, allowing them to adapt efficiently to solve the considered fractional-order initial value problems. These schemes employ variable step-size control based on error estimation, aiming to minimize computational costs while maintaining good accuracy and stability. We discuss the linear stability of the proposed numerical schemes and observe that a higher-stability region is obtained when the fractional parameter value equals one. We also discuss consistency and convergence analysis of the proposed methods and observe that as the fractional parameter values rise from 0 to 1, the scheme’s convergence rate improves and achieves its maximum at 1. Several numerical test problems are used to demonstrate the efficiency of the proposed methods in solving fractional-order initial value problems with either constant or variable step sizes. The proposed numerical schemes’ results demonstrate better accuracy and convergence behavior than the existing methods used for comparison. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)

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28 pages, 924 KB

Open AccessArticle

Hybrid Fuzzy Fractional for Multi-Phasic Epidemics: The Omicron–Malaria Case Study

by Mohamed S. Algolam, Ashraf A. Qurtam, Mohammed Almalahi, Khaled Aldwoah, Mesfer H. Alqahtani, Alawia Adam and Salahedden Omer Ali

Fractal Fract. 2025, 9(10), 643; https://doi.org/10.3390/fractalfract9100643 - 1 Oct 2025

Viewed by 200

Abstract

This study introduces a novel Fuzzy Piecewise Fractional Derivative (FPFD) framework to enhance epidemiological modeling, specifically for the multi-phasic co-infection dynamics of Omicron and malaria. We address the limitations of traditional models by incorporating two key realities. First, we use fuzzy set theory [...] Read more.

This study introduces a novel Fuzzy Piecewise Fractional Derivative (FPFD) framework to enhance epidemiological modeling, specifically for the multi-phasic co-infection dynamics of Omicron and malaria. We address the limitations of traditional models by incorporating two key realities. First, we use fuzzy set theory to manage the inherent uncertainty in biological parameters. Second, we employ piecewise fractional operators to capture the dynamic, phase-dependent nature of epidemics. The framework utilizes a fuzzy classical derivative for initial memoryless spread and transitions to a fuzzy Atangana–Baleanu–Caputo (ABC) fractional derivative to capture post-intervention memory effects. We establish the mathematical rigor of the FPFD model through proofs of positivity, boundedness, and stability of equilibrium points, including the basic reproductive number (R0). A hybrid numerical scheme, combining Fuzzy Runge–Kutta and Fuzzy Fractional Adams–Bashforth–Moulton algorithms, is developed for solving the system. Simulations show that the framework successfully models dynamic shifts while propagating uncertainty. This provides forecasts that are more robust and practical, directly informing public health interventions. Full article

(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods, and Applications—2nd Edition)

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13 pages, 251 KB

Open AccessArticle

Solution of Nonhomogeneous Linear System of Caputo Fractional Differential Equations with Initial Conditions

by Aghalaya S. Vatsala and Govinda Pageni

Axioms 2025, 14(10), 736; https://doi.org/10.3390/axioms14100736 - 29 Sep 2025

Viewed by 178

Abstract

The solution of a nonhomogeneous linear Caputo fractional differential equation of order

n q, (n - 1) < n q < n

with Caputo fractional initial conditions can be expressed using suitable Mittag–Leffler functions. In order to extend this result [...] Read more.

The solution of a nonhomogeneous linear Caputo fractional differential equation of order

n q, (n - 1) < n q < n

with Caputo fractional initial conditions can be expressed using suitable Mittag–Leffler functions. In order to extend this result to such a nonhomogeneous linear Caputo fractional differential equation of order

n q, (n - 1) < n q < n,

that also includes lower order fractional derivative terms, we can reduce such a problem to an n-system of Caputo fractional differential equations of order

q, 0 < q < 1,

with corresponding initial conditions. In this work, we use an approximation method to solve the resulting system of Caputo fractional differential equations of order q with initial conditions, using the fundamental matrix solutions involving the matrix Mittag–Leffler functions. Furthermore, we compute the fundamental matrix solution using the standard eigenvalue method. This fundamental matrix solution then allows us to express the component-wise solutions of the system using initial conditions, similar to the scalar case. As a consequence, we obtain solutions to linear nonhomogeneous Caputo fractional differential equations of order

n q, (n - 1) < n q < n,

with Caputo fractional initial conditions having lower-order Caputo derivative terms. We illustrate the method with several examples for two and three system, considering cases where the eigenvalues are real and distinct, real and repeated, or complex conjugates. Full article

17 pages, 915 KB

Open AccessFeature PaperArticle

Solutions for Linear Fractional Differential Equations with Multiple Constraints Using Fractional B-Poly Bases

by Md. Habibur Rahman, Muhammad I. Bhatti and Nicholas Dimakis

Mathematics 2025, 13(19), 3084; https://doi.org/10.3390/math13193084 - 25 Sep 2025

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Abstract

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and [...] Read more.

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and boundary value constraints. By integrating the LFDEs and approximating the solutions with modified fractional-order Bernstein polynomials, we derive operational matrices to solve the resulting system numerically. The method’s accuracy is validated through several examples, showing excellent agreement between numerical and exact solutions. Comparative analysis with existing data further confirms the reliability of the approach, with absolute errors ranging from 10⁻¹⁸ to 10⁻⁴. The results highlight the method’s efficiency and versatility in modeling complex systems governed by fractional dynamics. This work offers a computationally efficient and accurate tool for fractional calculus applications in science and engineering, helping to bridge existing gaps in numerical techniques. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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15 pages, 284 KB

Open AccessArticle

Existence and Stability Analysis of Anti-Periodic Boundary Value Problems with Generalized Tempered Fractional Derivatives

by Ricardo Almeida and Natália Martins

Mathematics 2025, 13(19), 3077; https://doi.org/10.3390/math13193077 - 24 Sep 2025

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Abstract

In this study, we investigate implicit fractional differential equations subject to anti-periodic boundary conditions. The fractional operator incorporates two distinct generalizations: the Caputo tempered fractional derivative and the Caputo fractional derivative with respect to a smooth function. We investigate the existence and uniqueness [...] Read more.

In this study, we investigate implicit fractional differential equations subject to anti-periodic boundary conditions. The fractional operator incorporates two distinct generalizations: the Caputo tempered fractional derivative and the Caputo fractional derivative with respect to a smooth function. We investigate the existence and uniqueness of solutions using fixed-point theorems. Stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias is also considered. Three detailed examples are presented to illustrate the applicability and scope of the theoretical results. Several existing results in the literature can be recovered as particular cases of the framework developed in this work. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications: 3rd Edition)

15 pages, 298 KB

Open AccessArticle

Solvability for Two-Point Boundary Value Problems for Nonlinear Variable-Order Fractional Differential Systems

by Yige Zhao and Rian Yan

Fractal Fract. 2025, 9(9), 615; https://doi.org/10.3390/fractalfract9090615 - 22 Sep 2025

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Abstract

A class of boundary value problems for fractional differential systems involving variable-order derivatives is considered. Such problems can be transformed into some boundary value problems for nonlinear Caputo fractional differential systems. Here, the relations between linear Caputo fractional differential equations and their corresponding [...] Read more.

A class of boundary value problems for fractional differential systems involving variable-order derivatives is considered. Such problems can be transformed into some boundary value problems for nonlinear Caputo fractional differential systems. Here, the relations between linear Caputo fractional differential equations and their corresponding linear integral equations are investigated, and the results demonstrate that a proper Lipschitz-type condition is needed for studying nonlinear Caputo fractional differential equations. Then, an existence and uniqueness result is established in some vector subspaces by Banach’s fixed-point theorem and

{∥ \cdot ∥}_{e}

norm. In addition, two examples are presented to illustrate the theoretical conclusions. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 3rd Edition)

16 pages, 984 KB

Open AccessArticle

The Effects of Shear Stress Memory and Variable Viscosity on Viscous Fluids Flowing Between Two Horizontal Parallel Plates

by Dumitru Vieru, Constantin Fetecau and Zulkhibri Ismail

Mathematics 2025, 13(18), 3043; https://doi.org/10.3390/math13183043 - 21 Sep 2025

Viewed by 267

Abstract

This article investigates a mathematical model with the Caputo derivative for the transient unidirectional flow of an incompressible viscous fluid with pressure-dependent viscosity. The fluid flows in the spatial domain bounded by two parallel plates extended to infinity. The plates translate in their [...] Read more.

This article investigates a mathematical model with the Caputo derivative for the transient unidirectional flow of an incompressible viscous fluid with pressure-dependent viscosity. The fluid flows in the spatial domain bounded by two parallel plates extended to infinity. The plates translate in their planes with time-dependent velocities, and the fluid adheres to the solid boundaries. The generalization of the model consists of formulating a fractional constitutive equation to introduce the memory effect into the mathematical model. In addition, the fluid’s viscosity is assumed to be pressure-dependent. More precisely, in this article, the viscosity is considered a power function of the vertical coordinate of the channel. Analytic solutions of the dimensionless initial and boundary value problems have been determined using the Laplace transform and Bessel equations. The inversion of Laplace transforms is conducted using both the methods of complex analysis and the Stehfest numerical algorithm. In addition, we discuss the explicit solution in some meaningful particular cases. Using numerical simulations and graphical representations, the results of the ordinary model

(α = 1)

are compared with those of the fractional model

(0 < α < 1)

, highlighting the influence of the memory parameter on fluid behavior. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems, 3rd Edition)

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43 pages, 2828 KB

Open AccessArticle

Efficient Hybrid Parallel Scheme for Caputo Time-Fractional PDEs on Multicore Architectures

by Mudassir Shams and Bruno Carpentieri

Fractal Fract. 2025, 9(9), 607; https://doi.org/10.3390/fractalfract9090607 - 19 Sep 2025

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Abstract

We present a hybrid parallel scheme for efficiently solving Caputo time-fractional partial differential equations (CTFPDEs) with integer-order spatial derivatives on multicore CPU and GPU platforms. The approach combines a second-order spatial discretization with the

L 1

time-stepping scheme and employs MATLAB parfor parallelization [...] Read more.

We present a hybrid parallel scheme for efficiently solving Caputo time-fractional partial differential equations (CTFPDEs) with integer-order spatial derivatives on multicore CPU and GPU platforms. The approach combines a second-order spatial discretization with the

L 1

time-stepping scheme and employs MATLAB parfor parallelization to achieve significant reductions in runtime and memory usage. A theoretical third-order convergence rate is established under smooth-solution assumptions, and the analysis also accounts for the loss of accuracy near the initial time

t = t_{0}

caused by weak singularities inherent in time-fractional models. Unlike many existing approaches that rely on locally convergent strategies, the proposed method ensures global convergence even for distant or randomly chosen initial guesses. Benchmark problems from fractional biological models—including glucose–insulin regulation, tumor growth under chemotherapy, and drug diffusion in tissue—are used to validate the robustness and reliability of the scheme. Numerical experiments confirm near-linear speedup on up to four CPU cores and show that the method outperforms conventional techniques in terms of convergence rate, residual error, iteration count, and efficiency. These results demonstrate the method’s suitability for large-scale CTFPDE simulations in scientific and engineering applications. Full article

(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs, Second Edition)

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26 pages, 556 KB

Open AccessArticle

Refined Error Estimates for Milne–Mercer-Type Inequalities for Three-Times-Differentiable Functions with Error Analysis and Their Applications

by Arslan Munir, Shumin Li, Hüseyin Budak, Artion Kashuri and Loredana Ciurdariu

Fractal Fract. 2025, 9(9), 606; https://doi.org/10.3390/fractalfract9090606 - 18 Sep 2025

Viewed by 313

Abstract

In this study, we examine the error bounds related to Milne-type inequalities and a widely recognized Newton–Cotes method, originally developed for three-times-differentiable convex functions within the context of Jensen–Mercer inequalities. Expanding on this foundation, we explore Milne–Mercer-type inequalities and their application to a [...] Read more.

In this study, we examine the error bounds related to Milne-type inequalities and a widely recognized Newton–Cotes method, originally developed for three-times-differentiable convex functions within the context of Jensen–Mercer inequalities. Expanding on this foundation, we explore Milne–Mercer-type inequalities and their application to a more refined class of three-times-differentiable s-convex functions. This work introduces a new identity involving such functions and Jensen–Mercer inequalities, which is then used to improve the error bounds for Milne-type inequalities in both Jensen–Mercer and classical calculus frameworks. Our research highlights the importance of convexity principles and incorporates the power mean inequality to derive novel inequalities. Furthermore, we provide a new lemma using Caputo–Fabrizio fractional integral operators and apply it to derive several results of Milne–Mercer-type inequalities pertaining to

(α, m)

-convex functions. Additionally, we extend our findings to various classes of functions, including bounded and Lipschitzian functions, and explore their applications to special means, the q-digamma function, the modified Bessel function, and quadrature formulas. We also provide clear mathematical examples to demonstrate the effectiveness of the newly derived bounds for Milne–Mercer-type inequalities. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)

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21 pages, 357 KB

Open AccessArticle

A New Study on the Approximate Controllability of Sobolev-Type Stochastic ABC-Fractional Impulsive Differential Inclusions with Clarke Sub-Differential and Poisson Jumps

by Yousef Alnafisah, Hamdy M. Ahmed and A. M. Sayed Ahmed

Fractal Fract. 2025, 9(9), 605; https://doi.org/10.3390/fractalfract9090605 - 18 Sep 2025

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Abstract

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape [...] Read more.

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape is further enriched by the incorporation of Clarke sub-differentials, facilitating the treatment of nonsmooth, nonconvex, and multivalued dynamics. The inherent complexity arising from the confluence of fractional memory, stochastic perturbations, and impulsive phenomena necessitates the deployment of a sophisticated apparatus from variational analysis, measurable selection theory, and multivalued fixed point frameworks within infinite-dimensional Banach spaces. This study delineates rigorous sufficient conditions, ensuring controllability under such hybrid influences, thereby generalizing classical paradigms to encompass nonlocal and discontinuous dynamical regimes. A precisely articulated exemplar is included to validate the theoretical constructs and demonstrate the operational efficacy of the proposed analytical methodology. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

12 pages, 442 KB

Open AccessArticle

Fundamental Solutions to Fractional Heat Conduction in Two Joint Half-Lines Under Conditions of Nonperfect Thermal Contact

by Yuriy Povstenko, Tamara Kyrylych, Viktor Dashkiiev and Andrzej Yatsko

Entropy 2025, 27(9), 965; https://doi.org/10.3390/e27090965 - 16 Sep 2025

Viewed by 280

Abstract

At the interface dividing two media, an area appears that has its own physical characteristics which differ from the properties of the bulk materials. The small width of the interface area permits considering this area as a two-dimensional median surface with the specified [...] Read more.

At the interface dividing two media, an area appears that has its own physical characteristics which differ from the properties of the bulk materials. The small width of the interface area permits considering this area as a two-dimensional median surface with the specified physical characteristics. The fundamental solutions to the Cauchy problem as well as to the source problem are considered for fractional heat conduction in two joint half-lines under conditions of nonperfect thermal contact. The specific example of classical heat conduction is also investigated. Full article

(This article belongs to the Special Issue Random Walks and Stochastic Processes in Complex Systems: From Physics to Socio-Economic Phenomena, Second Edition)

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22 pages, 570 KB

Open AccessArticle

Stability of Nonlinear Switched Fractional Differential Equations with Short Memory

by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan

Fractal Fract. 2025, 9(9), 598; https://doi.org/10.3390/fractalfract9090598 - 13 Sep 2025

Viewed by 333

Abstract

Nonlinear switched systems, which combine multiple subsystems with a switching rule, have garnered significant research interest due to their complex stability properties. In this paper we consider the case where the switching times, the switching rule and the family of functions defining the [...] Read more.

Nonlinear switched systems, which combine multiple subsystems with a switching rule, have garnered significant research interest due to their complex stability properties. In this paper we consider the case where the switching times, the switching rule and the family of functions defining the subsystems are given initially. Note that the switching rule could be such that it is not activated at any initially given switching time. When the switching rule is activated, then a subsystem from the given family is chosen. We study the case where the nonlinear subsystems consist of fractional differential equations. To be more concise we apply generalized Caputo fractional derivatives with respect to other functions. Lyapunov functions are used to analyze stability, and several sufficient conditions are obtained. The influence of the switching rule on the stability property of the solutions is discussed and illustrated with examples. It is shown that, in spite of the solutions of some of the subsystems being unstable, the zero solution of the switched system could be stable. Full article

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11 pages, 271 KB

Open AccessArticle

Legendre–Clebsch Condition for Functional Involving Fractional Derivatives with a General Analytic Kernel

by Faïçal Ndaïrou

Fractal Fract. 2025, 9(9), 588; https://doi.org/10.3390/fractalfract9090588 - 8 Sep 2025

Viewed by 430

Abstract

Fractional calculus of variations for a broad class of fractional operators with a general analytic kernel function is considered. Using techniques from variational analysis, we derive first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the Weierstrass necessary condition, the Legendre condition, [...] Read more.

Fractional calculus of variations for a broad class of fractional operators with a general analytic kernel function is considered. Using techniques from variational analysis, we derive first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the Weierstrass necessary condition, the Legendre condition, and finally the Legendre–Clebsch condition. Our results are new in the sense that the Euler–Lagrange equation is based on duality theory, and thus build up only with left fractional operators. The Weierstrass necessary condition is a variant of strong necessary optimality condition, and it is derived from maximum condition of Pontryagin for this general analytic kernels. The Legendre–Clebsch condition is obtained under normality assumptions on data because of equality constraints. Full article

(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels, Second Edition)

22 pages, 350 KB

Open AccessArticle

Coupled System of (k, ψ)-Hilfer and (k, ψ)-Caputo Sequential Fractional Differential Equations with Non-Separated Boundary Conditions

by Furkan Erkan, Nuket Aykut Hamal, Sotiris K. Ntouyas, Jessada Tariboon and Phollakrit Wongsantisuk

Axioms 2025, 14(9), 685; https://doi.org/10.3390/axioms14090685 - 7 Sep 2025

Viewed by 436

Abstract

This paper is concerned with the existence and uniqueness of solutions for a coupled system of

(k, ψ)

-Hilfer and

(k, ψ)

-Caputo sequential fractional differential equations with non-separated boundary conditions. We make use of the Banach [...] Read more.

This paper is concerned with the existence and uniqueness of solutions for a coupled system of

(k, ψ)

-Hilfer and

(k, ψ)

-Caputo sequential fractional differential equations with non-separated boundary conditions. We make use of the Banach contraction mapping principle to obtain the uniqueness result, while two existence results are proved by using Leray–Schauder nonlinear alternative and Krasnosel’skiĭ’s fixed point theorem. The obtained results are illustrated by numerical examples. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)

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