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19 pages, 474 KiB

Open AccessArticle

Differential and Integral Equations Involving Multivariate Special Polynomials with Applications in Computer Modeling

by Mohra Zayed, Taghreed Alqurashi, Shahid Ahmad Wani, Dixon Salcedo and Mohammad Esmael Samei

Fractal Fract. 2025, 9(8), 512; https://doi.org/10.3390/fractalfract9080512 - 5 Aug 2025

This work introduces a new family of multivariate hybrid special polynomials, motivated by their growing relevance in mathematical modeling, physics, and engineering. We explore their core properties, including recurrence relations and shift operators, within a unified structural framework. By employing the factorization method, [...] Read more.

This work introduces a new family of multivariate hybrid special polynomials, motivated by their growing relevance in mathematical modeling, physics, and engineering. We explore their core properties, including recurrence relations and shift operators, within a unified structural framework. By employing the factorization method, we derive various governing equations such as differential, partial differential, and integrodifferential equations. Additionally, we establish a related fractional Volterra integral equation, which broadens the theoretical foundation and potential applications of these polynomials. To support the theoretical development, we carry out computational simulations to approximate their roots and visualize the distribution of their zeros, offering practical insights into their analytical behavior. Full article

(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus and Mathematical Modelling)

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18 pages, 319 KiB

Open AccessArticle

On the Existence of Solutions and Ulam-Type Stability for a Nonlinear ψ-Hilfer Fractional-Order Delay Integro-Differential Equation

by Cemil Tunç, Fehaid Salem Alshammari and Fahir Talay Akyıldız

Fractal Fract. 2025, 9(7), 409; https://doi.org/10.3390/fractalfract9070409 - 24 Jun 2025

Viewed by 448

Abstract

In this work, we address a nonlinear

ψ

-Hilfer fractional-order Volterra integro-differential equation that incorporates n-multiple-variable time delays. Employing the

ψ

-Hilfer fractional derivative operator, we investigate the existence of a unique solution, as well as the Ulam–Hyers–Rassias stability, semi-Ulam–Hyers–Rassias stability, and [...] Read more.

In this work, we address a nonlinear

ψ

-Hilfer fractional-order Volterra integro-differential equation that incorporates n-multiple-variable time delays. Employing the

ψ

-Hilfer fractional derivative operator, we investigate the existence of a unique solution, as well as the Ulam–Hyers–Rassias stability, semi-Ulam–Hyers–Rassias stability, and Ulam–Hyers stability of the proposed

ψ

-Hilfer fractional-order Volterra integro-differential equation through the fixed-point approach. In this study, we enhance and generalize existing results in the literature on

ψ

-Hilfer fractional-order Volterra integro-differential equations, both including and excluding single delay, by establishing new findings for nonlinear

ψ

-Hilfer fractional-order Volterra integro-differential equations involving n-multiple-variable time delays. This study provides novel theoretical insights that deepen the qualitative understanding of fractional calculus. Full article

(This article belongs to the Special Issue Advances in Fractional Operators: Mathematical Perspectives and Multidisciplinary Applications)

22 pages, 303 KiB

Open AccessArticle

Remarks on a New Variable-Coefficient Integro-Differential Equation via Inverse Operators

by Chenkuan Li, Nate Fingas and Ying Ying Ou

Fractal Fract. 2025, 9(7), 404; https://doi.org/10.3390/fractalfract9070404 - 23 Jun 2025

Viewed by 1110

Abstract

In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary condition. To demonstrate the applicability [...] Read more.

In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary condition. To demonstrate the applicability of our main theorems, we provide several examples in which we compute values of the two-parameter Mittag–Leffler functions. The proposed approach is particularly effective for addressing a wide range of integral and fractional nonlinear differential equations with initial or boundary conditions—especially those involving variable coefficients, which are typically challenging to treat using classical integral transform methods. Finally, we demonstrate a significant application of the inverse operator approach by solving a Caputo fractional convection partial differential equation in

R^{n}

with an initial condition. Full article

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

25 pages, 5123 KiB

Open AccessArticle

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

by Kamel Al-Khaled, Isam Al-Darabsah, Amer Darweesh and Amro Alshare

Fractal Fract. 2025, 9(6), 392; https://doi.org/10.3390/fractalfract9060392 - 19 Jun 2025

Viewed by 445

Abstract

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 [...] Read more.

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 \in (0, 1)

and the memory-kernel parameter

β \in (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. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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26 pages, 332 KiB

Open AccessArticle

Uniqueness Methods and Stability Analysis for Coupled Fractional Integro-Differential Equations via Fixed Point Theorems on Product Space

by Nan Zhang, Emmanuel Addai and Hui Wang

Axioms 2025, 14(5), 377; https://doi.org/10.3390/axioms14050377 - 16 May 2025

Viewed by 315

Abstract

In this paper, we obtain unique solution and stability results for coupled fractional differential equations with p-Laplacian operator and Riemann–Stieltjes integral conditions that expand and improve the works of some of the literature. In order to obtain the existence and uniqueness of solutions [...] Read more.

In this paper, we obtain unique solution and stability results for coupled fractional differential equations with p-Laplacian operator and Riemann–Stieltjes integral conditions that expand and improve the works of some of the literature. In order to obtain the existence and uniqueness of solutions for coupled systems, several fixed point theorems for operators in ordered product spaces are given without requiring the existence conditions of upper–lower solutions or the compactness and continuity of operators. By applying the conclusions of the operator theorem studied, sufficient conditions for the unique solution of coupled fractional integro-differential equations and approximate iterative sequences for uniformly approximating unique solutions were obtained. In addition, the Hyers–Ulam stability of the coupled system is discussed. As applications, the corresponding results obtained are well demonstrated through some concrete examples. Full article

10 pages, 266 KiB

Open AccessArticle

Ulam–Hyers–Rassias Stability of ψ-Hilfer Volterra Integro-Differential Equations of Fractional Order Containing Multiple Variable Delays

by John R. Graef, Osman Tunç and Cemil Tunç

Fractal Fract. 2025, 9(5), 304; https://doi.org/10.3390/fractalfract9050304 - 6 May 2025

Cited by 2 | Viewed by 453

Abstract

The authors consider a nonlinear

ψ

-Hilfer fractional-order Volterra integro-differential equation (

ψ

-Hilfer FrOVIDE) that incorporates N-multiple variable time delays into the equation. By utilizing the

ψ

-Hilfer fractional derivative, they investigate the Ulam–Hyers–Rassias and Ulam–Hyers stability of the equation by [...] Read more.

The authors consider a nonlinear

ψ

-Hilfer fractional-order Volterra integro-differential equation (

ψ

-Hilfer FrOVIDE) that incorporates N-multiple variable time delays into the equation. By utilizing the

ψ

-Hilfer fractional derivative, they investigate the Ulam–Hyers–Rassias and Ulam–Hyers stability of the equation by using fixed-point methods. Their results improve existing ones both with and without delays by extending them to nonlinear

ψ

-Hilfer FrOVIDEs that incorporate N-multiple variable time delays. Full article

(This article belongs to the Special Issue Advances in Fractional Operators: Mathematical Perspectives and Multidisciplinary Applications)

31 pages, 476 KiB

Open AccessArticle

Strong Convergence of a Modified Euler—Maruyama Method for Mixed Stochastic Fractional Integro—Differential Equations with Local Lipschitz Coefficients

by Zhaoqiang Yang and Chenglong Xu

Fractal Fract. 2025, 9(5), 296; https://doi.org/10.3390/fractalfract9050296 - 1 May 2025

Viewed by 534

Abstract

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using [...] Read more.

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using a fractional calculus technique. Then, we establish the well—posedness of the analytical solutions of the mSVIEs. After that, a modified EM scheme is formulated to approximate the numerical solutions of the mSVIEs, and its strong convergence is proven based on local Lipschitz and linear growth conditions. Furthermore, we derive the modified EM scheme under the same conditions in the

L^{2}

sense, which is consistent with the strong convergence result of the corresponding EM scheme. Notably, the strong convergence order under local Lipschitz conditions is inherently lower than the corresponding order under global Lipschitz conditions. Finally, numerical experiments are presented to demonstrate that our approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the

L^{2}

sense. Full article

(This article belongs to the Section Numerical and Computational Methods)

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14 pages, 336 KiB

Open AccessArticle

The Existence and Stability of Integral Fractional Differential Equations

by Rahman Ullah Khan and Ioan-Lucian Popa

Fractal Fract. 2025, 9(5), 295; https://doi.org/10.3390/fractalfract9050295 - 1 May 2025

Viewed by 628

Abstract

The main goal of this research is to study integro-fractional differential equations and simulate their dynamic behavior using ABC-fractional derivatives. We investigate the Hyers–Ulam stability of the proposed system and further expand the prerequisites for the existence and uniqueness of the solutions. The [...] Read more.

The main goal of this research is to study integro-fractional differential equations and simulate their dynamic behavior using ABC-fractional derivatives. We investigate the Hyers–Ulam stability of the proposed system and further expand the prerequisites for the existence and uniqueness of the solutions. The Schauder fixed-point theorem and the Banach contraction principle are employed to obtain the results. Finally, we present an example to demonstrate the practical application of our theoretical conclusions. Full article

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

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18 pages, 286 KiB

Open AccessFeature PaperArticle

Existence of Mild Solutions for Fractional Integrodifferential Equations with Hilfer Derivatives

by Mian Zhou and Yong Zhou

Mathematics 2025, 13(9), 1369; https://doi.org/10.3390/math13091369 - 22 Apr 2025

Viewed by 263

Abstract

In this paper, we study the existence of solutions for fractional integrodifferential equations with Hilfer derivatives. We establish some new existence theorems for mild solutions by using Schaefer’s fixed-point theorem, a measure of noncompactness, and the resolvent operators associated with almost sectorial operators. [...] Read more.

In this paper, we study the existence of solutions for fractional integrodifferential equations with Hilfer derivatives. We establish some new existence theorems for mild solutions by using Schaefer’s fixed-point theorem, a measure of noncompactness, and the resolvent operators associated with almost sectorial operators. Our results improve and extend many known results in the relevant references by removing some strong assumptions. Furthermore, we propose new nonlocal initial conditions for Hilfer evolution equations and study the existence of mild solutions to nonlocal problems. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

23 pages, 380 KiB

Open AccessArticle

Generalized Grönwall Inequality and Ulam–Hyers Stability in ℒ^p Space for Fractional Stochastic Delay Integro-Differential Equations

by Abdelhamid Mohammed Djaouti and Muhammad Imran Liaqat

Mathematics 2025, 13(8), 1252; https://doi.org/10.3390/math13081252 - 10 Apr 2025

Viewed by 369

Abstract

In this work, we derive novel theoretical results concerning well-posedness and Ulam–Hyers stability. Specifically, we investigate the well-posedness of Caputo–Katugampola fractional stochastic delay integro-differential equations. Additionally, we develop a generalized Grönwall inequality and apply it to prove Ulam–Hyers stability in

L^{p}

space. [...] Read more.

In this work, we derive novel theoretical results concerning well-posedness and Ulam–Hyers stability. Specifically, we investigate the well-posedness of Caputo–Katugampola fractional stochastic delay integro-differential equations. Additionally, we develop a generalized Grönwall inequality and apply it to prove Ulam–Hyers stability in

L^{p}

space. Our findings generalize existing results for fractional derivatives and space, as we formulate them in the Caputo–Katugampola fractional derivative and

L^{p}

space. To support our theoretical results, we present an illustrative example. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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24 pages, 6901 KiB

Open AccessArticle

A Suitable Algorithm to Solve a Nonlinear Fractional Integro-Differential Equation with Extended Singular Kernel in (2+1) Dimensions

by Sameeha Ali Raad and Mohamed Abdella Abdou

Fractal Fract. 2025, 9(4), 239; https://doi.org/10.3390/fractalfract9040239 - 10 Apr 2025

Viewed by 377

Abstract

In this paper, the authors consider a problem with comprehensive properties in terms of form and content in the space

L_{2} [(a, b) \times (c, d)] \times C [0, T], T < 1

. In terms of time [...] Read more.

In this paper, the authors consider a problem with comprehensive properties in terms of form and content in the space

L_{2} [(a, b) \times (c, d)] \times C [0, T], T < 1

. In terms of time form, we assume that the time phase delay is implicitly contained in a nonlinear differential integral equation. The positional part is considered in two dimensions, and the position’s kernel is a general singular kernel, many different forms of which will be derived. In terms of content, all of the previously established numerical techniques are only appropriate for studying special cases of the kernel separately but are not suitable for studying the general kernel. This led to the use of the Toeplitz matrix method, which deals with the kernel in its extended nonlinear form and the special kernels will be studied as applications of the method. Moreover, this method has the advantage of converting all single integrals into regular integrals that can be easily solved. Additionally, the researchers examine the solution’s existence, uniqueness, and convergence in this paper. The error and its stability are also studied. At the end of the research, the authors studied some numerical applications of some of the singular kernels derived from the general kernel, examining the approximation error in each application separately. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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20 pages, 391 KiB

Open AccessArticle

Stability Analysis of a Fractional Epidemic Model Involving the Vaccination Effect

by Sümeyye Çakan

Fractal Fract. 2025, 9(4), 206; https://doi.org/10.3390/fractalfract9040206 - 27 Mar 2025

Viewed by 421

Abstract

This paper, by constructing a fractional epidemic model, analyzes the transmission dynamics of some infectious diseases under the effect of vaccination, which is one of the most effective and common control measures. In the model, considering that antibody formation by vaccination may not [...] Read more.

This paper, by constructing a fractional epidemic model, analyzes the transmission dynamics of some infectious diseases under the effect of vaccination, which is one of the most effective and common control measures. In the model, considering that antibody formation by vaccination may not cause permanent immunity, it has been taken into account that the protection period provided by the vaccine may be finite, in addition to the fact that this period may change according to individuals. The model differs from other

S V I R

models given in the literature in its progressive process with a distributed delay in the loss of the protective effect provided by the vaccine. To explain this process, the model was constructed by using a system of distributed delay nonlinear fractional integro-differential equations. Thus, the model aims to present a realistic approach to following the course of the disease. Additionally, an analysis was conducted regarding the minimum vaccination ratio of new members required for the elimination of the disease in the population by using the vaccine free basic reproduction number (

R_{0}^{v f}

). After providing examples for the selection of the distribution function, the variation of

R_{0}

was simulated for a specific selection of parameters in the model. Finally, the sensitivity indices of the parameters affecting

R_{0}

were calculated, and this situation is been visually supported. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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39 pages, 391 KiB

Open AccessArticle

Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations

by Chenkuan Li and Wenyuan Liao

Fractal Fract. 2025, 9(4), 200; https://doi.org/10.3390/fractalfract9040200 - 25 Mar 2025

Cited by 3 | Viewed by 390

Abstract

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in

R^{n}

with three-point conditions and variable coefficients in a Banach space using inverse operators containing multi-variable functions, a [...] Read more.

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in

R^{n}

with three-point conditions and variable coefficients in a Banach space using inverse operators containing multi-variable functions, a generalized Mittag-Leffler function, as well as a few popular fixed-point theorems. These studies have good applications in general since uniqueness, existence and stability are key and important topics in many fields. Several examples are presented to demonstrate applications of results obtained by computing approximate values of the generalized Mittag-Leffler functions. (ii) We use the inverse operator method and newly established spaces to find analytic solutions to a number of notable partial differential equations, such as a multi-term time-fractional convection problem and a generalized time-fractional diffusion-wave equation in

R^{n}

with initial conditions only, which have never been previously considered according to the best of our knowledge. In particular, we deduce the uniform solution to the non-homogeneous wave equation in n dimensions for all

n \geq 1

, which coincides with classical results such as d’Alembert and Kirchoff’s formulas but is much easier in the computation of finding solutions without any complicated integrals on balls or spheres. Full article

(This article belongs to the Special Issue Contemporary Methods of Fractional Order Differential and Differential-Operator Equations)

19 pages, 363 KiB

Open AccessArticle

Tempered Riemann–Liouville Fractional Operators: Stability Analysis and Their Role in Kinetic Equations

by Muhammad Umer, Muhammad Samraiz, Muath Awadalla and Meraa Arab

Fractal Fract. 2025, 9(3), 187; https://doi.org/10.3390/fractalfract9030187 - 18 Mar 2025

Viewed by 365

Abstract

Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial [...] Read more.

Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial fractional derivative is also defined, and its properties are discussed. The Laplace transforms (LTs) of the introduced fractional operators are evaluated. The Hyers–Ulam stability and the existence of a novel tempered fractional differential equation are examined. Moreover, a fractional integro-differential kinetic equation is formulated, and the LT is used to find its solution. A growth model and its graphical representation are established, highlighting the role of novel fractional operators in modeling complex dynamical systems. The developed mathematical framework offers valuable insights into solving a range of scenarios in mathematical physics. Full article

(This article belongs to the Special Issue General Fractional Calculus: Theory, Methods and Applications in Mathematical Physics)

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28 pages, 3393 KiB

Open AccessArticle

An Improved Numerical Scheme for 2D Nonlinear Time-Dependent Partial Integro-Differential Equations with Multi-Term Fractional Integral Items

by Fan Ouyang, Hongyan Liu and Yanying Ma

Fractal Fract. 2025, 9(3), 167; https://doi.org/10.3390/fractalfract9030167 - 11 Mar 2025

Viewed by 699

Abstract

This paper is dedicated to investigating a highly accurate numerical solution for a class of 2D nonlinear time-dependent partial integro-differential equations with multi-term fractional integral items. These integrals are weakly singular with respect to time, which are handled using the product integration rule [...] Read more.

This paper is dedicated to investigating a highly accurate numerical solution for a class of 2D nonlinear time-dependent partial integro-differential equations with multi-term fractional integral items. These integrals are weakly singular with respect to time, which are handled using the product integration rule on graded meshes to compensate for the influence generated by the initial weak singular nature of the exact solution. The temporal derivative is approximated by a generalized Crank–Nicolson difference scheme, while the nonlinear term is approximated by a linearized method. Furthermore, the stability and convergence of the derived time semi-discretization scheme are strictly proved by revising the finite discrete parameters. Meanwhile, the differential matrices of the spatial high-order derivatives based on barycentric rational interpolation are utilized to obtain the fully discrete scheme. Finally, the effectiveness and reliability of the proposed method are validated by means of several numerical experiments. 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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