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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 406

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)

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 420

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 310

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 444

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 514

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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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 692

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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12 pages, 269 KiB

Open AccessArticle

Four Different Ulam-Type Stability for Implicit Second-Order Fractional Integro-Differential Equation with M-Point Boundary Conditions

by Ilhem Nasrallah, Rabiaa Aouafi and Said Kouachi

Mathematics 2025, 13(1), 157; https://doi.org/10.3390/math13010157 - 3 Jan 2025

Viewed by 763

Abstract

In this paper, we discuss the existence and uniqueness of a solution for the implicit two-order fractional integro-differential equation with m-point boundary conditions by applying the Banach fixed point theorem. Moreover, in the paper we establish the four different varieties of Ulam stability [...] Read more.

In this paper, we discuss the existence and uniqueness of a solution for the implicit two-order fractional integro-differential equation with m-point boundary conditions by applying the Banach fixed point theorem. Moreover, in the paper we establish the four different varieties of Ulam stability (Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam-Rassias stability, and generalized Hyers–Ulam–Rassias stability) for the given problem. Full article

17 pages, 949 KiB

Open AccessArticle

Adaptive Control for Multi-Agent Systems Governed by Fractional-Order Space-Varying Partial Integro-Differential Equations

by Zhen Liu, Yingying Wen, Bin Zhao and Chengdong Yang

Mathematics 2025, 13(1), 112; https://doi.org/10.3390/math13010112 - 30 Dec 2024

Viewed by 787

Abstract

This paper investigates a class of multi-agent systems (MASs) governed by nonlinear fractional-order space-varying partial integro-differential equations (SVPIDEs), which incorporate both nonlinear state terms and integro terms. Firstly, a distributed adaptive control protocol is developed for leaderless fractional-order SVPIDE-based MASs, aiming to achieve [...] Read more.

This paper investigates a class of multi-agent systems (MASs) governed by nonlinear fractional-order space-varying partial integro-differential equations (SVPIDEs), which incorporate both nonlinear state terms and integro terms. Firstly, a distributed adaptive control protocol is developed for leaderless fractional-order SVPIDE-based MASs, aiming to achieve consensus among all agents without a leader. Then, for leader-following fractional-order SVPIDE-based MASs, the protocol is extended to account for communication between the leader and follower agents, ensuring that the followers reach consensus with the leader. Finally, three examples are presented to illustrate the effectiveness of the proposed distributed adaptive control protocols. Full article

(This article belongs to the Special Issue Dynamic Modeling and Simulation for Control Systems, 3rd Edition)

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19 pages, 375 KiB

Open AccessArticle

ADI Compact Difference Scheme for the Two-Dimensional Integro-Differential Equation with Two Fractional Riemann–Liouville Integral Kernels

by Ziyi Chen, Haixiang Zhang and Hu Chen

Fractal Fract. 2024, 8(12), 707; https://doi.org/10.3390/fractalfract8120707 - 29 Nov 2024

Cited by 8 | Viewed by 1015

Abstract

In this paper, a numerical method of a two-dimensional (2D) integro-differential equation with two fractional Riemann–Liouville (R-L) integral kernels is investigated. The compact difference method is employed in the spatial direction. The integral terms are approximated by a second-order convolution quadrature formula. The [...] Read more.

In this paper, a numerical method of a two-dimensional (2D) integro-differential equation with two fractional Riemann–Liouville (R-L) integral kernels is investigated. The compact difference method is employed in the spatial direction. The integral terms are approximated by a second-order convolution quadrature formula. The alternating direction implicit (ADI) compact difference scheme reduces the CPU time for two-dimensional problems. Simultaneously, the stability and convergence of the proposed ADI compact difference scheme are demonstrated. Finally, two numerical examples are provided to verify the established ADI compact difference scheme. Full article

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13 pages, 309 KiB

Open AccessArticle

On One Point Singular Nonlinear Initial Boundary Value Problem for a Fractional Integro-Differential Equation via Fixed Point Theory

by Said Mesloub, Eman Alhazzani and Hassan Eltayeb Gadain

Fractal Fract. 2024, 8(9), 526; https://doi.org/10.3390/fractalfract8090526 - 10 Sep 2024

Viewed by 1167

Abstract

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular [...] Read more.

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order

θ \in [0, 1]

. The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented. Full article

(This article belongs to the Special Issue Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods and Application)

12 pages, 292 KiB

Open AccessArticle

The Ulam Stability of High-Order Variable-Order φ-Hilfer Fractional Implicit Integro-Differential Equations

by Peiguang Wang, Bing Han and Junyan Bao

Fractal Fract. 2024, 8(9), 502; https://doi.org/10.3390/fractalfract8090502 - 26 Aug 2024

Viewed by 1026

Abstract

This study investigates the initial value problem of high-order variable-order

φ

-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional [...] Read more.

This study investigates the initial value problem of high-order variable-order

φ

-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions. Full article

(This article belongs to the Section General Mathematics, Analysis)

26 pages, 361 KiB

Open AccessArticle

Solutions of Second-Order Nonlinear Implicit ψ-Conformable Fractional Integro-Differential Equations with Nonlocal Fractional Integral Boundary Conditions in Banach Algebra

by Yahia Awad and Yousuf Alkhezi

Symmetry 2024, 16(9), 1097; https://doi.org/10.3390/sym16091097 - 23 Aug 2024

Cited by 2 | Viewed by 924

Abstract

In this paper, we introduce and thoroughly examine new generalized

ψ

-conformable fractional integral and derivative operators associated with the auxiliary function

ψ (t)

. We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, [...] Read more.

In this paper, we introduce and thoroughly examine new generalized

ψ

-conformable fractional integral and derivative operators associated with the auxiliary function

ψ (t)

. We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit

ψ

-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications. Full article

(This article belongs to the Special Issue New Trends on the Mathematical Models and Solitons Arising in Real-World Problems, 2nd Edition)

17 pages, 719 KiB

Open AccessArticle

Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions

by Xindong Zhang, Ziyang Luo, Quan Tang, Leilei Wei and Juan Liu

Fractal Fract. 2024, 8(8), 495; https://doi.org/10.3390/fractalfract8080495 - 22 Aug 2024

Cited by 1 | Viewed by 1017

Abstract

In this investigation, a new algorithm based on the compact difference method is proposed. The purpose of this investigation is to solve the 2D time-fractional integro-differential equation. The Riemann–Liouville derivative was utilized to define the time-fractional derivative. Meanwhile, the weighted and shifted Grünwald [...] Read more.

In this investigation, a new algorithm based on the compact difference method is proposed. The purpose of this investigation is to solve the 2D time-fractional integro-differential equation. The Riemann–Liouville derivative was utilized to define the time-fractional derivative. Meanwhile, the weighted and shifted Grünwald difference operator and product trapezoidal formula were utilized to construct a high-order numerical scheme. Also, we analyzed the stability and convergence. The convergence order was

O (τ^{2} + h_{x}^{4} + h_{y}^{4})

, where

τ

is the time step size,

h_{x}

and

h_{y}

are the spatial step sizes. Furthermore, several examples were provided to verify the correctness of our theoretical reasoning. Full article

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

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14 pages, 292 KiB

Open AccessArticle

Fractional Neutral Integro-Differential Equations with Nonlocal Initial Conditions

by Zhiyuan Yuan, Luyao Wang, Wenchang He, Ning Cai and Jia Mu

Mathematics 2024, 12(12), 1877; https://doi.org/10.3390/math12121877 - 16 Jun 2024

Viewed by 1144

Abstract

We primarily investigate the existence of solutions for fractional neutral integro-differential equations with nonlocal initial conditions, which are crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, [...] Read more.

We primarily investigate the existence of solutions for fractional neutral integro-differential equations with nonlocal initial conditions, which are crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by using fixed-point theorems, we establish the existence of mild solutions under more relaxed conditions. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Control: Challenges and Innovations)

20 pages, 414 KiB

Open AccessArticle

Contributions to the Numerical Solutions of a Caputo Fractional Differential and Integro-Differential System

by Abdelkader Moumen, Abdelaziz Mennouni and Mohamed Bouye

Fractal Fract. 2024, 8(4), 201; https://doi.org/10.3390/fractalfract8040201 - 29 Mar 2024

Cited by 1 | Viewed by 1309

Abstract

The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, [...] Read more.

The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, electrostatics, control theories, and the natural sciences. An effective approach solves the problem by reducing it to a pair of algebraically separated equations via a successful transformation. The proposed strategy uses first-order shifted Chebyshev polynomials and a projection method. Using the provided technique, the primary system is converted into a set of algebraic equations that can be solved effectively. Some theorems are proved and used to obtain the upper error bound for this method. Furthermore, various examples are provided to demonstrate the efficiency of the proposed algorithm when compared to existing approaches in the literature. Finally, the key conclusions are given. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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