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29 pages, 3058 KB

Open AccessArticle

Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation

by Sahar Abbas, Abdul Ahad Abro, Syed Muhammad Daniyal, Hanaa A. Abdallah, Sadique Ahmad, Abdelhamied Ashraf Ateya and Noman Bin Zahid

Fractal Fract. 2025, 9(8), 540; https://doi.org/10.3390/fractalfract9080540 - 16 Aug 2025

Cited by 1 | Viewed by 1062

Abstract

This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of solutions. Stability is analyzed in the Ulam–Hyers (UHS), generalized Ulam–Hyers (GUHS), and Ulam–Hyers–Rassias (UHRS) senses. A modified Adomian decomposition method (MADM) is introduced to derive explicit solutions without linearization, preserving the problem’s original structure. The first numerical example validates the theoretical findings on existence, uniqueness, and stability, supplemented by graphical results obtained via the MADM. Further examples illustrate fuzzy solutions by varying the uncertainty level (r), the variable (x), and both parameters simultaneously. The numerical results align with the theoretical analysis, demonstrating the efficacy and applicability of the proposed method. Full article

(This article belongs to the Special Issue Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations)

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27 pages, 665 KB

Open AccessArticle

Study of Stability and Simulation for Nonlinear (k, ψ)-Fractional Differential Coupled Laplacian Equations with Multi-Point Mixed (k, ψ)-Derivative and Symmetric Integral Boundary Conditions

by Xiaojun Lv and Kaihong Zhao

Symmetry 2025, 17(3), 472; https://doi.org/10.3390/sym17030472 - 20 Mar 2025

Cited by 2 | Viewed by 639

Abstract

The

(k, ψ)

-fractional derivative based on the k-gamma function is a more general version of the Hilfer fractional derivative. It is widely used in differential equations to describe physical phenomena, population dynamics, and biological genetic memory problems. In [...] Read more.

The

(k, ψ)

4 m + 2

-point symmetric integral boundary value problem of nonlinear

(k, ψ)

-fractional differential coupled Laplacian equations. The existence and uniqueness of solutions are obtained by the Krasnosel’skii fixed-point theorem and Banach’s contraction mapping principle. Furthermore, we also apply the calculus inequality techniques to discuss the stability of this system. Finally, three interesting examples and numerical simulations are given to further verify the correctness and effectiveness of the conclusions. Full article

(This article belongs to the Special Issue Symmetry in Fractional Derivatives, Fractional Equations and Fractional Order Systems)

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15 pages, 291 KB

Open AccessArticle

Ulam–Hyers and Generalized Ulam–Hyers Stability of Fractional Differential Equations with Deviating Arguments

by Natalia Dilna, Gusztáv Fekete, Martina Langerová and Balázs Tóth

Mathematics 2024, 12(21), 3418; https://doi.org/10.3390/math12213418 - 31 Oct 2024

Cited by 2 | Viewed by 1566

Abstract

In this paper, we study the initial value problem for the fractional differential equation with multiple deviating arguments. By using Krasnoselskii’s fixed point theorem, the conditions of solvability of the problem are obtained. Furthermore, we establish Ulam–Hyers and generalized Ulam–Hyers stability of the fractional functional differential problem. Finally, two examples are presented to illustrate our results, one is with a pantograph-type equation and the other is numerical. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

13 pages, 7840 KB

Open AccessArticle

Stability and Numerical Simulation of a Nonlinear Hadamard Fractional Coupling Laplacian System with Symmetric Periodic Boundary Conditions

by Xiaojun Lv, Kaihong Zhao and Haiping Xie

Symmetry 2024, 16(6), 774; https://doi.org/10.3390/sym16060774 - 20 Jun 2024

Cited by 5 | Viewed by 1622

Abstract

The Hadamard fractional derivative and integral are important parts of fractional calculus which have been widely used in engineering, biology, neural networks, control theory, and so on. In addition, the periodic boundary conditions are an important class of symmetric two-point boundary conditions for [...] Read more.

(p_{1}, p_{2})

-Laplacian systems with periodic boundary value conditions. Based on nonlinear analysis methods and the contraction mapping principle, we obtain some new and easily verifiable sufficient criteria for the existence and uniqueness of solutions to this system. Moreover, we further discuss the generalized Ulam–Hyers (GUH) stability of this problem by using some inequality techniques. Finally, three examples and simulations explain the correctness and availability of our main results. Full article

(This article belongs to the Special Issue Asymmetric and Symmetric Study on Applied Mathematics in ODE, PDE and FDE Model)

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13 pages, 353 KB

Open AccessEditor’s ChoiceArticle

Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with

p

-Laplacian

by Kaihong Zhao

Axioms 2023, 12(8), 733; https://doi.org/10.3390/axioms12080733 - 27 Jul 2023

Cited by 20 | Viewed by 2108

Abstract

The fractional order

p

-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary [...] Read more.

The fractional order

p

p

-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space. Full article

(This article belongs to the Special Issue Differential Equations in Applied Mathematics)

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16 pages, 356 KB

Open AccessArticle

On Fractional Langevin Equations with Stieltjes Integral Conditions

by Binlin Zhang, Rafia Majeed and Mehboob Alam

Mathematics 2022, 10(20), 3877; https://doi.org/10.3390/math10203877 - 19 Oct 2022

Cited by 6 | Viewed by 1914

Abstract

In this paper, we focus on the study of the implicit

FDE

involving Stieltjes integral boundary conditions. We first exploit some sufficient conditions to guarantee the existence and uniqueness of solutions for the above problems based on the Banach contraction principle and Schaefer’s [...] Read more.

In this paper, we focus on the study of the implicit

FDE

UHS

GUHS

UHRS

, and

GUHRS

by employing the classical techniques. In the end, the main results are demonstrated by two examples. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

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