Next Article in Journal
Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations
Previous Article in Journal
Automated First-Arrival Picking and Source Localization of Microseismic Events Using OVMD-WTD and Fractal Box Dimension Analysis
Previous Article in Special Issue
Ulam–Hyers Stability of Fractional Difference Equations with Hilfer Derivatives
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Fractal Fract
Volume 9
Issue 8
10.3390/fractalfract9080540
fractalfract-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

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

by
Sahar Abbas
1,
Abdul Ahad Abro
1,
Syed Muhammad Daniyal
1,*,
Hanaa A. Abdallah
2,
Sadique Ahmad
3,
Abdelhamied Ashraf Ateya
3 and
Noman Bin Zahid
1
1
The Faculty of Engineering, Sciences and Technology (FEST), Iqra University, Main Campus Karachi, Karachi City 75500, Pakistan
2
Department of Information Technology, College of Computer and Information Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3
EIAS Data Science Laboratory, College of Computer and Information Science, Prince Sultan University, Riyadh 11586, Saudi Arabia
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(8), 540; https://doi.org/10.3390/fractalfract9080540 (registering DOI)
Submission received: 4 July 2025 / Revised: 12 August 2025 / Accepted: 13 August 2025 / Published: 16 August 2025
(This article belongs to the Special Issue Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations)
Versions Notes

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.
Keywords: fuzzy fractional calculus; Volterra–Fredholm equations; weighted fractional operators; existence and uniqueness; Ulam–Hyers stability; Adomian decomposition method fuzzy fractional calculus; Volterra–Fredholm equations; weighted fractional operators; existence and uniqueness; Ulam–Hyers stability; Adomian decomposition method

Share and Cite

MDPI and ACS Style

Abbas, S.; Abro, A.A.; Daniyal, S.M.; Abdallah, H.A.; Ahmad, S.; Ateya, A.A.; Zahid, N.B. Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation. Fractal Fract. 2025, 9, 540. https://doi.org/10.3390/fractalfract9080540

AMA Style

Abbas S, Abro AA, Daniyal SM, Abdallah HA, Ahmad S, Ateya AA, Zahid NB. Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation. Fractal and Fractional. 2025; 9(8):540. https://doi.org/10.3390/fractalfract9080540

Chicago/Turabian Style

Abbas, Sahar, Abdul Ahad Abro, Syed Muhammad Daniyal, Hanaa A. Abdallah, Sadique Ahmad, Abdelhamied Ashraf Ateya, and Noman Bin Zahid. 2025. "Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation" Fractal and Fractional 9, no. 8: 540. https://doi.org/10.3390/fractalfract9080540

APA Style

Abbas, S., Abro, A. A., Daniyal, S. M., Abdallah, H. A., Ahmad, S., Ateya, A. A., & Zahid, N. B. (2025). Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation. Fractal and Fractional, 9(8), 540. https://doi.org/10.3390/fractalfract9080540

Article Metrics

Article metric data becomes available approximately 24 hours after publication online.
Back to TopTop