Picard and Adomian decomposition methods for a fractional quadratic integral equation via generalized fractional integral

Alan Jalal  Abdulqader; Saleh S.  Redhwan; Ali Hasan Ali; Omar  Bazighifan; Awad T.  Alabdala

doi:10.52866/ijcsm.2024.05.03.008

Authors

Alan Jalal Abdulqader Mathematical Department, College of Education, Al-Mustansiriyah University, Iraq https://orcid.org/0000-0002-5758-214X
Saleh S. Redhwan Department of Mathematics, Al-Mahweet University, Yemen https://orcid.org/0000-0002-4779-7657
Ali Hasan Ali University of Debrecen
Omar Bazighifan Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Roma, Italy https://orcid.org/0000-0002-7251-9608
Awad T. Alabdala Management Department - Université Française d’Égypte, Egypt https://orcid.org/0000-0002-3123-2360

DOI:

https://doi.org/10.52866/ijcsm.2024.05.03.008

Keywords:

fractional operator, Fractional differential equations, Monotone operator, fixed point theorems

Abstract

The primary focus of this paper is to thoroughly examine and analyze a class of a fractional quadratic
integral equation via generalized fractional integral. To achieve this, we introduce an operator that possesses
fixed points corresponding to the solutions of the fractional quadratic integral equation, effectively transforming the
given equation into an equivalent fixed-point problem. By applying the Banach fixed-point theorems, we prove the
uniqueness of solutions to fractional quadratic integral equation. Additionally, The Adomian decomposition method
is used, to solve the resulting fractional quadratic integral equation. This technique rapidly provides convergent
successive approximations of the exact solution to the given fractional quadratic integral equation, therefore, we
investigate the convergence of approximate solutions, using the Adomian decomposition method. Finally, we provide
some examples, to demonstrate our results. Our findings contribute to the current understanding of fractional
quadratic integral equation and their solutions and have the potential to inform future research in this area.