Picard and Adomian decomposition methods for a fractional quadratic integral equation via ζ-generalized ξ-fractional integral.

The primary focus of this paper is to thoroughly examine and analyze a class of a fractional quadratic integral equation via ζ-generalized ξ-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. © 2024 College of Education, Al-Iraqia University. All rights reserved.