Analyzing the stability of coupled nonlinear fractional Volterra–Fredholm integro-differential equations with a modern method for numerical solutions

In this study, we examine the coupled nonlinear fractional Volterra–Fredholm integro-differential equations that utilize Caputo fractional derivatives. The primary objective of this research is to explore the existence, uniqueness, stability of solutions, and convergence analysis using the fractional-order biorthogonal flatlet multiwavelet collocation method (FBFMCM) for the specified coupled integro-differential equations. We begin by demonstrating the existence and uniqueness of the solution to the problem through the application of the well-established Krasnoselskii theorem and the Banach contraction principle. Next, we analyze the Ulam–Hyers and Ulam–Hyers–Rassias stability for the problem at hand. Finally, we present the implementation of the proposed method along with a convergence analysis. Additionally, we compute the operational matrix for fractional integration and provide an example to illustrate our main findings.