Solutions of nonlinear integro-differential equations of fractional order defined using fractional boundary conditions

In this paper, we explore the application of fractional nonlocal integral boundary conditions to nonlinear Volterra–Fredholm integro-differential equations with variable coefficients. Utilizing the Adomian decomposition method and the homotopy analysis method, we develop an approximate numerical solution. The effectiveness and reliability of these methods are illustrated through graphical representations. We discuss the essential criteria to ensure the existence and uniqueness of the solution for various kernel modifications. Numerical examples showcase the implementation of our key theorems, validate the accuracy of the proposed methods, and demonstrate the convergence result. Our findings are substantiated using standard fixed point theorems, including the Krasnoselskii fixed point theorem and the Arzelà–Ascoli theorem.