On an approximate solution of a boundary value problem for a nonlinear integro-differential equation

الملخص EN

The aim of this work is to discuss the solvability of a boundary value problem for a nonlinear integro-differential equation.

First, we derive an equivalent nonlinear Fredholm integral equation (NFIE) to this problem.

Second, we prove the existence of a solution to the NFIE using the Krasnosel’skii fixed point theorem under verifying some sufficient conditions.

Third, we solve the NFIE numerically and study the convergence rate via methods based upon applying the modified Adomian decomposition method and Liao’s homotopy analysis method.

As applications, some examples are illustrated to support our work.

The results in this work refer to both methods are efficient and converge rapidly, but the homotopy analysis method may converge faster when we succeed in choosing the optimal homotopy control parameter.