An effective approach for numerical solutions of high-order Fredholm integro-differential equations

摘要

We propose an effective method to solve high-order linear Fredholm integro-differential equations having a weak or strong kernel. The target is to construct fast and accurate analytic approximations via an easy, elegant and powerful algorithm based on the power series representation via ordinary polynomials. Employing such polynomials leads to algebraic equations to be solved regarding the treated integro-differential equations. A mathematical proof for the numerical analysis is also provided. In practice, the convergence of the power series solution can be easily pursued via the ratio test. Exact solutions are obtained when the solutions are themselves polynomials. Better accuracies are also achieved within the method by increasing the number of polynomials. The introduced approach is applied to already worked problems in the literature by means of different numerical methods. Comparisons clearly show that our scheme is better and even more superior as compared to the existing ones.