A Combined Arithmetic Average Discretization and TAGE Iterative Method for Non-linear Two Point Boundary Value Problems with a Source Function in Integral Form

Abstract

In this article, we report the derivation of high accuracy numerical method based on arithmetic average discretization for the solution of $${u^{\prime\prime} = F(x, u, {u}^{\prime})+\int_0^1 {K(x, s)ds}, 0 < x < 1, 0 < s < 1}$$ subject to natural boundary conditions and the application of two parameter alternating group explicit (TAGE) iterative method on a non-uniform mesh. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compare to the method discussed in Mohanty and Dhall (Appl Math Comput 215:2024–2034, 2009). The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of the method is briefly discussed. The advantage of using this new variable mesh strategy is highlighted computationally.