High Accuracy Arithmetic Average Discretization for Non-Linear Two Point Boundary Value Problems with a Source Function in Integral Form

Ranjan K. Mohanty, Deepika Dhall

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DOI: 10.4236/am.2011.210173                

Abstract

In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Uⁿ=F(x,u,u´)+∫K(x,s)ds , 0 <x < 1, 0 < s < 1 subject to natural boundary conditions on a non-uniform mesh. 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 a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.

Keywords

Variable Mesh, Arithmetic Average Discretization, Non-Linear Integro-Differential Equation, Diffusion Equation, Simpson’s 1/3 Rd Rule, Singular Coefficients, Burgers’ Equation, Maximum Absolute Errors

Conflicts of Interest

The authors declare no conflicts of interest.

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