The Convergence of Geometric Mesh Cubic Spline Finite Difference Scheme for Nonlinear Higher Order Two-Point Boundary Value Problems

Navnit Jha,Vinod Chauhan,R K Mohanty

doi:10.1155/2014/527924

Abstract

An efficient algorithm for the numerical solution of higher (even) orders two-point nonlinear boundary value problems has been developed. The method is third order accurate and applicable to both singular and nonsingular cases. We have used cubic spline polynomial basis and geometric mesh finite difference technique for the generation of this new scheme. The irreducibility and monotone property of the iteration matrix have been established and the convergence analysis of the proposed method has been discussed. Some numerical experiments have been carried out to demonstrate the computational efficiency in terms of convergence order, maximum absolute errors, and root mean square errors. The numerical results justify the reliability and efficiency of the method in terms of both order and accuracy.

Highlights

Consider the following nonlinear two-point boundary value problems of order 2m, m ∈ Z+:− u(2m) (x) + g (x, u (x), u(1) (x), u(2) (x), . . . , u(2m−1) (x))= 0, −∞ < a ≤ x ≤ b < ∞, u(2l) (a) = al, u(2l) (b) = bl, l = 0 (1) m − 1, (1)where g ∈ C2m+2[a, b], gu > 0, and al, bl are finite real constants and −∞ < a ≤ x ≤ b < ∞.The higher order two-point boundary value problems play an important role in various areas of mathematical physics and engineering
The mathematical modeling of geological folding of rock layers [1], theory of plates and shell [2], waves on a suspension bridge [3], reaction diffusion equation [4], astrophysical narrow convection layers bounded by stable layers [5], viscoelastic and inelastic flows, deformation of beam and plate deflection theory [6,7,8], and so forth are some of the modeling problems in mathematical physics
This paper aims to develop a third order accurate numerical method based on cubic polynomial spline basis and geometric mesh finite difference approximations for the numerical solution of second order two-point boundary value problems

Summary

Introduction

The higher order two-point boundary value problems play an important role in various areas of mathematical physics and engineering. The approximate solution for the second, fourth, and/or sixth order twopoint boundary value problems has been discussed using homotopy analysis by Liang and Jeffrey [13], reproducing kernel space by Yao and Lin [14], spline solution by Siddiqi and Twizell [15], and the Sinc-Galerkin and Sinc-Collocation methods by Rashidinia and Nabati [16]. The geometric mesh technique gains its importance from the theory of electrochemical reaction-convection-diffusion problems in one-dimensional space geometry [18]. The formulation of the geometric mesh finite difference approximations for the two-point singular perturbation problems was developed by Jain et al [19] and Iyengar et al [20]. The applications of geometric mesh in the context of second order two-point boundary value

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Conclusion