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

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. 1. Introduction Consider the following nonlinear two-point boundary value problems of order : where , , and , are finite real constants and . 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 [6–8], and so forth are some of the modeling problems in mathematical physics. The analytical solution of (1) for the arbitrary choice of is difficult and thus we attempt to develop an economical computational method. The existence and uniqueness of the solutions of higher order boundary value problems have been discussed by Agarwal and Krishnamoorthy [9], O’Regan [10], Aftabizadeh [11], and Wei [12]. In the past, the approximate solution for the second, fourth, and/or sixth order two-point 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 monotone iterative technique and quasilinearization method for the higher order ordinary differential equations have been analysed by Koleva and Vulkov [17]. 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