%0 Journal Article
%T Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System
%A Navnit Jha
%A R. K. Mohanty
%A Vinod Chauhan
%J Advances in Numerical Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/614508
%X Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method. 1. Introduction Consider the fourth-order boundary value problem: subject to the necessary boundary conditions: where and , , , and are real constants and . Or equivalently subject to the natural boundary conditions: Fourth-order differential equations occur in various areas of mathematics such as viscoelastic and inelastic flows, beam theory, Lifshitz point in phase transition physics (e.g., nematic liquid crystal, crystals, and ferroelectric crystals) [1], the rolls in a Rayleigh-Benard convection cell (two parallel plates of different temperature with a liquid in between) [2], spontaneous pattern formation in second-order materials (e.g., polymeric fibres) [3], the waves on a suspension bridge [4, 5], geological folding of rock layers [6], buckling of a strut on a nonlinear elastic foundation [7], traveling water waves in a shallow channel [8], pulse propagation in optical fibers [9], system of two reaction diffusion equation [10], and so forth. The existence and uniqueness of the solution for the fourth and higher-order boundary value problems have been discussed in [11¨C14]. In the recent past, the numerical solution of fourth-order differential equations has been developed using multiderivative, finite element method, Ritz method, spline collocation, and finite difference method [15¨C18]. The determination of eigen values of self adjoint fourth-order differential equations was developed in [19] using finite difference scheme. The motivation of variable mesh technique for differential equations arises from the theory of electrochemical reaction-convection-diffusion problems in one-dimensional space geometry [20]. The geometric mesh method for self-adjoint singular perturbation problems using finite difference approximations was discussed in [21]. The use of geometric mesh in the context of boundary value problems was studied extensively in [22¨C24]. In this paper, we derive a geometric mesh finite difference method for the solution of fourth- and sixth-order differential boundary value problems with order
%U http://www.hindawi.com/journals/ana/2013/614508/