Advances in Numerical Analysis
Volume 2012 (2012), Article ID 626419, 20 pages
http://dx.doi.org/10.1155/2012/626419
Research Article

A Class of Numerical Methods for the Solution of Fourth-Order Ordinary Differential Equations in Polar Coordinates

Jyoti Talwar and R. K. Mohanty

Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, 110007 Delhi, India

Received 16 August 2011; Revised 11 January 2012; Accepted 18 January 2012

Academic Editor: Alfredo Bermudez De Castro

Copyright © 2012 Jyoti Talwar and R. K. Mohanty. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this piece of work using only three grid points, we propose two sets of numerical methods in a coupled manner for the solution of fourth-order ordinary differential equation 𝑢 𝑖 𝑣 ( 𝑥 ) = 𝑓 ( 𝑥 , 𝑢 ( 𝑥 ) , 𝑢 ( 𝑥 ) , 𝑢 ( 𝑥 ) , 𝑢 ( 𝑥 ) ) , 𝑎 < 𝑥 < 𝑏 , subject to boundary conditions 𝑢 ( 𝑎 ) = 𝐴 0 , 𝑢 ( 𝑎 ) = 𝐴 1 , 𝑢 ( 𝑏 ) = 𝐵 0 , and 𝑢 ( 𝑏 ) = 𝐵 1 , where 𝐴 0 , 𝐴 1 , 𝐵 0 , and 𝐵 1 are real constants. We do not require to discretize the boundary conditions. The derivative of the solution is obtained as a byproduct of the discretization procedure. We use block iterative method and tridiagonal solver to obtain the solution in both cases. Convergence analysis is discussed and numerical results are provided to show the accuracy and usefulness of the proposed methods.