%0 Journal Article
%T A Class of Numerical Methods for the Solution of Fourth-Order Ordinary Differential Equations in Polar Coordinates
%A Jyoti Talwar
%A R. K. Mohanty
%J Advances in Numerical Analysis
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/626419
%X 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 , , subject to boundary conditions , , , and , where , , , and 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. 1. Introduction Consider the fourth-order boundary value problem subject to the prescribed natural boundary conditions or equivalently, for , subject to the natural boundary conditions where , , , and are real constants and . Fourth-order differential equations occur in a number of areas of applied mathematics, such as in beam theory, viscoelastic and inelastic flows, and electric circuits. Some of them describe certain phenomena related to the theory of elastic stability. A classical fourth-order equation arising in the beam-column theory is the following (see Timoshenko [1]): where is the lateral deflection, is the intensity of a distributed lateral load, is the axial compressive force applied to the beam, and represents the flexural rigidity in the plane of bending. Various generalizations of the equation describing the deformation of an elastic beam with different types of two-point boundary conditions have been extensively studied via a broad range of methods. The existence and uniqueness of solutions of boundary value problems are discussed in the papers and book of Agarwal and Krishnamoorthy, Agarwal and Akrivis (see [2¨C5]). Several authors have investigated solving fourth-order boundary value problem by some numerical techniques, which include the cubic spline method, Ritz method, finite difference method, multiderivative methods, and finite element methods (see [6¨C16]). In the 1980s, Usmani et al. (see [17¨C19]) worked on finite difference methods for solving and finite difference methods for computing eigenvalues of fourth-order linear boundary value problem. In 1984, Twizell and Tirmizi (see [20]) developed multi-derivative methods for linear fourth-order boundary value problems. In 1984, Agarwal and Chow (see [21]) developed iterative methods for a fourth-order boundary value problem. In 1991, O¡¯Regan (see [13]) worked on the solvability of some fourth-(and higher) order singular boundary value problems. In 1994, Cabada (see
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