Numerov's method for non-linear two-point boundary value problems II. monotone approximations

Abstract

For the numerical solution of non-linear two-point boundary value problems: y″ + f(x y) = 0, 0≧x≧1y(0) = α,y(1) = β, the well-known Numerov's method leads to a non-linear system of finite difference equations for the approximate solution. In [1], Chawla and Shivakumar considered in detail the application of Newton's method for the solution of the resulting non-linear system. But Newton's method requires the setting up of the Jacobian matrix for the non-linear system. As an alternative to the use of Newton's method, in the present paper we present an iteration scheme which provides a monotonically decreasing (or increasing) sequence of approximations which converges to the solution of the Numerov discretization equations. In contrast with Newton's method the present iteration scheme does not require the setting up of the Jacobian matrix of the non-linear system of equations, but instead it requires sup f and inf f over a suitably determined interval [V 0,U 0] containing the true solution of the non-linear discretized system. The present iteration method for solving the Numerov discretization equations is illustrated by considering two non-linear examples; it compares quite favourably with the Newton's solution of the non-linear systems. In the following we assume familiarity with the notation and discussion in Chawla and Shivakumar [1].