A self-starting linear multistep method for a direct solution of the general second-order initial value problem

Abstract

In this paper, we propose a linear multistep method of order 5 that is self-starting for the direct solution of the general second-order initial value problem (IVP). The method is derived by the interpolation and collocation of the assumed approximate solution and its second derivative at x=x _n+j , j=1, 2, …, r−1, and x=x _n+j , j=1, 2, …, s−1, respectively, where r and s are the number of interpolation and collocation points, respectively. The interpolation and collocation procedures lead to a system of (r+s) equations involving (r+s) unknown coefficients, which are determined by the matrix inversion approach. The resulting coefficients are used to construct the approximate solution from which multiple finite difference methods (MFDMs) are obtained and simultaneously applied to provide a direct solution to IVPs. In particular, the method is implemented without the need for either predictors or starting values from other methods. Numerical examples are given to illustrate the efficiency of the method.

Keywords:

• collocation
• interpolation
• second order
• initial value problem
• matrix inversion
• multiple finite difference methods

AMS Subject Classification :

• 65L05
• 65L06
• 65L12