Explicit and implicit methods for second order ordinary differential equations

Abstract

A family of explicit formulas is developed for solving a system of second order linear ordinary differential equations with constant coefficients and with initial conditions specified. A family of implicit formulas for solving the same system with specified boundary conditions is also developed. Both families are based on Padé approximants to the exponential function and for each formula developed the order of the formula is seen to be one higher than the order of the Padé approximant used. In the case of the family of implicit formulas it is seen that the order of the formula is made arbitrarily high by using an appropriate Padé approximant. It is shown that the families are readily applicable to the numerical solution of second order hyperbolic partial differential equations with constant coefficients. The formulas developed are tested on four problems.