Computer algebra programmes for the construction of a family of Numerov-type exponentially-fitted methods for the numerical solution of the Schrödinger equation

Abstract

In this paper a computer algebra programme, written in the MAPLE (and REDUCE) language(s), is presented for the production of exponentially-fitted methods. By using this programme a family of predictor–corrector exponential Numerov-type methods is obtained for the numerical solution of the coupled equations arising from the Schrödinger equation. The Numerov-type methods considered contain free parameters which allow them to be fitted to exponential functions. These new fourth algebraic order methods are very simple and integrate more exponential functions than both the well-known fourth order Numerov-type exponentially-fitted methods and the sixth algebraic order Runge–Kutta type methods. From the exponentially-fitted methods obtained using this computer algebra programme, a variable-step exponentially-fitted method is constructed. Numerical results indicate that the new variable-step method is much more efficient than other well-known methods for the numerical solution of the coupled equations arising from the Schrödinger equation.