Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation

Abstract

In the present paper we compare the two methodologies for the development of exponentially and trigonometrically fitted methods. One is based on the exact integration of the functions of the form: {1,x,x2,…,xp,exp (±wx),xexp (±wx),…,xmexp (±wx)} and the second is based on the exact integration of the functions of the form: {1,x,x2,…,xp,exp (±wx),exp (±2wx),…,exp (±mwx)}. The above functions are used in order to improve the efficiency of the classical methods of any kind (i.e. the method (5) with constant coefficients) for the numerical solution of ordinary differential equations of the form of the Schrodinger equation. We mention here that the above sets of exponential functions are the two most common sets of exponential functions for the development of the special methods for the efficient solution of the Schrodinger equation. It is first time in the literature in which the efficiency of the above sets of functions are studied and compared together for the approximate solution of the Schrodinger equation. We present the error analysis of the above two approaches for the numerical solution of the one-dimensional Schrodinger equation. Finally, numerical results for the resonance problem of the radial Schrodinger equation are presented.