A new family of exponentially fitted methods

An exponentially-fitted method is developed in this paper. This is a higher-order extension of the dissipative (i.e., nonsymmetric) two-step method first described by Simos and Williams in [1], for the numerical integration of the Schrِdinger equation. An application to the bound-states problem and the resonance problem of the radial Schrِdinger equation indicates that the new method is more efficient than the classical dissipative method and other well-known methods. Based on the new method and the method of Raptis and Allison [2] a new variable-step method is obtained. The application of the new variable-step method to some coupled differential equations arising from the Schrِdinger equation indicates the efficiency of the new approach.