A step-by-step method with Laguerre functions for solving time-dependent problems

In this paper, a step-by-step modification of a well-known approach of Mikhailenko and Konyukh for solving dynamic problems is proposed. The approach is based on the Laguerre transform with respect to time. In this modification the Laguerre transform is applied to a sequence of finite time intervals. The solution obtained at the end of a time interval is used as the initial data for problem solving on the next time interval. The method is illustrated by the examples of a harmonic oscillator problem and a 1D wave equation. The accuracy and stability of the method are analyzed. This approach allows obtaining a solution of high accuracy on large time intervals.