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

A step-by-step modification of the well-known approach proposed by Mikhaylenko and Konyukh to 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 solving the problem on the next time interval. The method is illustrated by examples for the harmonic oscillator problem and the 1D wave equation. Accuracy and stability of the method are analyzed. This approach allows obtaining a solution of high accuracy on large time intervals.