A method of solving evolutionary problems based on the Laguerre step-by-step transform

Abstract

In [1], Mikhailenko proposed a method of solving dynamic problems of elasticity theory. The method is based on the Laguerre transform with respect to time. In this paper, we propose a modification of this approach, applying the Laguerre transform to a sequence of finite time intervals. The solution obtained at the end of one time interval is used as initial data for solving the problem on the next time interval. To implement the approach, four parameters are chosen: a scale factor to approximate the solution by Laguerre functions, an exponential coefficient of a weight function that is used for finding a solution on a finite time interval, the duration of this interval, and the number of projections of the Laguerre transform. A way to find parameters that provide stability of calculations is proposed. The effect of the parameters on the accuracy of calculations when using second-and fourth-order difference schemes is studied. It is shown that the approach makes it possible to obtain a high-accuracy solution on large time intervals.To simulate the dynamics of seismic waves, Mikhailenko [1] proposed to use the integral Laguerre transform in time. The solution u(t) is sought for aswhere the coefficients u m are calculated by the formulaThe Laguerre functions l α m (t) are expressed in terms of the Laguerre polynomials L α m (t):A review of applications of this method was done by Reshetova in her doctoral dissertation [2]. Since the zeros of the Laguerre polynomials L α m (t) (and, hence, the functions l α m (t)) are located on an interval that depends on m, to obtain good approximation of u(t) with increasing t in formula (1) the number of terms must be increasing. To avoid this effect, we propose to approximate with formula (1) only for t on a finite interval 0 ≤ t ≤ τ and use the thus obtained approximate value of u(τ ) as an initial one for the interval τ ≤ t ≤ 2τ , etc. This is an analog of the Euler method for numerical integration of dynamic problems.Since it is assumed in (1) that u(0) = 0, because l α m (0) = 0 at α > 0, and the thus obtained value of u(τ ) may turn out to be nonzero, here we can only take α = 0.A preliminary study of the algorithm was done in paper [3]. *