The usage of wave polynomials in solving direct and inverse problems for two-dimensional wave equation

Abstract

The paper presents a new relatively simple yet very effective method to obtain an approximate solution of the direct and inverse problems for two-dimensional wave equation (two space variables and time). Such a equation describes, for example, the vibration of a membrane. To obtain an approximate solution, the wave polynomials (Trefftz functions for wave equation) were used. It is shown how to get these polynomials and their derivatives. The method of solving the functions is described and it is proved that the approximation error decreases when taking more polynomials in approximation. A new approach for solving 2D direct and inverse problems of elasticity is described. In order to improve the quality of the solution, a physical regularization was proposed. Moreover, the paper shows a new technique of smoothing the noisy data by using wave polynomials. The quality of the approximate solutions was verified on test examples. In these cases the direct and inverse problems were taken into consideration. Copyright © 2009 John Wiley & Sons, Ltd.