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The usage of Trefftz functions and Picard's iteration for solving different problems of a two-dimensional wave equation

Krzyszkowska P., Maciąg A., Walaszczyk M.

Journal of Applied Mathematics and Computational Mechanics

2014

Vol. 13, nr 3

129--140

The paper presents a method of solving two-dimensional wave equations which describe vibrations of the membrane with variable thickness and with damping. The differential operator is decomposed into two parts. The first one describes vibrations of the membrane with constant thickness without damping. The second contains the rest of the original operator and is treated as inhomogeneity for the first one. Picard’s iterations are used to calculate a successive approximation of the exact solution. Trefftz functions (wave polynomials) are used to solve the problem in each iteration. The presented examples show the usefulness of the method. The approach described in this paper can be used also for solving nonlinear problems for a wave equation.