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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.

Application of the Trefftz method to nonlinear potential problems

Uściłowska A.

Computer Assisted Mechanics and Engineering Sciences

2008

Vol. 15, No. 3/4

377-390

In this paper some types of nonlinear potential problems are discussed and some of these problems are solved by the Trefftz method. The attention is paid to Fundamental Solutions Method (FSM) supported by Radial Basis Functions (RBF) approximation. Application of FSM to nonlinear boundary problem requires certain modifications and special algorithms. In this paper two methods of treating the nonlinearity are proposed, One on them is Picard iteration. Due to some problems of application of this method the Homotopj Analysis Method (HAM) is implemented for nonlinear boundary-value problems. The results of numerical experiment arc presented and discussed. The '(inclusion is that the method based on FSM for solving nonlinear boundary-value problem gives result with demanded accuracy.