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SUMMARYThe 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.

The paper presents a new approximate method of solving non-Fourier heat conduction problems. The approach described here is suitable for solving both direct and inverse problems. The way of generating Trefftz functions for non-Fourier heat conduction equation has been shown. Obtained functions have been used for solving direct and boundary inverse problems (identification of boundary condition). As a rule, inverse problems are ill-posed. Therefore, each method of solving these problems has to be checked according to disturbance of the input data. Presented examples confirm high usability of the presented approach for solving direct and inverse non-Fourier heat conduction problems.

We demonstrate a specific power series expansion technique to solve the three-dimensional homogeneous and inhomogeneous wave equations. As solving functions, so-called wave polynomials are used. The presented method is useful for a finite body of certain shape. Recurrent formulas to improve efficiency are obtained for the wave polynomials and their derivatives in a Cartesian, spherical, and cylindrical coordinate system. Formulas for a particular solution of the inhomogeneous wave equation are derived. The accuracy of the method is discussed and some typical examples are shown.

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