The article presents an approximate method of solving direct and inverse problems described by Bernoulli–Euler inhomogeneous equation of vibrations of a beam. A semianalytical solution is approximated by a linear combination of the Trefftz functions (T-functions, solving functions), which satisfies identically the homogenous equation describing the vibrations of a beam. In the paper, the properties of the solving functions have been investigated, theorems concerning their linear independence have been formulated and proved. A method of obtaining the particular solution of the inhomogeneous equation has been shown. To get this solution, recurrent formulas enabling us to determine the inverse operator for monomials have been derived. The paper discusses two kinds of inverse problems. The first one is a boundary inverse problem, in which the boundary conditions are to be determined, based on known displacements within the area. In the second one, the load on the beam needs to be found (identification of the source). The solving functions can be used as a finite element method base functions. This approach is tested for solving inverse problems. The paper includes examples which illustrate the usefulness of the method.

The Trefftz functions method has been developed very quickly. The paper presents the application of this method to solving direct and inverse problems of elasticity and thermoelasticity. The system of equations for displacements is reduced to a system of wave equations. Then the wave polynomials (Trefftz functions for wave equation) as base functions for several variants of Finite Element Method are used. In the paper, continuous FEMT and substructuring are considered. In the case of thermoelasticity, the temperature field occurs as inhomogeneity in one of the wave equations. It is shown how to get the particular solution in 2D and 3D. When using FEMT, the difference of solutions between the elements has to be minimized. The mechanical energy of the body depends on the velocity of the displacements. Therefore, the difference of the velocities between the elements is also minimized – it is a kind of physical regularization. The quality of the approximate solutions of direct and inverse problem was verified on the test example.

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.