A boundary method of Trefftz type for PDEs with scattered data

This paper presents the application of a Trefftz type method for partial differential equations (PDEs) of the elliptic type with inhomogeneous term given by a set of scattered data. The method of particular solutions is used. Basis functions of a new type were introduced to approximate the scattered data. Using these basis functions, we get the approximation in the form of series over some orthogonal system of eigenfunctions. The particular case of the trigonometric eigenfunctions is considered. The corresponding approximation of the inhomogeneous term allows to get a particular solution for PDEs with constant coefficients or for the systems of such PDEs easily. We test our basis functions on recovering well-known Franke's and PEAKS functions given by scattered data. We also present results of solution Helnholtz PDE, PDE with differential operator of 4th order and system of PDEs arising in shell deflection problems. A comparison of the numerical solutions with analytic solutions is performed for all the problems.