Orthogonal Franklin system and orthogonal system of finite functions in numerical methods of boundary problems solving

Possibilities of classical trigonometric Fourier series are substantially limited in 2-D and 3-D boundary value problems. Boundary conditions of such problems for areas with curvilinear boundaries often fails when using the classical Fourier series. The solution of this problem is the use of orthogonal finite functions. However, orthogonal Haar basis functions are not continuous. The orthogonal Daubechies wavelets have compact supports, but is not written in analytical form and have low smoothness. Continuous finite Schauder–Faber functions are not orthogonal. Orthogonal Franklin continuous functions are not finite. The connection of the orthogonal Franklin functions with a sequence of grid groups of piecewise linear orthogonal finite basis functions (OFF) is established here. The Fourier-OFF series on the basis of such continuous OFF is formed. Such series allows to execute boundary conditions of Dirichlet's type on curvilinear boundaries in integral performances of boundary value problems. A similar problem is connected with a satisfaction of Neumann boundary conditions and also is eliminated in the integral mixed performances of boundary value problems. Fourier-OFF series increases the effectiveness of mixed numerical methods for boundary value problems solving.