Mathematical Modeling, Numerical Methods and Software Complexes

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

V. L. Leontiev

Ulyanovsk State University, Ulyanovsk, 432017, Russian Federation

Abstract: 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.

Keywords: orthogonal system of functions, orthogonal finite functions, Fourier series, mixed numerical methods for boundary problems solving.

UDC: 519.63, 519.651

MSC: 65M60, 41A30

Original article submitted 03/IX/2014
revision submitted – 09/XI/2014

DOI: 10.14498/vsgtu1414

Bibliographic databases:

• 
• 
• 
• 
• 
• 
•