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The Application of R-Functions Theory and Wavelets to the Boundary Value Problems Solving of the Elliptic Type

Keywords:

V.F. Kravchenko, A.V. Yurin


The new approach of solving boundary value problems for differential equations of the elliptic type partial derivatives is represented. It is based on Galerkin classic variation diagram which is converted with the help of R-functions structural method [1. V.F. Kravchenko, V.L. Rvachov. Boolean algebra, atomic functions and wavelets in physical applications, Moscow, Fizmatlit, 2006] and wavelet-basis properties. The main point of such approach is the construction of the computational algorithm concerning the wavelet approximation of the analytic and geometric components of the boundary value problem. To convert the geometric information into analytic one as well as to satisfy the boundary conditions of the problem using the structures of solution helps the R-functions body of mathematics. The basic elements of the obtained functional is the wavelet-basis [6. H.L. Resnikoff, R.O. Wells. Wavelet analysis: the scalable structure of information. New York, Springer, 1998], the expansion coefficients of the domain function, the function of the right part of the equation and the function of boundary conditions of the wavelet-basis. As a result while matrix system compiling we obtain some calculating advantages: matrixes of the system are discharged, the calculation of matrix elements does not demand the integration and is carried out with the help of finite number of elementary mathematical operations over the coupling coefficients of the corresponding wavelet system. New fast computational algorithms based on fundamental wavelet properties for coupling coefficients are also introduced and founded in this work.
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