Questions of using multidimensional Pade-type approximations for the generalized summary of the decisions of the boundary-value problems
Journal Title: Комп’ютерне моделювання: аналіз, управління, оптимізація - Year 2018, Vol 1, Issue 1
Abstract
The aim of this work is to develop a methodology for constructing multidimensional Pade approximants for solving boundary value problems and determining the conditions for their convergence. In the theory of multidimensional fractional-rational function approximations, two problem aspects are distinguished. First, it is the definition of the very concept of such approximations and the method of constructing the approximation, and secondly, the choice of the class of approximable functions and the proof of the convergence of the chosen scheme. We propose the development of the theory of multidimensional fractional-rational approximations for the approximation of power series defined on certain systems of basis functions. The choice of the form of the basis functions and the method of constructing the approximation makes it possible in a number of cases to achieve a substantial improvement in the useful properties of approximants for certain distinguished classes of functions. We consider the creation of a method for constructing such multidimensional fractional-rational approximations and determining the set of coefficients of the series necessary for constructing an approximation of a given structure. Conditions for the convergence of approximants of approximate multidimensional functions constructed according to the method developed by the authors are determined for various basic functions. The development of a modified method of continuation by the parameter based on the results obtained is proposed. The difference between the proposed method is the use of a combination of asymptotic methods for solving boundary value problems with a generalized summation of the obtained series on the basis of multidimensional Padé-type approximations, which will allow constructing alternative solutions to existing numerical methods that are not inferior in accuracy. Another important difference is the complexity of the research – from the creation of the method of constructing approximations, the justification of its convergence, to its implementation in the solution of boundary value problems and the creation of application programs. Thus, the present work is aimed at developing new methods for calculating the boundary value problems of mathematical physics and developing the theory of approximation of functions of several variables.
Authors and Affiliations
В. И. Олевский, Ю. Б. Олевская, Т. С. Науменко, И. В. Шапка
Keywords
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