Computer approach to the R-functions method of solution of boundary value problems in arbitrary domains

Abstract

An analytically numerical method of solution of boundary-value problems is considered in arbitrary domains that may be concave and/or multiconnected. An essential feature of this so-called R-functions method (RFM) is a conversion of logical operations performed on sets (relevant to subdomains of which the considered domain is composed) into algebraic operations performed on elementary functions. The solution by the RFM is realized in two phases. In the first phase, an analytical formula for the so-called “general structure of solution” (GSS) is derived. GSS is a mapping that still contains undetermined function(s) but exactly satisfies all the prescribed boundary conditions. In the second phase, which is usually of numerical character, such function(s) is approximately evaluated by means of any suitable discrete method in order to satisfy the governing differential equation, which we consider or to minimize a relevant functional. Numerous tedious analytical operations, especially differentiations of complicated elementary functions, are necessary to derive GSS. In the original version of the method these had to be manually performed. This prevented many potential users from applying the RFM. Thus the main object of this work is to use the symbolic programming in order to obtain a fully computerized approach to the R-functions method. Both GSS itself and the results of all required operations performed on it are automatically obtained by the computer in an analytical form and written as FORTRAN subroutines ready for use in calculations. The Tschebychev approximation of undetermined function(s) and the least squares procedure complete this approach. As implemented only a few simple dates are required from the user. Some numerical examples are presented and discussed here. Suggestions are made as to areas of further research. The RFM may be applied to a wide class of linear and nonlinear boundary value problems in mechanics with the linear boundary conditions.