TheR-function method in boundary-value problems with geometric and physical symmetry

Abstract

The traditional mathematical models of fields of various physical nature are boundary-value problems for partial differential equations. The usual statement of a boundary-value problem has the following form:where ffZ is the region in which the solution u is being sought; OfZi, i = 1, 2 ..... m, is a covering of the boundary t?~ of the region f) (the parts c~f~ i are not necessarily different and may be equal to Of)); f and (Pi are known functions (perturbing the field), vector-valued functions, tensors, or possibly elements of some other type. In the Rfunction method developed in the present article geometric information is taken into account by functions co, co i, chosen as a rule from among the elementary functions, so that 0~2 and 0f~i are the sets of their zeros, and the solution u is sought in the pencilHere r is a known function; B is an operator depending on the shape of the boundary Og2 and its parts OO i, i = 1, 2 ..... m, formed so that for any choice of an undetermined component ~ from a set M formula (3) satisfies the boundary conditions (2) exactly. If this formula has the property of completeness relative to the boundary-value problem (1)-(2), that is, it is possible to choose an undetermined component 9 that converts (3) into a solution of the problem with any prescribed precision (in some sense), it is called a general structure solution of the problem in question. -We recall first how the R-function method solves the problem of constructing the functions co and col that occur in formulas of the form (3). A complicated locus (also called a geometric object or a drawing) which the region f2, and hence also its boundary 0f~, may be, can be described by a logical formula connecting simple loci (primitives) after which transition to the usual equations (resp. inequalities) of the form to(x, y)= 0 (resp. co(x, y)> 0) recognized in analytic geometry is accomplished using R-functions. The technique of constructing them will be shown through examples below.In the majority of cases the following R-functions are used to construct the functions to, co/:

Simulation of Surface Heating for Arbitrary Shape’s Moving Bodies/Sources by Using R-Functions

“…Such relation can be described using the Rfunctions [10]. The R-function method was developed as an improvement of Ritz methods for solving boundary-value problems.…”

Semi-analytic geometry with R-functions

“…The simplest function µ satisfying this requirement is the 45 • saw-tooth pattern, but smooth approximations may be constructed using Fourier series, or by methods described by Rvachev (1974). Rvachev, Sheiko and Shapiro (1999) observed that more complex coordinate transformation functions µ i may be constructed as semi-analytic compositions of the above primitive coordinate transformations and Rfunctions. Such coordinate transformations may prescribe periodic, symmetric, and random properties that explicitly depend on the values of the coordinates x i and additional external parameters.…”

R-functions in mathematical modeling of geometric objects with symmetry

“…The papers [4][5][6] propose new approaches that make it possible to apply R-functions to set up normalized equations of complex GOs with symmetry, to be used both in solving boundary-value problems for partial derivative equations modeling various physical fields (it is natural that if geometrical information appears in solution structures, w ( , , )…”