Journal of Mathematical Sciences

, Volume 97, Issue 1, pp 3777–3795 | Cite as

On local singularities in mathematical models of physical fields

  • V. T. Grinchenko
  • A. F. Ulitko
Article
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Abstract

In constructing mathematical models of physical fields there are various widely used hypotheses that include assumptions on the smoothness of the field, the nature of the boundary surfaces of the region of existence of the field and the properties of the interaction between the object being studied and surrounding objects. In many cases the use of convenient mathematical model representations leads to a situation in which some characteristics of the field do not have bounded values at individual points. The problem of the appearance of such local singularities in physical fields of different natures is of fundamental importance in the study of these fields by the methods of mathematical modeling, from the point of view of general understanding of the possibilities and content of mathematical modeling. The appearance of singularities in the characteristics of the field is a consequence of the contradictions arising when the properties of the medium and the nature of the boundary and the boundary conditions are modeled independently. In the present work we discuss a unified methodological approach to determining the nature of the singularity in various types of physical fields. The approach is based on introducing the concept of a general solution of the boundary-value problem and the use of such a solution in the vicinity of singular points. It is shown that the solutions with singularities give a reliable qualitative and quantitative description of the fields outside a small zone in a neighborhood of the singularity. Analysis of the solutions of boundary-value problems with local singularities shows the nature of the difficulties in interpreting the physical content of the solution near singularities. Typical situations are those in which interpenetration of different parts of the body is observed, the values of the displacements or velocities of certain points of the boundary depend on the direction of the limiting approach, and others. In a number of cases a priori knowledge of the nature of the singularity makes it possible to develop new approaches to the construction of effective solutions of boundary-value problems.

Keywords

Boundary Condition Mathematical Model General Solution Singular Point Model Representation 

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© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • V. T. Grinchenko
  • A. F. Ulitko

There are no affiliations available

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