Behoudwetformulerings vir randvoorwaardes

In the study of field-theoretical problems a good deal of effort is devoted to the formulation of the governing equations. In addition to these equations, subsidiary conditions, such as boundary conditions, are required to make the problem well-posed. The precise formulation of boundary conditions for field-theoretical problems in terms of physical interpretations has been largely neglected in the past. Conditions such as the classical Dirichlet and Neumann boundary conditions are often assumed with little motivation as regards their realisability. In addition certain conditions which strictly do not qualify as boundary conditions, are often classified as such. Thus there are large gaps in the literature concerning the formulation and physical meaning of boundary conditions. Despite the existence of proper formulations in certain ad hoc cases, there is no general theory for the meaningful formulation of boundary conditions. In this dissertation a general theory for the formulation of boundary conditions is presented. The main premise from which the theory is developed, is the fact that conditions which hold at the boundary of a domain result from contact between the domain and a boundary medium. Boundary conditions must be a mathematical description of the interaction between a domain and its boundary medium. The formulation employs also conservation laws and constitutive equations as applied to the boundary medium. The first chapter gives a background to the present study and introduces terminology to be used. In Chapter 2 boundary conditions at stationary boundaries are obtained by the application of the main assumption. One-, two- and three-dimensional models are developed for thick and thin boundary media respectively. The general theory is illustrated by applying it to specific problems. In Chapter 3 the above results are extended to problems in domains with moving boundaries. In Chapter 4 criteria for the applicability of thick or thin boundary models are developed. Again examples are used to illustrate the theory. In the final chapter an evaluation of the thesis is given together with an indication of the prospects for further research.Thesis (DSc)--University of Pretoria, 1983.Mathematics and Applied MathematicsDScUnrestricte