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Difference methods for ordinary differential equations with applications to parabolic equations Doedel, Eusebius Jacobus

Abstract

The first chapter of the thesis is concerned with the construction of finite difference approximations to boundary value problems in linear nth order ordinary differential equations. This construction is based upon a local collocation procedure with polynomials, which is equivalent to a method of undetermined coefficients. It is shown that the coefficients of these finite difference approximations can be expressed as the determinants of matrices of relatively small dimension. A basic theorem states that these approximations are consistent, provided only that a certain normalization factor does not vanish. This is the case for compact difference equations and for difference equations with only one collocation point. The order of consistency may be improved by suitable choice of the collocation points. Several examples of known, as well as new difference approximations are given. Approximations to boundary conditions are also treated in detail. The stability theory of H. O. Kreiss is applied to investigate the stability of finite difference schemes based upon these approximations. A number of numerical examples are also given. In the second chapter it is shown how the construction method of the first chapter can be extended to initial value problems for systems of linear first order ordinary differential equations. Specific examples are 'included and the well-known stability theory for these difference equations is summarized. It is then shown how these difference methods may be applied to linear parabolic partial differential equations in one space variable after first discretizing in space by a suitable method from the first chapter. The stability of such difference schemes for parabolic equations is investigated using an eigenvalue-eigenvector analysis. In particular, the effect of various approximations to the boundary conditions is considered. The relation of this analysis to the stability theory of J. M. Varah is indicated. Numerical examples are also included.

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