Abstract

CLOSED FORM SOLUTION OF A DIRICHLET HARMONIC PROBLEMS USING COMPLEX VARIABLE TECHNIQUES

NDUBUEZE GEORGE EMENOGU & ANYANWU JUDE ABARA

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One of the most interesting examples of applying the theory of harmonic functions to physical modelling is the study of two-dimensional steady state temperature fields. When steady state prevails, the temperature function inside a two-dimensional body ? ( x y, ) is readily seen to be harmonic as we explore the intimate connection between complex analysis and solutions to the Laplace equation in solving harmonic Dirichlet problems of heat flow in two- dimensional solids whose boundaries are maintained at prescribed values. The problem is formulated by complex variable techniques and solved by conformal mapping method. The method uses the appropriate mapping function to transform the domain D and boundary ? of the given problem in the z -plane onto one in the upper half of the w -plane and the appropriate portions of the y ? axis where its solutions for steady state temperature is easily identified as the imaginary part of some branch of the logarithmic function. Key words: Closed form solution, steady state temperature, analytic functions, conformal mapping and Dirichlet problem.

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