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Integral Transform Solutions of Di usion and Eigenvalue Problems Within Arbitrarily Shaped Domains

Leandro A. Sphaier
Department of Mechanical Engineering – PGMEC, Universidade Federal Fluminense, Rua Passo da Patria 156, bloco E, sala 216, Niteroi, RJ, 24210-240, Brazil

Renato M. Cotta
Laboratory of Transmission and Technology of Heat, LTTC, Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, POLI&COPPE/UFRJ, Cidade Universitaria, Cx. Postal 68503, 21945-970, Rio de Janeiro, RJ, Brazil

Abstract

Di usion and Sturm-Liouville type problems de ned within irregular multidimensional regions are analytically and automatically solved through integral transforms, by making use of the mixed symbolic-numerical Mathematica platform. The approach is illustrated for linear two-dimensional di usion and eigenvalue problems, and formal exact solutions are obtained. Two alternative approaches are considered, by either handling the intermediate exact eigenvalue problem, or by direct integral transformation of the original diffusion problem. A cylindrical region test case with known exact solution is rst considered, and treated as an irregular region in the cartesian coordinate system. Then, a more complex con guration, represented by the Brazilian map as provided by a set of points in the plane, is analyzed for both the associated eigenvalue and di usion problems.

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