A Hybrid Time/Laplace Domain Method Based on Numerical Green's Functions Applied to Parabolic and Hyperbolic Bioheat Transfer Problems

Abstract

There are some applications of extremely short time duration or at very low temperature for which the parabolic Pennes bioheat equation, which assumes an infinite thermal speed of propagation according to Fourier’s law, is not adequate and the mathematical model may be more accurately described by the hyperbolic bioheat equation. Hence, the purpose of the present paper is to describe the numerical solution of both parabolic and hyperbolic bioheat equations in a unified manner by a hybrid time/Laplace domain method. Starting from the hyperbolic bioheat equation, which includes the parabolic one as a special case, the Explicit Green’s Approach method that adopts numerical Green’s function matrices in its formulation is employed to compute the solution on time. The Green’s function equation is firstly discretized in the Laplace domain by the finite element method and then Green’s function matrices are computed in the time domain through standard Laplace inversion algorithms. Finally, a numerical example is analyzed in order to illustrate the accuracy and potentialities of the proposed unified method.