# Numerical Simulation for Unsteady Anisotropic-Diffusion Convection Equation of Spatially Variable Coefficients and Incompressible Flow

## DOI:

https://doi.org/10.5556/j.tkjm.54.2023.4069## Keywords:

variable coefficients, anisotropic functionally graded materials, unsteady diffusion convection equation, Laplace transform, boundary element method## Abstract

The anisotropic-diffusion convection equation of spatially

variable coefficients which is relevant for functionally graded media

is discussed in this paper to find numerical solutions by using a

combined Laplace transform and boundary element method. The variable

coefficients equation is transformed to a constant coefficients equation.

The constant coefficients equation is then Laplace-transformed so

that the time variable vanishes. The Laplace-transformed equation

is consequently written in a pure boundary integral equation which

involves a time-free fundamental solution. The boundary integral equation

is therefore employed to find numerical solutions using a standard

boundary element method. Finally the results obtained are inversely

transformed numerically using the Stehfest formula to get solutions

in the time variable. The combined Laplace transform and boundary

element method is easy to be implemented, efficient and accurate for

solving unsteady problems of anisotropic functionally graded media

governed by the diffusion convection equation.

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*Tamkang Journal of Mathematics*,

*54*. https://doi.org/10.5556/j.tkjm.54.2023.4069

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