The eigenfunction expansion method (EEM) is used to solve diffusion equations in non-homogeneous materials. The problem under deliberation is a square plate with a circular inclusion, the plate and the inclusion having different thermal conductivities. In order to validate the effectiveness of EEM, 1-D and 2-D diffusion equations in homogeneous materials are considered, for which the Fourier series solution (FSS) already exists. A generalized procedure that involves the Galerkin method and formulation of the final solution in terms of the procured eigenfunctions, is adopted. The Galerkin method basically includes expressing the given BVP in terms of a standard mathematical relation, generating a set of continuous base functions, formulating the S-L problem (eigenvalue problem), and determining the eigenvalues and the corresponding orthonormal eigenfunctions.;For the non-homogeneous material, a set of functions for the plate and inclusion are determined separately, through which an independent set of eigenfunctions are ascertained for the entire domain. EEM involves tedious and time-consuming computations, which is facilitated with the aid of a computer algebra system, Mathematica.
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