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A boundary element method (BEM) is utilized to find numerical solutions to boundary value problems of quadratically graded media governed by a spatially varying coefficients anisotropic-diffusion convection-reaction equation. The variable coefficients equation is firstly transformed into a constant coefficients equation for which a boundary integral equation can be formulated. A BEM is then derived from the boundary integral equation. Some problems are considered. A FORTRAN script is developed for the computation of the solutions. The numerical solutions verify the validity of the analysis used to derive the boundary element method with accurate and consistent solutions. The computation shows that the BEM procedure elapses very efficient time in producing the solutions. In addition, results obtained for the considered examples show the effect of anisotropy and inhomogeneity of the media on the solutions. An example of a layered material is presented as an illustration of the application.