A domain-boundary integral equation method for the anisotropic DR equation of spatio-temporal coefficients

A combined Laplace transform (LT) and domain-boundary element method (DBEM) were utilized to obtain numerical solutions for an unsteady spatio-temporal coefficient diffusion–reaction (DR) equation with arbitrary initial conditions and source terms, addressing problems in a distinct class of anisotropic functionally graded materials (FGMs). The procedure begins by transforming the equation of the time-space variable coefficients into one of the time-variable coefficients. The time variable of the integral equation was reduced by utilizing the Laplace transform and its convolution theorem to obtain a domain-boundary integral equation. Numerical solutions within the framework of the Laplace transform were then obtained using the standard domain-boundary element method. These numerical solutions were inverted using the Stehfest method to obtain solutions for the original time variable. Several problems involving trigonometric, exponential, and quadratic spatial gradation function coefficients were solved. The validity of the analysis used to derive the domain boundary integral equation was verified, and accurate BEM solutions were obtained.