Laplace transform-based BEM for unsteady heat conduction in space-time anisotropic functionally graded materials with heat source

This study investigates transient heat conduction in a separable class of anisotropic functionally graded materials with spatially and temporally varying thermal properties and an internal heat source term. The governing transient heat equation with variable coefficients is first transformed into an equivalent form with constant coefficients through an appropriate variable transformation under prescribed separability conditions on the material properties. The resulting equation is then mapped into the Laplace domain to eliminate the time derivative and simplify the transient formulation. A fundamental solution in the Laplace space is employed to reformulate the transformed equation into a boundary integral representation containing source-related domain integrals, which serves as the basis for the boundary element formulation. The resulting integral equations are discretized using boundary elements, while the domain integrals are evaluated numerically through domain discretization in conjunction with particular solutions. The transient solution in the physical time domain is recovered by applying the Stehfest numerical inversion technique for the Laplace transform. Several benchmark test problems with known analytical solutions are presented to assess the accuracy, convergence, and stability of the proposed approach. Numerical results demonstrate that the developed boundary element formulation provides accurate and consistent solutions for transient heat conduction problems in anisotropic functionally graded materials satisfying the prescribed transformation conditions.