The Laplace transform boundary element method for diffusion-type problems

Diffusion-type problems are described by parabolic partial differential equations; they are defined on a domain involving both time and space. The usual method of solution is to use a finite difference time-stepping process which leads to an elliptic equation in the space variable. The major drawback with the finite difference method in time is the possibility of severe stability restrictions. An alternative process is to use the Laplace transform. The transformed problem can be solved using a suitable partial differential equation solver and the solution is transformed back into the time domain using a suitable inversion process. In all practical situations a numerical inversion is required. For problems with discontinuous or periodic boundary conditions, the numerical inversion is not straightforward and we show how to overcome these difficulties. The boundary element method is a well-established technique for solving elliptic problems. One of the procedures required is the evaluation of singular integrals which arise in the solution process and a new formulation is developed to handle these integrals. For the solution of non-homogeneous equations an additional technique is required and the dual reciprocity method used in conjunction with the boundary element method provides a way forward. The Laplace transform is a linear operator and as such cannot handle non-linear terms. We address this problem by a linearisation process together with a suitable iterative scheme. We apply such a procedure to a non-linear coupled electromagnetic heating problem with electrical and thermal properties exhibiting temperature dependencies.