CHAPTER 1 Diffusion–convection problems A.E. Taigbenu School of Civil and Environmental Engineering, University of the Witwatersrand, South Africa. Abstract This chapter presents some solutions to the diffusion–convection equation that are based on the boundary element theory. Four formulations are discussed, but solu- tions from only three of them are presented. The formulations represent different replications of the differential equation along the lines of the singular integral the- ory. Their fundamental solutions come from different linear parts of the differential operator. The elliptic diffusion (ED) formulation that is based on the ED opera- tor shows the most promise. Attempts at avoiding domain integrations through the dual reciprocity method are discussed, while full domain discretization through the Green element method for ease of evaluation of domain integrations and in solving heterogeneous and nonlinear transport is presented. The solutions from the three formulations to the nonlinear Burgers’ equation are also presented, with the ED formulation exhibiting superior performance. 1 Introduction There are many transport phenomena of theoretical and practical interest in a number of fields of science and engineering that are governed by the diffusion– convection equation. It can, under certain conditions, describe transport of mass, momentum, vorticity, and energy when mechanisms of diffusion or dispersion and convection or advection are of importance. Its solution continues to attract consid- erable interest in numerical circles because of its unique feature of being either a parabolic or a hyperbolic equation, depending on the values of the parameters of www.witpress.com, ISSN 1755-8336 (on-line) © 2007 WIT Press WIT Transactions on State of the Art in Science and Engineering, Vol 14, doi:10.2495/978-1-84564-100-9/01