Distributed Systems Modelling with Partial Differential Equations

Abstract

This chapter presents an overview of partial differential equations (PDEs) for modelling distributed systems. Differential operators such as the gradient, divergence, curl and Laplacian are introduced, as well as the divergence theorem for modelling conserved physical quantities. An overview of basic analytical and numerical solution techniques for PDEs is then provided, including separation of variables, the finite difference method and the method of lines. Examples of PDEs solved analytically are 1D diffusion as well as the electrical potential distribution around a disc electrode in a 3D semi-infinite volume conductor. Explicit and implicit finite difference methods are used to solve for 1D diffusion and the method of lines is used for solving neural propagation along an axonal fibre. All numerical examples provide complete Matlab code listings. The chapter ends with ten problems with fully-worked solutions provided in the solutions section of the text.