Numerical solutions to elliptic differential equations - finite difference approach.

ABSTRACT

This dissertation contains materials on numerical solutions based on elliptic differential equations only appropriate for senior level undergraduate or beginning level graduate students. The reader based on this dissertation should have had introductory courses in Calculus, linear algebra and general numerical analysis. A formal course in ordinary or partial differential equations would be useful. In our study, it should be understood that, there are many procedures that come under the name numerical methods. We shall see how the very popular finite-difference methods can be used to solve elliptic equations. To begin, we introduce the idea of finite differences. We then show how to use these finite differences to solve a Dirichlet Problem inside a square. However, the numerical algorithm for solving the Dirichlet Problem (Liebmann‟s method ) has been included. Moreover, systems of algebraic equations have been solved numerically by an iterative process in order to obtain an approximate solution to the partial differential equation. The iterative methods such as Gauss Seidel iteration method, Gauss Jacobi iteration method and Liebmann iteration method have been discussed. It is also pointed out that the reader will find how numerical solutions to elliptic differential equations are applicable in daily life experience. DEFINITIONS : Conduction: This is the mode of heat transfer that occurs at molecular level when a temperature gradient exists in medium, which can be either solid or fluid. Heat is transferred along that temperature gradient by conduction. Convection: It is the mode of heat transfer that happens in fluids in one of the mechanism. Random molecular motion which is termed diffusion or the bulk of motion of fluid carries energy from one place to another. Convection can be either forced, for example pushing the flow along the surface or natural as that which happens due to buoyancy forces.