Numerical Treatment of Laplace Equation

Abstract

This thesis addresses Laplace’s equation, one of the most fundamental partial differential equations (PDEs), under Dirichlet boundary conditions. Two main approaches are employed: - The analytical solution using the method of separation of variables. - And the numerical approximation using the Finite Difference Method (FDM). In Chapter 2, the equation is studied over a rectangular domain. The analytical solution is derived through variable separation, while the numerical solution is constructed via FDM. Both models are implemented in MATLAB to evaluate and compare their accuracy and efficiency. Chapter 3 extends the analysis to a circular domain. By switching to polar coordinates, the same analytical and numerical techniques are applied, with a focus on analyzing the outcomes and comparing the numerical performance to the theoretical solution.