MAT6262PArtial Differential Equations with their Numerical Solutions

Partial differential Equations (PDEs) are used to construct models of the most basic theories underlying physics and engineering. In general, the construction of a mathematical model is based on two main ingredients: general laws and constitutive relations. This module presents methods of solving first order linear equations and IV/BVP for the well-known second order PDEs. The module presents also numerical methods of solving IV/BVP for PDEs: Finite difference methods and Finite elements methods.
 This course id divided into two parts:

Part I: This part begins with an introduction to PDEs, their definitions, classifications and their use. It gives the three origins of some PDEs (Diffusion equation, wave equation). The second part deals with the Boundary value problems and well - posedness of introducing function spaces. The third part deals with Laplace’s and Poisson’s equation (a maximum principle, Uniqueness for the Dirichlet problem). The fourth chapter deals with the heat and wave equations.

Part II: Numerical solutions for PDEs

Chapter 5: This chapter describes the finite difference methods for solutions of PDEs: Finite differences, Approximating solution to the diffusion equation, order, stability and convergence, the crank-Nicholson scheme, Approximation of Laplace’s equation, the discrete mean value property, stability analysis.

Chapter 6: This chapter describes the Finite element methods for solving IV/BVP for PDEs: Galerkin method, Approximation of elliptic problems. Finite approximation of initial boundary value problems. Energy dissipation, conservation and stability. Analysis of finite element methods for evolution problems.