To provide an introduction to a range of analytical and numerical methods for partial differential equations arising in many areas of applied mathematics.  

To provide a focus on advanced analytical techniques for solution of both elliptic and parabolic partial differential equations in two and three dimensions, and then on numerical discretization methods of finite differences and finite elements. 

To review Matrix analysis methods to make the module self-contained.

Apply a range of standard numerical methods for solution of PDEs and should have an understanding of relevant practical issues.

obtain solutions to certain important PDEs using a variety of analytical techniques and should be familiar with important properties of the solution.

Understand and be able to apply standard approaches for the numerical solution of linear equations     

Have a basic understanding of the variation approach to inverse problems.

4 Lectures

Elliptic partial differential equations (PDEs) in two independent variables: Harmonic functions. Mean value property. Maximum principle. Dirichlet and Neumann boundary value problems. Well-posed and ill-posed problems.

10 Lectures

Free-space Green''s functions of Laplace and Helmholtz equations. Representation of solutions in terms of Green''s functions and Boundary integral reformulations.  Continuous dependence of data for Dirichlet boundary value problem.  Nonhomogeneous problems: Fourier series,  integral transforms, and the discrete Fourier Transform (and FFT).

PART 2 -------------------------------

6 Lectures

Discretiz ation by finite difference methods: Continuous functions to discrete vectors. Operators to matrices.  Error analysis for elliptic and parabolic PDEs.

PART 3 -------------------------------

6 Lectures

Matrix Analysis:  Ill/well-conditioning.   Nodal ordering and sparsity. Matrix free conjugate gradient methods: Preconditioning methods. Analysis of iterative solvers for elliptic PDEs by local Fourier analysis.

PART 4 -------------------------------

4 Lectures

Inverse (ill-posed) problems: Least squares and variational formulation. Image denoising. Tikhonov regularization and Lagrange multipliers. Euler-Lagrange equations. Parabolic PDEs and Gradient descent method.

PART 5 -------------------------------

6 Lectures

Introduction to finite element methods: variational reformulation, Sobolev spaces, piecewise linear interpolants,reference element, matrix assembly and sparse linear systems. 1D and 2D problems.