Module Details
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title ANALYTICAL & COMPUTATIONAL METHODS FOR APPLIED MATHEMATICS
Code MATH424
Coordinator Dr T Valkonen
Mathematical Sciences
Tuomo.Valkonen@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2017-18 Level 7 FHEQ Second Semester 15

Aims

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, and then on numerical discretisation methods of finite differences and finite elements. 

To provide the algorithms for solving the linear equations arising from the above discretisation techniques.

Learning Outcomes

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.

Syllabus
1 

1. Explicit and transform methods [8h]
  1.1 Introduction to PDEs; classification and boundary conditions
1.2. Fundamental solutions and Green’s functions; Laplace and Poisson equations
1.3. Solution by change of variables; Helmholtz equation in spherical coordinates
1.4. Fourier transform; convolution, Bessel potent ials, heat equation 
1.5. Fourier series, DFT & FFT; separation of variables, wave equation
2. Finite difference methods [6h]
2.1. Derivation from Taylor expansions
2.2. Implementation of boundary conditions
2.3. Convergence, stability, consistency
2.4. Nodal ordering and other implementation details
2.5. Time-dependent problems
3. Numerical solution of linear systems [6h]
3.1. Splitting methods; Gauß-Seidel and Jacobi iterations
3.2. Fixed point theorems, convergence analysis
3.3. Solution by optimisation; conjugate gradient method
3.4. Application to finite difference schemes
3.5. Preconditioning
4. Sobolev spaces and the weak form of a PDE [4h]
4.1. Weak derivatives
4.2. Sobolev spaces; basic definitions, approximation, traces, Sobolev and Poincaré inequalities, compact embedding
4.3. The weak form of a PDE; boundary conditions and H_0
5. Finite element methods [6h]
5.1. Galerkin’s method; Céa’s lemma, finite-dimensional subspaces of Sobolev spaces, the stiffness matrix
5.2. Construction of finite-dimensional subspaces; triangulation
5.3. Linear and polynomial elements; local bases
5.4. Numerical examples
6. Variational methods and inverse problems [6h]
6.1. The calculus of variations; PDEs from energy minimisation
6.2. Inverse problems theory
6.3. Examples from image processing; brief account of functions of bounded variation

Recommended Texts
Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
Explanation of Reading List:

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):
MATH102; MATH103  

Co-requisite modules:
 

Modules for which this module is a pre-requisite:
 

Programme(s) (including Year of Study) to which this module is available on a required basis:
 

Programme(s) (including Year of Study) to which this module is available on an optional basis:
G101 (Year 3 and 4) FGH1 (Year 3 and 4) F344 (Year 3 and 4) MMAS (Year 1)

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Unseen Written Exam  2.5 hours  Second semester  100  Yes  Standard UoL penalty applies  Written Exam Notes (applying to all assessments) Students will be required to answer 4 out of 4 questions  
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes