### 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 ASYMPTOTIC METHODS FOR DIFFERENTIAL EQUATIONS Code MATH433 Coordinator Prof A Movchan Mathematical Sciences Abm@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2018-19 Level 7 FHEQ Second Semester 15

### Aims

This module provides an introduction into the perturbation theory for  partial differential equations. We consider singularly and regularly perturbed problems and applications in electro-magnetism, elasticity, heat conduction and propagation of waves.

### Learning Outcomes

The ability to make appropriate use of asymptotic approximations.

The ability to analyse boundary layer effects.

The ability to use the method of compound asymptotic expansions in the analysis of singularly perturbed problems.

### Syllabus

Asymptotic expansions. Definition and examples.

Singular and regular perturbations. Definitions and one-dimensional examples of singularly and regularly perturbed boundary value problems.

Asymptotic behaviour ofsolutions of boundary value problems in non-smooth domains. Dirichlet and Neumann boundary value problems in domains with conical points on the boundary. Evaluation of coefficients in the asymptotic expansions.

Asymptotic approximations for solutions of Dirichlet and Neumann problems in domains withsmall holes. Examples in electro-statics and elasticity (anti-plane shear).

Asymptotics in thin domains. Dirichlet and mixed boundary value problems for the Laplacian in a thin rectangle. Boundary layer. The limit one-dimensional model.

Asymptotic analysis of  fields  in multi-structures. Mixed boundary value problems for the Laplacian in domains with junctions.

Asymptotic theory of wave propagation. Long-wave approximations. Models of dynamic cracks. Dynamic problems for multi-structures (domains with junctions).

Asymptotics for heat conduction problems. Thermal crack in a half-plane. Examples.

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

MATH101; MATH102; MATH103; MATH201

### Programme(s) (including Year of Study) to which this module is available on an optional basis:

Programme:MMAS Year:1 Programme:G101 Year:3,4 Programme:F344 Year:3,4 Programme:FGH1 Year:3,4

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
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opportunity
Penalty for late
submission
Notes
Written Exam  3 hours  second semester  100  Standard university policy    Assessment 1 Notes (applying to all assessments) Full marks will be awarded for complete answers to FOUR questions. Only the best FOUR answers will be taken into account.
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes