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 2019-20 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

(LO1) The ability to make appropriate use of asymptotic approximations.

(LO2) The ability to analyse boundary layer effects.

(LO3) The ability to use the method of compound asymptotic expansions in the analysis of singularly perturbed problems.

(S1) Problem solving skills

(S2) Numeracy

Syllabus
 

Asymptotic expansions. Definition and examples.

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

Asymptotic behaviour ofsolutions of boundary value problems in non-smoothdomains. Dirichlet and Neumann boundary value problems in domains withconical points on the boundary. Evaluation of coefficients in the asymptoticexpansions.

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

Asymptotics in thin domains. Dirichlet and mixed boundary value problems forthe Laplacian in a thin rectangle. Boundary layer. The limit one-dimensionalmodel.

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

Asymptotic theory of wave propagation. Long-wave approximations. Models ofdynamic cracks. Dyna mic problems for multi-structures (domains withjunctions).

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

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

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

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:
 

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Assessment 1 Assessment Schedule (When) :second semester  3 hours    100       
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes