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.

(LO1) Solve routine problems involving asymptotic approximations

(LO2) Perform elementary analysis of boundary layer effects.

(LO3) Practice to distinguish between regular perturbation and singular perturbation problems.

(LO4) For ordinary differential equations, use the method of compound asymptotic expansions in the analysis of singularly perturbed boundary value problems.

(LO5) For routine applications in problems of electrostatics, apply asymptotic approximations for domains with perturbed boundaries.

(LO6) Apply asymptotic approximations to boundary value problems for partial differential equations.

(LO7) Analyse boundary layer effects for Dirichlet and Neumann singularly perturbed boundary value problems for Laplace’s operator.

(LO8) Analyse Dirichlet and Neumann boundary value problems for small regular perturbations of the boundary.

(LO9) Apply the method of compound asymptotic expansions in the analysis of singularly perturbed boundary value problems in domains with small inclusions.

(LO10) Apply asymptotic approximations for perturbation problems of elasticity and wave propagation.

(S1) Problem solving skills

(S2) Numeracy

Asymptotic expansions.
Definition and examples.
Singular and regular perturbations.
Definitions and one-dimensional examples of singularly and regularly perturbed boundary value problems.
Asymptotic behaviour of solutions 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 with small 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.