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.

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.

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.