Advanced Techniques in Applied Mathematics

The following sections are included:

• Preface
• Contents

This chapter runs through some techniques that can be used to tackle partial differential equations (PDEs) in practice. It is not a theoretical work — there will be no proofs — instead I will demonstrate a range of tools that you might want to try. We begin with first-order PDEs and the method of characteristics; classification of second-order PDEs and solution of the wave equation; and separation of variables. Finally, there is a section on perturbation methods which can be applicable to both ordinary differential equations (ODEs) and PDEs of any order as long as there is a small parameter.

This chapter provides an introduction into the theory of resonant scattering of waves, beginning with classical fields and going over to nonrelativistic quantum scattering. Basic concepts of resonances are first introduced using model examples and then extended to a more general setting. An emphasis is placed on the connection between causality and analyticity of scattering amplitudes, and on dispersion relations arising.

Modelling is at the heart of applied mathematics. Here, we look at some of the basic ideas and methods that underlie many mathematical models such as conservation laws and approximation. Such approximations should be rational and can lead to simpler systems of equations that can be solved more readily. Simplifications can lead to loss of accuracy, but can also give insight into the underlying phenomena. We look at a variety of classic models, some of which have made a significant impact beyond their initial areas of application.

This chapter is concerned with the numerical treatment of partial differential equations using the Finite Element Method (FEM). We will start with the concept of variational formulation, introduce the Galerkin method, discrete (polynomial) spaces, uniform, algebraically graded and adaptively refined meshes, prove some interpolation results, discuss a priori and a posteriori error estimates, and present several algorithms for implementing the FEM.

These notes provide a basic introduction to the main ideas and methods of Random Matrix Theory, aimed at a broad audience of scientists and engineers. Most of the technical details are restricted to the simplest case of unitary symmetry.

In this chapter, we discuss applications of symmetry analysis (sometimes called similarity analysis) to find symmetry reductions and exact analytical solutions of nonlinear partial differential equation.