MSc Applied Mathematics

This course is intended to provide an overview of a range of techniques that can be used to analyse and solve partial differential equations (PDEs). The methods discussed include characteristics for first and second order scalar PDEs, Fourier and other series solutions, Fourier transforms, and some examples of transform methods for nonlinear PDEs.

The focus is on developing methods that will be applicable in applied mathematics, and implementing these in practical examples.

To provide a practical overview of analytical methods for solving partial differential equations.

On successful completion of this course unit students will be able to:

• Derive and apply the method of characteristics to solve linear, semi-linear and quasilinear first order PDEs, and analyse the uniqueness and existence of these solutions.
• Classify second order linear PDEs, choose suitable canonical variables, and transform them to canonical form.
• Use direct integration to solve hyperbolic and parabolic PDEs and calculate solutions obeying given initial and boundary conditions.
• Define Fourier full and half-range series, and use them to solve linear, constant-coefficient PDEs in rectangular domains.
• Derive and apply Sturm-Liouville properties to justify the completeness of separable solutions, and to bound eigenvalues within the complex plane.
• Construct series solutions for general boundary value and initial value problems, with homogeneous or inhomogeneous boundary conditions.
• Use direct integration, contour deformation and convolution methods to evaluate Fourier transforms and inverse transforms.
• Select and apply Fourier full and half-range transforms to solve linear PDEs in infinite or semi-infinite domains.
• Use Cole-Hopf and Backlund transform methods to relate the solutions of different nonlinear PDEs, and when related to linear PDEs, solve initial and boundary value problems.
• Calculate the 1D wave scattering reflection and transmission coefficients for a given potential, and determine the corresponding bound states.
• Determine the time evolution of scattering coefficients for the KdV equation and apply the Gelfand-Levitan-Marchenko equation to solve the KdV equation with one or two solitons.

 1. Introduction

Basic notation. Classification of PDEs, examples of common PDEs.

  2. First order PDEs

Construction of solutions to linear and nonlinear first order PDEs via method of characteristics. Application of Cauchy data. Examples of shock formation.

  3. Linear second order PDEs

Characteristics of second order PDEs, classification, reduction to normal form. Well-posedness of boundary conditions.

  4. Fourier series

Properties of full and half range Fourier series, and discussion of orthogonality. Use of separable solutions in constructing series solutions for appropriate BVPs and IVPs.

  5. Sturm-Liouville systems

Definition of Sturm-Liouville systems, and proofs of main properties for regular S-L systems. Further discussion of applicability of series solutions.

  6. Fourier transforms

Connection to Fourier series. Summary of main properties of Fourier transforms, and examples of calculation. Inversion via contour integration, and relation to convolution properties. Examples of solution of linear PDEs in infinite domains, and use of sine and cosine transforms in semi-infinite domains.

  7. Nonlinear PDEs

Failure of superposition principle. Cole-Hopf transform for Burgers' equation. Examples of Backlund transforms. Inverse scattering methods for the KdV equation.

• Coursework weighting 20%
• End of semester examination: weighting 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  In-class tests also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Applied Partial Differential Equations, revised edition. Ockendon, Howison, Lacey and Movchan, Oxford University Press, 2003.
• Partial Differential Equations, second edition. J. Kevorkian, Springer, 1999.
   

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further updates.