## PDEs: Theory and Practice (MAGIC058)

 Unit code: MATH64062 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

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#### Aims

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

#### Overview

This course unit develops certain aspects of the theory and practice of pdes and systems of pdes, in particular existence and uniqueness of solutions together with discretisation methods for approximating such solutions.

The course will introduce basic definitions and concepts pertaining to pdes and develop both classical and modern methods of proving existence and uniqueness results. In particular analytic function theory will be used to prove the classical Cauchy-Kowalevskaya theorem and functional analysis (including the Lebesgue integral) will form the basis for proving theorems for second and higher order elliptic equations via the Riesz representation theorem.

The basic methods of discretisation of partial derivatives will be presented for various types of pde, for example, elliptic, parabolic and hyperbolic equations. Examples will be presented of boundary and initial value problems of interest in Applied Mathematics, Industrial Mathematics, Numerical Analysis, various branches of Engineering and Physics (including geophysics).

#### Learning outcomes

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.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• In-class test: weighting 20%
• End of semester examination: three hours weighting 80%

#### Syllabus

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.

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

#### Feedback methods

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.

#### Study hours

• Lectures - 24 hours
• Tutorials - 9 hours
• Independent study hours - 117 hours

#### Teaching staff

Alice Thompson - Unit coordinator