# Applied Partial Differential Equations 2

## Unit aims

To provide the student with the necessary mathematical tools in order to model a wide variety of different physical problems, ranging from waves on strings, the propagation of signals, the diffusion of heat in solids and chemicals in solution, traffic flow and the vibrations of membranes and surfaces.

## Unit description

Partial differential equations (PDEs) are differential equations involving partial derivatives of functions of several variables. They are essential for understanding many physical processes including the behaviour of ocean waves, the flow of rivers, the diffusion of pollutants, aerodynamics, the operation of musical instruments, atomic physics, and many other branches of science. This unit will give an introduction to simple PDEs and how they arise in physical problems; it will develop techniques for solving them and understanding the behaviour of the solutions.

The unit will develop students' understanding of first year multivariable calculus and linear algebra. It will introduce Fourier series, the Fourier integral, the delta function and other methods for solving linear and nonlinear PDEs, (such as the method of characteristics) and will show how eigenvalues play a central role in applied mathematics. The course emphasises techniques and broad understanding rather than proofs.

## Relation to other units

This unit will be a prerequisite for Mathematical Methods, Fluid Dynamics, Quantum Mechanics and other applied mathematics units. It gives applications of the vector calculus, complex variable methods and other material in Multivariable Calculus and Methods of Complex Functions.

## Learning objectives

At the end of the course the student should should be able to:

- Understand the physical models and derive PDE's representing diffusion and wave propagation;
- Identify appropriate boundary conditions for simple linear PDEs;
- Solve linear two-dimensional PDEs on bounded spatial domains by separation of variables using Sturm-Liouville Theory and Fourier series;
- Calculate and manipulate Fourier transforms, and use them to solve simple linear PDEs on unbounded spatial domains;
- Use the method of characteristics to solve simple linear and nonlinear first order PDEs;
- Describe some differences between linear and nonlinear PDEs;
- Solve multi-dimensional linear PDE's using separation of variables in a variety of coordinate systems

## Transferable skills

Clear thinking; mathematical modelling of physical situations; skill in mathematical manipulation.

## Syllabus

**Part I: Introduction**Review of linear ODE's, Boundary and Initial Conditions. Introduction of Eigenvalues and Eigenfunctions for linear homogeneous problems.

Introduction to PDE's: simple examples of linear and nonlinear. Derivation of the wave equation in 1D and 2D. Derivation of the diffusion equation in 1D and 2/3D. Conservation Laws, flux. Meaning of Boundary conditions.

**Bounded spatial domain. Boundary conditions, separation of variables, solution of initial-value problem by Fourier series.**

Part II: PDEs in One Space Dimension

Part II: PDEs in One Space Dimension

Infinite spatial domain. Fourier transforms and its properties: derivatives, convolution. Use of contour integration. Introduction and use of the delta "function" as a point source and its Fourier transform. Heaviside functions.

Application of Fourier transforms to solve PDE problems on an infinite domain, including diffusion, wave and Laplace's equation.

Solution of wave equation by separation of variables and Fourier series. Solution of the initial-value problem for an infinite string by Fourier transforms. D'Alembert's solution, interpretation as travelling waves.

Linear and Nonlinear first-order PDE's. Characteristic methods for solution of initial-value problems. Shock waves, expansion fans. Derivation of traffic flow models and solution.

**Time-independent solutions of the diffusion equation. Laplace's equation. Solution in Cartesian and plane polar coordinates by separation of variables and Fourier series.**

Part III: PDEs in two and three space dimensions.

Part III: PDEs in two and three space dimensions.

Separation of variables in cylindrical and spheical coordinates. Eigenfunction expansions. Bessel functions and Legendre functions. Laplace and diffusion equations in cylindrical and spherical coordinates.

Properties of harmonic functions. Maximum Principle for Laplace equation.

## Reading and References

There are many books covering the subject. Reecommended texts to compliment the course are

- S. J. Farlow,
*Partial Differential Equations for Scientists and Engineers*, Dover Publications 1993, QA377 FAR (Queen's Building) approx. £14. - R. Haberman,
*Applied Partial Differential Equations,*Pearson/Prentice-Hall 2004. QA377 HAB (Queen's Building) approx £60

Farlow is cheap, written in a straightforward and clear style, and covers most ot the course, rather briefly. Haberman has much more detail, and includes more up to date and advanced material beyond the scope of this unit, but would be useful to students planning to take more advanced courses in applied mathematics.

**Unit code:** MATH20402

**Level of study:** I/5

**Credit points:** 20

**Teaching block (weeks):** 2 (13-24)

**Lecturer:** Professor Jens Eggers

## Pre-requisites

MATH20901 Multivariable Calculus and MATH20001 Methods of Complex Functions

## Co-requisites

None

## Methods of teaching

Three lectures and one problems class per week. Regular problem sheets will be distributed which will test the students' understanding of the material through a variety of problems ranging from elementary to difficult. Set questions will be marked promptly and returned with comments. Full solutions of all problems will be distributed.Problems classes will go through examples that compliment both the lectures and the worksheets.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in May/June

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.