Partial Differential Equations and Integral Transforms
MODULE CODE
CREDIT VALUE
DELIVERY
Module Aims
Aim 1
To give students a range of techniques for solving partial differential equations analytically.
Aim 2
To explain the significance of particular partial differential equations in solving 'real-world' problems.
Aim 3
To present to students a coherent development of the theory of integral transforms.
Aim 4
To develop students’ skills in handling the techniques of integral transforms.
Aim 5
To develop students’ skills in applying integral transforms to solving partial differential equations.
Module Content
Recap: Fourier series. Sturm-Liouville problems.
Introduction to PDEs: Order. Classification of linear, non-linear etc. Classification of elliptic, hyperbolic and parabolic.
First order PDEs: Functions of integration. Method of characteristics.
Second Order Parabolic PDEs: The 1-D heat equation. Separation of variables in Cartesian coordinates. Homogeneous and non-homogeneous Dirichlet boundary conditions. Existence and uniqueness of solutions.
Second Order Hyperbolic PDEs: The 1-D wave equation. Separation of variables in Cartesian Coordinates. D’Alembert’s solution. Existence and uniqueness of solutions.
Second Order Elliptic PDEs: Laplace’s equation in 2-D Cartesian and polar coordinates. Laplace’s equation in 3-D Cartesian and spherical polar coordinates. Legendre polynomials and their properties, including orthogonality and generating functions.
Fourier transforms: Fourier Integral. Derivation of Fourier transform. Definition of Fourier transform and inverse Fourier transform. Properties of the Fourier transform (including linearity, shifting, scaling etc.). Fourier transform of the delta function, exponential, Gaussian and unit step function. Convolution. Using Fourier transforms to solve PDEs (including the heat equation, wave equation and Laplace’s equation).
Laplace transforms: Definition of the Laplace transform and inverse Laplace transform. Existence of the Laplace transform. Properties of the Laplace transform (including linearity, shifting properties, scaling, multiplication, division by t). Convolution. Laplace transform of the unit step function and periodic functions. Using Laplace transforms to solve PDEs (including the heat equation and the wave equation).
Visualisation and interpretation of solutions to PDEs.
Learning Outcomes
On successful completion of this module, a student will be able to:
Teaching Methods
The module will be delivered on campus, with weekly lecture and tutorial sessions.
Printed notes will be given ahead of time for each section of the course, to support and enhance students’ preparation and engagement during class sessions. Lectures will follow the notes, with discussions of the main theoretical topics, and study of examples of the applications of the theory. There will be a strong emphasis on student involvement in discussions in lectures, to encourage a more active approach to learning the material, and to allow the delivery to be tailored to build on the students’ current understanding.
Regular formative work in tutorial sessions will allow students to internalise the mathematical ideas and methods developed in the lectures, and lead to the development of problem-solving skills. This formative work will also feed back into the delivery of lectures and tutorials.
Assessment Methods
The module is assessed through a portfolio of exercises and an examination.
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Compulsory
Optional
