# Mathematics / Applied Mathematics MSc

# 5E: Partial Differential Equations MATHS5076

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**20**Level:**Level 5 (SCQF level 11)**Typically Offered:**Semester 1**Available to Visiting Students:**No**Available to Erasmus Students:**No

#### Short Description

The course introduces basic classes of linear and nonlinear partial differential equations. Different methods of solving the equations are considered. Relevant applications where these equations may arise are also

discussed.

#### Timetable

34 x 1 hr lectures and 12 x 1 hr tutorials in a semester

#### Requirements of Entry

Mandatory Entry Requirements

3H Mathematical Methods (MATHS4075)

Recommended Entry Requirements

#### Excluded Courses

4H: Partial Differential Equations (MATHS4110)

#### Assessment

Assessment

90% Examination, 10% Coursework.

Reassessment

Resit opportunities available for MSc students.

**Main Assessment In:** April/May

#### Course Aims

The main aim of this course is to introduce concepts and techniques for the solution of partial differential equations (PDEs). The second-order linear PDEs of elliptic, hyperbolic and parabolic types are considered as well as some nonlinear PDEs. Some particular aims are to present the method of characteristics for hyperbolic PDEs, separation of variables, the method of Green's functions and transform methods.

#### Intended Learning Outcomes of Course

By the end of this course students will be able to:

a) Formulate and solve the Monge equations for a first order quasi-linear PDE in 2D;

b) Calculate the earliest time at which solutions become multi-valued;

c) Construct shock solutions and study their behaviour, in particular calculate shock speeds;

d) Apply the theory to canonical examples in traffic flow;

e) State uniqueness and existence theorems for different types of initial and boundary value problems;

f) Apply separation of variables to construct solutions to Laplace's and Poisson's equation by making use of special functions and their properties, including associated Legendre polynomials and Bessel functions;

g) Apply the Fourier transform method to solve initial value problems for the heat equation;

h) Derive and the Cole-Hopf transformation appropriate to solving a Burgers' equation and use it to derive exact solutions or solve IVPs for this equation;

i) Formulate and solve initial value problems for the wave equation in 1D, 2D and 3D;

j) Derive and apply Green's functions for second order linear PDEs, including Laplace's equation.

#### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.