## MATH3414 Analytic Solutions of Partial Differential Equations

### 15 creditsClass Size: 229

Module manager: Dr Benjamin Sharp
Email: B.G.Sharp@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2021/22

### Pre-requisite qualifications

(MATH1012 or MATH1400) and MATH2365, or equivalent.

This module is not approved as a discovery module

### Module summary

The majority of physical phenomena can be described by partial differential equations (e.g. the Navier-Stokes equations of fluid dynamics, Maxwell's equations of electromagnetism).This module considers the properties of, and analytical methods of solution for some of the most common first and second order PDEs of Mathematical Physics. In particular, we shall look in detail at elliptic equations (Laplace's equation), parabolic equations (heat equations) and hyperbolic equations (wave equations), and discuss their physical interpretation.

### Objectives

To provide an understanding of, and methods of solution for, the most important types of partial differential equations that arise in Mathematical Physics.

On completion of this module, students should be able to:
a) use the method of characteristics to solve first-order partial differential equations and systems of first-order partial differential equations;
b) classify a second order PDE as elliptic, parabolic or hyperbolic;
c) use Fourier transform and separation of variables for suitable initial-boundary value problems;
d) have a basic understanding of diffusion and waves;
e) use Green's functions to solve elliptic equations.

### Syllabus

- First order PDEs.
- Semi-linear and quasi-linear PDEs; method of characteristics.
- Second order PDEs; Classification and standard forms.
- Separation of variables: Laplace's, heat and wave equations.
- Fourier transform and generalised functions; Dirac's delta function, its properties and its physical interpretation.
- Parabolic equations: exemplified by solutions of the diffusion equation.
- Hyperbolic equations: exemplified by the d' Alembert's formula for the solution of the wave equation; domains of dependence and influence and their physical interpretation.
- Elliptic equations: maximum / minimum and mean value principles; boundary value problems, including Green's function techniques.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Lecture 17 1.00 17.00 Private study hours 133.00 Total Contact hours 17.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated