## MATH3414 Analytic Solutions of Partial Differential Equations

### 15 creditsClass Size: 100

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

MATH2360 or MATH2420 or equivalent.

This module is approved as an Elective

### 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 hyperbolic equations; b) classify a second order PDE as elliptic, parabolic or hyperbolic; c) use Green?s functions to solve elliptic equations; d) have a basic understanding of diffusion; e) obtain a priori bounds for reaction-diffusion equations.

### Syllabus

The majority of physical phenomena can be described by partial differential equations (e.g. the Navier-Stokes equation 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), describing steady-state phenomena and the diffusion / heat conduction equation describing the slow spread of concentration or heat. The topics covered are: First order PDEs. Semi-linear and quasi-linear PDEs; method of characteristics. Characteristics crossing. Second order PDEs. Classification and standard forms. Elliptic equations: weak and strong minimum and maximum principles; Green's functions. Parabolic equations: exemplified by solutions of the diffusion equation. Bounds on solutions of reaction-diffusion equations.

### Teaching methods

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

Lectures: 26 hours. 7 examples classes.

### Methods of assessment

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

One 3 hour examination at end of semester (100%).