# MAGIC058: Theory of Partial Differential Equations

## Course details

A core MAGIC course

### Semester

Spring 2023
Monday, January 23rd to Friday, March 31st

### Hours

Live lecture hours
10
Recorded lecture hours
10
80

Tuesdays
15:05 - 15:55

### Course forum

Visit the MAGIC058 forum

## Description

This course unit surveys analytical methods for linear and nonlinear first and second order PDEs.

We will discuss exact solutions, series solutions, Fourier transforms and nonlinear transforms, with a view to developing, applying and analysing a broad toolbox of methods to solve problems in applied mathematics.

### Prerequisites

No prior knowledge of PDEs is required, but experience with vector calculus and general undergraduate methods courses would be very helpful.

### Syllabus

1. Introduction
Basic notation. Classification of PDEs, examples of common PDEs.

2. First order PDEs
Construction of solutions to linear and nonlinear first order PDEs via method of characteristics. Application of Cauchy data. Examples of shock formation.

3. Linear second order PDEs
Characteristics of second order PDEs, classification, reduction to normal form. Well-posedness of boundary conditions.

4. Fourier series
Properties of full and half range Fourier series, and discussion of orthogonality. Use of separable solutions in constructing series solutions for appropriate BVPs and IVPs.

5. Sturm-Liouville systems
Definition of Sturm-Liouville systems, and proofs of main properties for regular S-L systems. Further discussion of applicability of series solutions.

6. Fourier transforms
Connection to Fourier series. Summary of main properties of Fourier transforms, and examples of calculation. Inversion via contour integration, and relation to convolution properties. Examples of solution of linear PDEs in infinite domains, and use of sine and cosine transforms in semi-infinite domains.

7. Nonlinear PDEs
Failure of superposition principle. Cole-Hopf transform for Burgers' equation. Examples of Backlund transforms. Inverse scattering methods for the KdV equation.

## Lecturer

• JR

### Dr Jonathan Rawlinson

University
University of Manchester

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

• An Introduction to Partial Differential Equations, 2nd. edition (M. Renardy and R.C. Rogers, book)

## Assessment

The assessment for this course will be released on Monday 1st May 2023 and is due in by Sunday 14th May 2023 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hoursâ€™ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.