MAGIC058: Theory of Partial Differential Equations

Course details

A core MAGIC course


Spring 2020
Monday, January 20th to Friday, March 27th


Live lecture hours
Recorded lecture hours
Total advised study hours


15:05 - 15:55 (UK)
15:05 - 15:55 (UK)


There will be no lectures on 20th and 21st January 2020. The first lecture for the course will be on Monday 27th January 2020.


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.


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

Related courses


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.


  • AT

    Dr Alice Thompson

    University of Manchester


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, )


The assessment for this course will be released on Monday 20th April 2020 at 00:00 and is due in before Monday 4th May 2020 at 11:00.

The assessment for this course will be via a single take-home paper with 2 weeks to complete and submit online. There will be five questions, which may be of different lengths. The questions total to 80 marks, and you will need to gain at least 40 marks to pass.

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


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