MAGIC058: Theory of Partial Differential Equations

Course details

A core MAGIC course


Spring 2021
Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th


Live lecture hours
Recorded lecture hours
Total advised study hours


09:05 - 09:55

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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.


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


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


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