Section: Course Materials | B5.2 Applied Partial Differential Equations (2025-26)

Lecturer: Profile: Peter Howell

Course information

General prerequisites:

Differential Equations 1 and Differential Equations 2 from Part A are prerequisites, and the material in these courses will be assumed to be known. Calculus of Variations and Fluids and Waves from Part A are desirable but not essential. Integral Transforms from Part A is strongly desirable.

Course term: Michaelmas

Course lecture information: 16 lectures

Course weight: 1

Course level: H

Assessment type: Written Examination

Course overview:

This course continues the Part A Differential Equations courses. In particular, first-order conservation laws are solved and the idea of a shock is introduced; general nonlinear and quasi-linear first-order partial differential equations are solved, the classification of second-order partial differential equations is extended to systems, with hyperbolic systems being solved by characteristic variables. Then Riemann's function, Green's function and similarity variable methods are demonstrated.

Learning outcomes:

Students will know a range of techniques to characterise and solve PDEs including non-linear first-order systems, and second-order. They will be able to demonstrate various principles for solving PDEs including the method of characteristics, Green's functions, similarity solutions and Riemann functions.

Course synopsis:

First-order equations; applications. Characteristics, domain of definition. [2 lectures]
Weak solutions, conservation laws, shocks. [2 lectures]
Non-linear equations; Charpit's equations; eikonal equation. [3 lectures]
Systems of partial differential equations, characteristics. Shocks; weak solutions. [3 lectures]
2nd order semilinear equations. Hyperbolic equations, Riemann functions. [2 lectures]
Elliptic equations, parabolic equations. Well-posed problems, Green's function, similarity solutions. [4 lectures]

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Section outline