Revision
Main content blocks
- 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]
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]
Section outline
-
Revision lecture
- 9-10am on Wednesday in week 2 (6 May) in L4
- I will go through the 2025 B5.2 exam paper.
Consultation sessions
- 9-10am on Wednesday in weeks 3 and 4 (13 and 20 May) in L4
- I am happy to discuss any questions on the lecture notes, problem sheets or past papers.
- Please email me at least 24 hours in advance with any questions.
- I will post a list of questions/topics to be discussed here.