B5.2 Applied Partial Differential Equations - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Prof. Andreas Muench
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: 
1.00 unit(s)
Course Level: 
H

Assessment type:

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]

Reading List: 
  1. J. R. Ockendon, S. D. Howison, A. A. Lacey and A. B. Movchan, Applied Partial Differential Equations (revised edition, Oxford University Press, Oxford, 2003).
  2. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations (Springer-Verlag, New York, 2004).
  3. J. P. Keener, Principles of Applied Mathematics: Transformation and Approximation (revised edition, Perseus Books, Cambridge, Mass., 2000).