# B5.2 Applied Partial Differential Equations - Material for the year 2020-2021

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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.

16 lectures

### Assessment type:

- Written Examination

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.

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.

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]

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