Note: the module details from the 2023/24 academic year can be found here.

Lecturer: Marie-Therese Wolfram

Term(s): Term 1

Status for Mathematics students: Core for Maths

Commitment: 30 one hour lectures plus assignments

Assessment: 85% by 2 hour examination, 15% coursework

Formal registration prerequisites: None

Assumed knowledge: We will use facts and techniques of the following topics, knowledge of which may have been obtained in the modules stated below or otherwise: Solutions to linear first and second order ODEs, separation of variables techniques - MA146 Methods of Mathematical Modelling 1 or MA147 Mathematical Methods and Modelling 1 or MA133 Differential Equations.

Differentiation along a curve, Leibniz' rule and the divergence theorem (MA144 Methods of Mathematical Modelling 2 or MA145 Mathematical Methods and Modelling 2). Diagonalisation of symmetric matrices (MA150 Algebra 2 or MA149 Linear Algebra or MA148 Vectors and Matrices).

Synergies: Sometimes, solving partial differential equations comes down to having to solve ordinary differential equations, and then computational techniques as discussed in MA2K4 Numerical Methods and Computing can prove useful. Eigenvalue problems for second order ordinary differential equations are investigated in MA254 Theory of ODEs in more depth. There are many applications of partial differential equations, for example, those discussed in PX263 Electromagnetic Theory and Optics and PX264 Physics of Fluids.

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA3D1 Fluid Dynamics
• MA3J4 Mathematical Modelling with PDE
• MA3G1 Theory of Partial Differential Equations
• MA390 Topics in Mathematical Biology
• MA3H0 Numerical Analysis and PDEs
• MA4M2 Mathematics of Inverse Problems
• MA4L0 Advanced Topics in Fluids
• MA482 Stochastic Analysis
• MA4L3 Large Deviation Theory
• MA4M9 Mathematics of Neuronal Networks

Content: The theory of partial differential equations (PDE) is important both in pure and applied mathematics. On the one hand they are used to mathematically formulate many phenomena from the natural sciences (electromagnetism, Maxwell's equations) or social sciences (financial markets, Black-Scholes model). On the other hand since the pioneering work on surfaces and manifolds by Gauss and Riemann partial differential equations have been at the centre of many important developments on other areas of mathematics (geometry, Poincare-conjecture).

Subject of the module are four significant partial differential equations (PDEs) which feature as basic components in many applications: The transport equation, the wave equation, the heat equation, and the Laplace equation. We will discuss the qualitative behaviour of solutions and, thus, be able to classify the most important partial differential equations into elliptic, parabolic, and hyperbolic type. Possible initial and boundary conditions and their impact on the solutions will be investigated. Solution techniques comprise the method of characteristics, Green's functions, and Fourier series.

Aims: To introduce the basic phenomenology of partial differential equations and their solutions. To construct solutions using classical methods.

Objectives: At the end, you will be familiar with the notion of well-posed PDE problems and have an idea what kind of initial or boundary conditions may be imposed for this purpose. You will have studied some techniques which enable you to solve some simple PDE problems. You will also understand that properties of solutions to PDEs sensitively depend on the its type.

Books:
A script based on the lecturer's notes will be provided. For further reading you may find the following books useful (sections of relevance will be pointed out in the script or in the lectures):

S Salsa, Partial Differential Equations in Action, From Modelling to Theory. Springer (2008)
A Tveito and R Winther, Introduction to Partial Differential Equations, A Computational Approach. Springer TAM 29 (2005)
W Strauss, Partial Differential Equations, An Introduction. John Wiley (1992)
JD Logan, Applied Partial Differential Equations. 2nd edt. Springer (2004)
MP Coleman, An Introduction to Partial Differential Equations with MATLAB. Chapman and Hall (2005)
M Renardy and RC Rogers, An Introduction to Partial Differential Equations, Springer TAM 13 (2004)
LC Evans, Partial Differential Equations. 2nd edt. American Mathematical Society GMS 19 (2010)

 

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