# MA250 Introduction to Partial Differential Equations

**Lecturer:** Hugo van den Berg

**Term(s):** Term 1 (weeks 6-10), Term 2 (weeks 15-19)

**Status for Mathematics students:** List A

**Commitment:** 30 lectures

**Assessment:** 2 hour exam.

**Prerequisites:** Analytical knowledge as obtained in MA131 Analysis is required. Some techniques on ordinary differential equations as seen in MA133 Differential Equations, on uniform convergence of series as taught in MA244 Analysis III, and on the divergence theorem as presented in MA231 Vector Analysis will be needed and only briefly introduced in the lectures.

**Leads To: **MA254 Theory of ODEs, MA3G7 Functional Analysis I, MA3D1 Fluid Dynamics, MA3G1 Theory of Partial Differential Equations, MA3G8 Functional Analysis II, MA3H0 Numerical Analysis and PDEs, MA3H7 Control Theory, MA3J4 Mathematical modelling with PDE and MA4L3 Large Deviation theory.

**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).