Linear Partial Differential Equations (MATH0070)

Partial differential equations are important in many fields of mathematics and are the essential language of physical applied mathematics, where they are used to model phenomena including wave propagation, heat flow, as well as soap films and soap bubbles. This course provides an introduction to some of the mathematical techniques needed to study linear partial differential equations and serves as a foundation for more advanced work on nonlinear PDE and PDE on manifolds. Tools such as the theory of distributions and the Fourier transform are of wide applicability beyond the theory of PDEs and are of great interest in their own right. The objectives of the course are to introduce test functions (smooth functions with compact support) and distributions, the Schwartz space, and then study the Fourier transform in the Schwartz space and L2. Using these tools we shall then be able to write down fundamental solutions for a large class of linear differential operators and will be able to study the qualitative differences between elliptic, parabolic and hyperbolic partial differential equations. The Fourier transform gives access to the simplest L2-based Sobolev spaces and allows us to give basic versions of elliptic regularity for constant-coefficient operators. The course will conclude with a detailed study of harmonic functions, emphasising the parallels with complex function theory and will use Perron¿s method to study the Dirichlet problem for domains in Rn..

Throughout the last fifty years fast numerical methods have revolutionised the applications of mathematics, making it possible to simulate huge problems in...

Computational Methods and Programming (MATH0058)

Throughout the last fifty years fast numerical methods have revolutionised the applications of mathematics, making it possible to simulate huge problems in science and engineering that are intractable by analytic methods. This course is designed to give an overview of the design, analysis and implementation of the most fundamental numerical techniques in numerical linear algebra, the interpolation of functions, and the evaluation of integrals. The course will be heavily based on Python as programming language. While we present many Python examples throughout the course, students are strongly advised to have either some previous programming experience in any language or to have taken MATH0011, which includes Python programming.