ICMS/EMS Postgraduate Courses
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Topics in computational mathematics (I)
Solving ODEs and simple reaction-diffusion equations numerically
Dr Dugald Duncan (Heriot-Watt)
The numerical solution of ordinary differential equations is a key tool for finding quantitative information from mathematical models of a multitude of phenomena and processes. A little knowledge and thought about the ODE problems being tackled, leading to a sensible choice of solution technique, can save much wasted effort and can mean the difference between getting something useful or not. ODE methods can also be an efficient way to solve some types of reaction-diffusion equations. This talk aims to provide nonspecialists with a basic knowledge of ODE solvers to help them avoid the most common pitfalls.
Topics in computational mathematics (II)
Subspace decompositions and applications in numerical analysis
Professor Mark Ainsworth (Strathclyde)
Domain decomposition and multigrid methods represent fundamental techniques for the solution of problems arising in numerical analysis of partial differential equations and boundary integral equations. The past decade has seen the emergence of an elegant abstract framework for the analysis of such methods. The idea is based on splitting or decomposing the underlying solution space, and replacing the original (large) problem by a sequence of smaller problems posed on the subspaces. Examples of simple numerical methods that fit into this abstract framework include the Jacobi and Gauss-Seidel methods for iterative solution of matrix equations. The talk will give an overview of the abstract theory and show how it is applied to the analysis and derivation of some domain decomposition and multigrid methods.
Fourier analysis and PDE I & II
Dr Jim Wright (Edinburgh)
The interplay between Fourier analysis and differential equations has a long history which dates back to the beginning days of both subjects. One can use the Fourier transform (series) not only to solve specific partial differential equations (PDE) with explicit solutions, but also to prove general existence results (e.g. local solvability for constant coefficient PDE). In these lectures we will examine aspects of both types (specific/general) of uses of Fourier analysis in the theory of PDE and arrive at the fundamental functional equation for the Riemann zeta function, the location of whose zeros is now a $1,000,000 prize offered by the Clay Mathematics Institute!
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