
Module Details M Level Second Semester 15
Aims This module (divided into 5 parts) introduces a range of analytical and numerical methods for partial differential equations arising in many areas of applied mathematics. It focuses initially on advanced analytical techniques for solution of both elliptic and parabolic PDEs and qualitative features of the solutions. This is followed by an introduction to discretization methods and matrix analysis i.e. * Applied Analysis * Iterative Solution of Ax=b * Finite Difference Method for PDEs * Finite Element Method for PDEs The module is one of the few closely related to research work. Here the Green identities are used in Parts 1, 4, 5. The ability and skill of working with normal derivatives is emphasized in applying Green identities and in deriving a weak formulation for finite elements solution. In part 4, The derivation of an EulerLagrange equation in context of Calculus of Variations should be challenging and of interests to all students. The second part of a finite difference method and the third part of an iterative method are quite standard for numerical solution of PDEs. In contrast, the final part of using piecewise functions over triangles for working out the finite element equations should be both challenging and fun to a mathematician.