# IST PDE : Theory and Computation (2018) - Speakers and Syllabus

PDEs arise naturally in many mathematical modelling describing physical phenomenon and engineering applications. It is therefore important to understand the PDEs and their solution process. The theory of PDEs consists of studying the existence, uniqueness of the solution and further to establish its stability and regularity. The solution process includes to develop methodologies to find solutions either in closed form or by an approximation. The instructional school aims to provide the participants a tour to classical theory of second order PDEs and to introduce to them the advanced PDEs through variational approach. Furthermore, the school aims to demonstrate numerical analysis and techniques to solve the PDEs, in particular, finite difference methods and finite element methods . The topics in the school are designed in such way that the participants get exposure to topics in both classical and modern approach in the theory and computation of PDEs which are also part of the curriculum in the universities and post graduate colleges.

Syllabus:

 Name of the Speaker with affiliation No. of Lectures Detailed Syllabus Sandeep Kunnath TIFR-CAM Bangalore 6 Classical PDE: Laplace equation: Harmonic functions, maximum principle, fundamental solution, Green’s function and the solvability of Dirichlet problem. Heat equation: Weak and strong maximum principles for the heat equation, fundamental solution, existence and uniqueness for the initial value problem. A. K. Nandakumaran Dept. Math. IISc Bangalore 6 Variational Approach for PDE:  Quick review of distributions, mainly the concept of weak derivative, Fourier transform and convolution. Sobolev spaces, mainly H^1 space and important inequalities, imbedding and Trace theorems. Variational approach to elliptic PDEs, existence, uniqueness, regularity results and eigenvalue problem A.S. Vasudevamurthy TIFR-CAM Bangalore 6 Finite Difference Methods: Analysis(consistency, stability and error estimate) of finite difference schemes for  Heat equation (1d in space)  Wave equation (1d in space) Laplace equation(1d and 2d) Some nonlinear equations like semi-linear heat, Burgers will also be discussed Highlighting new phenomenon discovered through numerics. Thirupathi Gudi Dept. Math. IISc Bangalore 6 Finite Element Methods: Introduction to finite element methods, construction of finite element spaces, Interpolation theory, applications to elliptic and parabolic PDEs. Error analysis and implementation aspects.

References:

1. L. C. Evans. Partial Differential Equations, AMS.
2. S. Kesavan. Topics in Functional Analysis and Applications, New Age Int. Publ.
3. P. G. Ciarlet. The finite element method for elliptic problems
4. S. Larsson and V. Thomee. Partial differential equations with numerical methods. Springer.
5. J. C. Strikwerda. Finite difference schemes and PDEs.

Time-Table (with initials of speakers and course associates for entire days of the programme ):