# IST - Partial Differential Equations : From Theory to Computation (2019)

## Speakers and Syllabus

Many physical phenomena and engineering problems are governed by Partial Differential Equations(PDEs). It is therefore important to understand the mathematical aspects of PDEs and their solution process. The theory of PDEs mainly deals with the existence and uniqueness of the solution and itsstability and regularity. The solution process includes methodologies to find solutions, either in closed form or by an approximation. Proposed instructional school aims to provide the participants a taste of classical theory of second order PDEs and to introduce them the advanced PDEs through variational approach. Furthermore, the school aims to demonstrate numerical 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 approaches 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 K. Sreenadh, IIT Delhi 6 Classical PDEs: 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. T. Muthukumar, IIT Kanpur 6 Variational Approach for PDEs: 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. Rajendra K. Ray, IIT Mandi 6 Finite Difference Methods: Analysis (consistency, convergence, stability and error estimate) of finite difference schemes for  Heat equation (1D and 2D)  Wave equation (1D and 2D) Laplace equation(1D and 2D) Some nonlinear equations like semi-linear heat, Burgers will also be discussed Highlighting new phenomenon discovered through numerics. Saumya Bajpai, IIT Goa 6 Finite Element Methods: Introduction to finite element methods, construction of finite element spaces, Interpolation theory, applications to elliptic and parabolic PDEs. Error analysis. implementation aspects.

References:
1. L.C. Evans, Partial Differential Equations, AMS.
2. J.C. Strikwerda, Finite difference schemes and PDEs, SIAM
3. J.N. Reddy, An introduction to Finite Element Method, McGraw Hill.
4. S. Larsson and V. Thomee. Partial differential equations with numerical methods, Springer.

## Time Table

 Day Date Lecture 1 (9.30to11.00) Tea (11.05 to11.25) Lecture 2 (11.30 to1.00) Lunch (1.05 to2.25) Tutorial (2.30 to 3.30) Tea (3.35 to3.55) Tutorial (4.00 to5.00) Snacks (5.05 to5.30) (name of the speaker) (name of the speaker) (name of the speaker + tutors) (name of the speaker + tutors) Mon 03/06/2019 KS KS KS, RR, AK TM, RR, SJ Tues 04/06/2019 KS KS KS, RR, AK TM, RR, SJ Wed 05/06/2019 KS TM KS, RR, AK TM, RR, SJ Thu 06/06/2019 KS TM KS, RR, AK TM, RR, SJ Fri 07/06/2019 TM TM TM, RR, SJ TM, RR, AK Sat 08/06/2019 TM TM TM, RR, SJ TM, RR, AK SUNDAY : OFF Mon 10/06/2019 RR RR RR, AK, SJ RR, AK, SJ Tues 11/06/2019 RR RR RR, AK, SJ RR, AK, SJ Wed 12/06/2019 SB RR SB, TM, AK SB, TM, AK Thu 13/06/2019 SB RR SB, RR, SJ SB, RR, SJ Fri 14/06/2019 SB SB SB, RR, AK SB, RR, AK Sat 15/06/2019 SB SB SB, RR, SJ SB, RR, SJ

KS: K. Sreenadh
TM: T. Muthukumar
RR: Rajendra K. Ray
SB: Saumya Bajpai
SB: Subit K Jain
AK: Atendra Kumar

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