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

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Venue: IIT,  Mandi, Himachal Pradesh
Date:  3rd, Jun 2019 to 15th, Jun 2019

 

School Convener(s)

Name Dr. Rajendra K Ray Dr. Prasanth P. Jose
Mailing Address Associate Professor
School of Basic Sciences
Indian Institute of Technology Mandi
Assistant Professor
School of Basic Sciences
Indian Institute of Technology Mandi

 

 


The list of speakers and course associates is given below

sr.n Name Affiliation
1. K. Sreenadh IIT Delhi
2. T. Muthukumar, IIT Kanpur
3. Rajendra K. Ray IIT Mandi
4. Saumya Bajpai IIT Goa
5. Sudeb Majee IIT Mandi
6. Manotosh Kumbhakar  IIT Mandi

 

 


 Syllabus:

 Dr. T. Muthukumar from IIT Kanpur delivered six lectures of 90 minutes each on Variational Approach for PDEs. First, he gave a quick review of distributions, mainly the concept of weak derivative, Fourier transform and convolution. After that he discuss the Sobolev spaces; mainly H^1 space and important inequalities, imbedding and Trace theorems. Then, variational approach to elliptic PDEs, was discussed along with existence, uniqueness, regularity results and eigenvalue problems. The following topics were discussed during the lectures and corresponding problems were solved during tutorial classes;

1 Theory of Distribution

  • 1.1 Need to relax the notion of Solution of a DE
  • 1.2 Space of Test Functions
    • 1.2.1 Smooth Non-Analytic (Bump) Functions
    • 1.2.2 Mollifiers
    • 1.2.3 Inductive Limit Topology on Cc  ()
    • 1.2.4 Regularization and Cut-off Technique
  • 1.3 Space of Distributions
    • 1.3.1 Functions as Distributions
    • 1.3.2 Measures as Distributions
    • 1.3.3 Multipole Distributions
    • 1.3.4 Infinite Order Distributions
    • 1.3.5 Topology on Distributions
  • 1.4 Operations with Distributions
    • 1.4.1 Differentiation

2 Sobolev Spaces

  • 2.1 Wk;p(Ω) Spaces
  • 2.2 Smooth Approximations
  • 2.3 Characterisation of W k , p ()
    • 2.3.1 Poincar e Inequality
  • 2.4 Extension Operators
  • 2.5 Topological Dual of Sobolev Spaces
  • 2.6 Fractional Order Sobolev Space
  • 2.7 Imbedding Results
    • 2.7.1 Sobolev Inequality (1 ≤ p < n)
    • 2.7.2 Equality case, p = n
    • 2.7.3 Morrey’s Inequality (n < p <  )
    • 2.7.4 Generalised Sobolev Imbedding
    • 2.7.5 Compact Imbedding
  • 2.8 Trace Theory
  • 3 Variational Inequality
  • 4 Second Order Elliptic PDE
    • 4.1 Homogeneous Dirichlet Problem
    • 4.2 Inhomogeneous Dirichlet problem
    • 4.3 Linear Elliptic Operators
    • 4.4 Periodic Boundary Conditions
    • 4.5 Coercive Neumann Problem
    • 4.6 Semi-coercive Neumann Problem
    • 4.7 Mixed and Robin Problem
    Appendices

Prof. K. Sreenadh from IIT Delhi delivered six lectures of 90 minutes each on Classical PDEs. He basically discussed the Laplace equation, Harmonic functions, maximum principle, fundamental solution, Green’s function and the solvability of Dirichlet problem. After that Prof. Sreenadh discussed the heat equation in detail with weak and strong maximum principles for the heat equation, fundamental solution, existence and uniqueness for the initial value problem. The following topics were discussed during the lectures and corresponding problems were solved during tutorial classes;

1 Elliptic Equations

  • 1.1 Potential Theory
    • 1.1.1 Fundamental Solution
    • 1.1.2 Dirichlet Problem
    • 1.1.3 Properties of Harmonic Function
    • 1.1.4 Regularization and Cut-off Technique
  • 1.2 Maximum Principles
  • 1.3 Existence of Harmonic Functions: Perron’s Method
  • 1.4 Eigenvalue Problems
  • 1.5 Problems

Dr. Rajendra K. Ray from IIT Mandi delivered six lectures of 90 minutes each on Finite Difference Methods. He first discussed the basics of numerical computations and different examples in the field of fluid dynamics, solid mechanics, and other related areas to grow the interest among the participants in this topic. After that the basic approaches of finite difference approximations of first and second order derivatives were discussed. Then those approximations were used to discretize some differential equations. After that the consistency, convergence and stability for general finite difference methods were discussed. Finally, he discussed the stability and error
estimates of different finite difference methods and its usefulness to discretize the following equation:

  • Heat equation (1D and 2D)
  • Wave equation (1D and 2D)
  • Laplace equation (1D and 2D)

After discussing the above three equations representing three difference classes, some nonlinear equations like semi-linear heat equation, Burger’s equation and unsteady convection-diffusion equation were also discussed. At the end some advance numerical methods like higher order accurate (HOC) finite difference scheme and advance iterative solver like SOR, Bi-Conjugate gradient algorithm was discussed and highlighted some new phenomenon discovered through numerics from the speaker’s own research work. The following topics were discussed during the lectures and corresponding problems were solved during tutorial/computer lab classes;

1 Introduction

  • 1.1 Characteristics of Numerical Computing
  • 1.2 Partial Differential Equations and their Classifications
    • 1.2.1 Physical Classification
    • 1.2.2 Mathematical Classification

2 Finite Difference approximations for Partial Derivatives

  • 2.1 Simple Method
  • 2.2 General Method
  • 2.3 Polynomial Fitting
  • 2.4 Non-uniform Mesh

3 Accuracy of Finite Difference Solution

  • 3.1 Consistency
  • 3.2 Stability
  • 3.3 Convergence

4 Finite Difference Schemes

  • 4.1 Heat equation
  • 4.2 Wave equation
  • 4.3 Laplace equation

5 Some Nonlinear Equations

5.1 Convection Diffusion Equation
5.2 Burgers Equation

Dr. Saumya Bajpai from IIT Goa delivered six lectures of 90 minutes each on Finite Element Methods. First she introduce the basic idea behind the development of finite element methods and the construction of finite element spaces. After that she discussed the interpolation theory and the applications of finite element method to discretize and solve elliptic and parabolic PDEs. Finally, she discussed the error
analysis and some implementation aspects. The following topics were discussed during the lectures and corresponding problems were solved during tutorial/computer lab classes;

1 Introduction

  • 1.1 Background
  • 1.2 Finite Difference Methods (FDM)
  • 1.3 Finite Element Methods (FEM)

2 Theory of Distribution and Sobolev Spaces

  • 2.1 Review of L2- Space
  • 2.2 A Quick Tour to the Theory of Distributions
  • 2.3 Elements of Sobolev Spaces

3 Abstract Elliptic Theory
3.1 Abstract Variational Formulation
3.2 Some Examples

4 Elliptic Equations
4.1 Finite Dimensional Approximation to the Abstract Variational Problems
4.2 Examples
4.3 Computational Issues
4.4 Adaptive Methods

Bibliography

Dr. Manotosh Kumbhakar from IIT Mandi assisted in fifteen tutorials of one hour each. Besides this he also attended all the lectures and helped participants with their doubts after the classes and tutorials. The participants
connected with Dr. Manotosh quite naturally and his presence was of a great help to the participants. They learnt a lot from him. He added a lot of value to the IST.

Mr. Sudeb Majee from IIT Mandi assisted in fifteen tutorials of one hour each. Besides this he also attended all the lectures and helped participants with their doubts after the classes and tutorials. The participants connected with Mr. Sudeb quite naturally and his presence was of a great help to the participants. They learnt a lot from him. He added a lot of value to the IST.


 Time Table
Tea and Snack Breaks: 11.00 - 11.30 and 4.00 - 4.30
Lunch Break: 1.00 - 2.00

  9.30 - 11.00
Lecture 1
11.30 - 1.00
Lecture 2
2.00 - 3.00
Tutorial 1
3.00 - 4.00
Tutorial 2
June 3 TM TM TM, RR, MK TM, RR, SM
June 4 TM TM TM, RR, MK TM, RR, SM
June 5 TM TM TM, RR, MK TM, RR, SM
June 6 KS KS KS, RR, MK KS, RR, SM
June 7 KS KS KS, RR, SM KS, RR, MK
June 8 KS KS KS, RR, SM KS, RR, MK
June 9 RR RR RR, MK, SM RR, MK, SM
June 11 RR RR RR, MK, SM RR, MK, SM
June 12 SB RR RR, MK, SM SB, RR, MK
June 13 SB RR SB, RR, SM RR, MK, SM
June 14 SB SB SB, RR, MK SB, RR, MK
June 15 SB SB SB, RR, SM SB, RR, SM

 

TM: T. Muthukumar
KS: K. Sreenadh
RR: Rajendra K. Ray
SB: Saumya Bajpai
SM: Sudeb Majee
MK: Manotosh Kumbhakar

 


List of actual Participants

Sr. SID Full Name Gender Affiliation Position in College/ University University/Institute M.Sc./ M.A. Year of Passing M.Sc./ M.A Ph.D. Degree Date
1 24581 Mr Akmal Raza Male Jamia Millia Islamia PhD (student) Jamia Millia Islamia 2014  
2 24660 Mr Kumar Rajeev Ranjan Male NIT Patna Ph.D IIT Roorkee 2012  
3 24848 Mr. Yuvaraj P Male Bharthiyar University Asst.Professor Bharthiyar University 2011  
4 24901 Mr. Akshay Kumar Male University of Hyderabad Ph.D Banaras Hindu University 2017  
5 25200 Dr. Jitender Singh Male Guru Nanak Dev University Amritsar Assisatnt Professor Panjab University Chandigarh 2002 7/4/2008
6 25322 Mr. Saurabh Kumar Male The LNM Institute of Information Techonolgy Research Scholor C.C.S University Meerut 2016  
7 26399 Mr. Guru Prasad N S Male National Institute Of Engineering, Mysuru Assistant Professor Mysore University/ M.Sc. 2015  
8 26748 Ms. Pratima Tiwari Female IIT BHU Research scholar NIT Jamshedpur 2017  
9 26806 Ms. Swati Yadav Female IIT (BHU), Varanasi PhD university of Luchnow 2015  
10 26889 Mr. S.Shankar Narayan Male CMR Institute of Technology Asst.Professor Bangalore University 2016  
11 26959 Ms Nandhini Rajendran Female PSG College of Arts & Science Assistant Professor PSG College of Arts & Science, M.Sc 2016  
12 26979 Ms Anu Rani Female Bennett university PhD SLIET, Longowal(M.Sc.) 2017  
13 27068 Mr. Mahesh Udupa Male Dayananda Sagar University Junior Research Fellow St. Joseph's College 2018  
14 27082 Mr. Tamil Selvan T Male PSG College of Arts & Science Assistant Professor PSG College of Arts & Science 2016  
15 27163 Mr. Soura Sana Male IIT Bhubaneswar phD The University of Burdwan 2015  
16 27199 Dr. Subburayan Veerasamy Male SRM Institute of Science and Technology, Assistant Professor M.Sc. 2009 25/11/2013
17 27336 Dr. Rajendra Kumar Bairwa Male Department of Mathematics, University of Rajasthan , Jaipur Assistant Professor University of Rajasthan, Jaipur 2007 10/11/2016
18 27350 Ms S Yaazhini Female Anna University PhD scholar Central University Of Tamilnadu 2016  
19 27468 Dr Gunvant Achutrao Birajdar Male Tata Institute Of Social Sciences Assistant Professor Dr B A M University Aurangabad 2008 18/1/2014
20 27541 Ms Pavithra Kumar Female Anna University, CEG Campus PH.D RIASM University Of Madras 2014  
21 27570 Ms Soumyarani Mishra Female S'O'A University PhD STUDENT CET, BBSR 2017  
22 27613 Mrs Sonali Mandar Narsale Female Modern education society's college of engineering,pune01 Assistant Professor Pune 2006  
23 27647 Mr. Aniruddha Palit Male Surya Sen Mahavidyalaya Assistant Professor University Of North Bengal 2004 14/11/2017
24 27686 Mr. Divay Divay Male IIT Delhi PhD Student IIT Madras 2017  
25 27691 Ms. Manisha Chowdhury Female Indian Institute of Technology Kanpur, Kanpur PhD IIT Kanpur 2014  
26 27751 Ms. Sangita Dey Female IIT Kanpur PHD IIT Madras 2013  
27 27755 Dr. Tarul Garg Female The Northcap University Assistant Professor IIT Roorkee 2013 6/9/2018
28 27757 Ms Megha P Female National Institute of Technology Karnataka PhD Mahatma Gandhi College Iritty, Kannur. 2016  
29 27830 Dr Halle Dattu Male Rajiv Gandhi University of Knowledge Technologies Asst prof Pondicherry University 2009 12/12/2016
30 27850 Ms. Tanvi Wadhawan Female IIT Delhi PhD. IIT Roorkee 2018 27/12/2018
31 27987 Dr Deepak Grover Male Lovely Professional University Phagwara Assistant professor Guru Jambheshwar University Hisar 2006 28/3/2014
32 25055 Ms. Ravina Shokeen Female The LNM Institute of Information
Technology, Jaipur
       
33   Dipayan Sanpui Male Jadavpur University, West-Bengal        
34   Utkarsh Godwal Male U.I.E.T Kurukshetra University, Haryana        

 


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