FINAL REPORT ON NATIONAL PROGRAMME ON DIFFERENTIAL EQUATIONS-THEORY ...

365 Pages·2012·42.88 MB·English

FINAL REPORT
ON
NATIONAL PROGRAMME ON DIFFERENTIAL
EQUATIONS-THEORY, COMPUTATION & APPLICATIONS
(NPDE-TCA)
During the Period 2012-13 to 2016-17
(Project No: SERB/F/1279/2011-2012 dated 09/01/2012)
(Started on 10/02/2012)
Department of Mathematics
IIT Bombay
Powai Mumbai - 400076
Project No: SERB/F/1279/2011-2012 dated 09/01/2012
Name of the PI: Prof. Amiya Kumar Pani
Name of the Co-PIs: Prof. Neela Nataraj, Prof. S. Baskar, Prof. Sivaji Ganesh
Date of start of the project: 10 February 2012 ( Academic Programme started from April 2012)
Completion date: 9 February 2017
NATIONAL PROGRAMME ON DIFFERENTIAL EQUATIONS-THEORY,
COMPUTATION & APPLICATIONS
(NPDE-TCA)
National Programme on Differential Equations: Theory, Computation & Applications (NPDE-TCA)
was sanctioned by the Department of Science & Technology (DST) in February, 2012 for a period
of five years with a sanctioned outlay of Rs. 4.5 crores. However, all academic programmes started
in the financial year 2012-2013. The planning and implementation of different programmes were
executed by a local executive committee, which were then approved from time to time by the
National Scientific Organizing Committee.

With a mission “To Create Human Resources and Generate Knowledge Source for Academia and
Industry in the areas of Differential Equations”, the main goal of this programme is to create a
work force at the national level in broad areas of Applied Mathematics, specifically in Differential
Equations, Scientific Computing and Modelling. One of the key components of the proposed
activity is to form a national forum on Differential Equations : Theory, Computation & Applications
for academia and industry.
Objectives
The major objectives of the proposed National programme are :
To attract young talented students to study and to pursue research in the area of Differential
Equations.
To create a pool of trained mathematicians to support the advancement of science and
technology in general and reap its benefits for the development of society.
To promote fruitful interaction between academia and industry.
To provide a platform for academic interaction and collaborative research amongst
mathematicians and scientists in the country.
To expand and strengthen existing expertise in the area of Differential Equations at various
institutions / universities in the country.
Major Outcomes

The major outcomes of this programme are summarized as :
(a) One of the intangible benefits of the programme is the mathematical maturity which the
participants gain as a result of attending the lectures by resource persons who are experts in
the area. This helps tremendously in their overall understanding and outlook of the current
trends in PDEs and Numerical Analysis, (see special feedback in Appendix- III for more
comments).
(b) Students in rural areas are chosen for this programme and this has helped in giving them
exposure to DE and has possibly changed their career paths, (refer special feedback in
Appendix-III.).
(c) A large number of girl students have been trained in all our programmes. In our country, the
number of women who have exposure to higher education is quite less and NPDE
programmes do support the idea of giving ample opportunities to women.
(d) The visit of eminent Professors as a part of this programme has benefitted many faculty
members who are working in Applied Mathematics. The constant interaction has not only
resulted in many joint publications with reputed faculty from abroad but also has been
playing a role in mentoring research scholars.
(e) An informal forum on Differential Equations: Theory, Computations and Applications has
been helping the local groups to organise two CIMPA Schools, already one international
conference in South Asian University (SAU), New Delhi, one will be in Punjab University
in December 2016, and a couple of advanced level workshops ( one in Kashmir University
and other in SAU ) not financed by NPDE programme.
(f) Formation of different groups such as Control and Optimal control, Evolution Equations:
Theory and Computation, Finite Element Analysis, Multiscale Problems, Biomathematics,
Industrial Maths (see activities under Thematic programme in Appendix-I).
(g) Through Modelling and Study Group Meetings with industry every year, for the last 4 years,
participants have started getting a feel for industrial problems and in a similar manner,
industries have started looking forward to greater interaction, (see, Appendix-I for the
programme each year). Now, there are problems coming from industries, but there is a
dearth of manpower.
(h) Through the training programmes, several lecture notes are put on the website and a couple
of those notes may be put in book form.
(i) Apart from Ph.D theses under the supervision of PI & Co- PI's, more Ph.D theses were
written (each in IITK, Panjab University, Ravenshaw University, LNMIIT, South Asian
University) in the areas different from areas of expertise of the PI's is a positive outcome of
the National Programme.
(j) On Visitors Programme: Total (43 visitors) A good number of international visitors visited
various universities and institutes to initiate new interaction and strengthen the ongoing
research collaborations. For more details on visitors, see Appendix-I. As an outcome, 54
research publications in top ranking journals were published, see the attached list of
publications in the Appendix-II.
(k) On publications: There are 54 journal publications. Most of these publications are in the top
ranking journals like SIAM J. Numer. Anal. IMA J. Numer. Anal., Math.comp., J. Sci.
Comp., SIAM J. Sci. Comp., M2AN, J. Comp. Phy., etc. For more detailed publications, see
Appendix-II.
(l) The Internship programme with a motto “Learn mathematics while working on a real life
problem”, is becoming more popular year after year with the number of applicants of around
200-300 every year. Sample feedback for this programme is given in Appendix-III. For more
detailed feedback, see the link:
https://drive.google.com/open?id=0B-bLZm3A4-6KVGFZak51MV9aMDg

Manpower Trained: Total No - 130
Sr. No. Year of Programme Participants
1 2012 – Summer Internship 4
Wintership 10
2 2013 – Summer Internship 14
Wintership 8
3 2014 – Summer Internship 27
Wintership 12
4 2015 – Summer Internship 21
Wintership 18
5 2016 – Summer Internship 07
Wintership 9

(m) On Advanced Level Thematic Programme. These advanced level workshops were organised
on the basis of a proposal by a group which were scrutinized by Local Executive Committee
and then approved by National Scientific Committee. A good number of advanced programmes
have been organized in different areas on Evolution Equations, Hyperbolic PDE, Adaptive and
Non-standard FEM, Homogenization, Control Theory and Biomathematics etc. For some
selective consolidated feedback (see , Appendix III). For more detailed feedback, see the link:
https://drive.google.com/open?id=0B-bLZm3A4-6KVGFZak51MV9aMDg
Manpower Trained: Total No :1118
Sr. No. Year Name of the Programme Participants
1. 2012- 1. Advanced Level Workshop on “Differential Equations in 34
2013 Ecology & Epidemiology”
2. Advanced Workshop on “Mathematical Theory of Control 38
and Numerics” (MTCN – 2012)
3. Advanced Workshop on “Evolution Equations: Theory, 45
Methods & Applications” (AWEETMA-2012)
4. Advanced Workshop on “Non-Standard Finite Element 32
Methods”(AWNSFEM 2013)
5. 3rd Indo-German Workshop on “Adaptive Finite Element 24
Methods”(AWAFEM-2013)
6.Workshop on Recent Developments on “Numerical Methods 30
for Evolution Equations” (RDNMEE-2013)
2. 2013- 1. Advanced Workshop on “Theoretical and Computational 32
2014 Aspects of Nonlinear Waves”(TCANW-2013)
2. Advanced Workshop on “Mathematical Epidemiology and 33
Differential Equations” (AWMEDE-2013)
3. Advanced Workshop on “Nonlinear Differential Equations 50
: Dynamics of Complex Systems” (NDEDCS-2013)
4. Advanced Workshop “on Homogenization 2013” 29
5.Advanced Workshop on “Computational Methods for 29
Integral Equations & Applications”(AWCMIEA-2014)
6. Advanced Workshop & Symposium on “Stability Analysis 31
of Differential Equations with Application to Fluid Flow
Problems”(SADEAFFP-2014)
7. Advanced Level Workshop on “Stabilization Methods for 37
Singularly Perturbed Differential Equations” (SPDE- 2013)
8. Advanced Level Workshop on “Orthogonal Spline 33
Collocation Methods” (OSCM – 2014)
9. Advanced Workshop on “Nonlinear Functional Analysis & 45
Its Applications”(NFAA-2013)
10. Advanced Workshop on “Mathematical Foundation of 30
Advanced Finite Element Methods (MFAFEM-2013)”
3. 2014- 1. Advanced Level Workshop on “Finite element methods for 41
2015 Navier-Stokes equations”
2. Advanced Workshop on “Variational Analysis And 42
Optimization”
3. Advanced Workshop on “Computational Methods For 21
Control Problems”
62
4. Advanced Workshop on “Finite Difference Methods for
Differential equations”
5. Advanced Workshop on “Mimetic Spectral Element 41
Methods”
6. Advanced Workshop on “Biomathematics” 51
7. Advanced Workshop on “Hyperbolic PDEs: Theory, 46
Numerics and Applications (HPTNA-2014)”
4. 2015- 1. Advanced Level Workshop on “Integral Equations – 31
2016 Solvability, Analysis and Applications”
2. Advanced Workshop on “Controllability of Heat and Wave 18
equations”
3. Advanced Workshop on “Variational Inequality (VI) & 36
Applications”
4. Advanced Workshop on “Homogenization and Control: 29
Theory & Application” (AWHCTA-2016)
5. Advanced Workshop on “Dynamical Models in Ecological, 40
Epidemiological Sciences and Engineering – Theory,
Computations, and Real World Applications”
6. Advanced Level Workshop On “Singularly Perturbed PDEs: 42
Theory, Computation And Applications”
5 2016- 1.Advanced workshop on “Research Workshop on 45
2017 Mathematical Modeling of Environmental, Ecological and
Epidemiological systems”
2.“Advanced workshop on Diffusion and Sub-diffusion 21
Problems: Theory, Numerics and Applications”
(n) On Under Graduate Level Training Programme (UGTP): Main theme- Modelling, ODE,
Analysis, Linear Algebra, Numerical analysis with hands on computations. Based on
academic performance and recommendations, participants are selected from all over India.
Care has been taken to give more representation from rural areas. While programme
feedback was taken at the end of each UGTP (see, Appendix-I), special feedback was saught
recently through Google. For representative feedback, see Appendix- III. For more detailed
feedback, see the link:
https://drive.google.com/open ?id=0B-bLZm3A4-6KVGFZak51MV9aMDg
Manpower trained: Total - 271
Sr. Year of the Programme Institute Participants
No.
1 UG Programme 2012 IMA, Bhubaneswar, Odisha 45
(May 7th -26th, 2012)
2 UG Programme 2013 Birla Institute Of Applied 55
Sciences, Bhimtal,
(June 3rd -22nd , 2013)
Uttarakhand
3 UG Programme 2014 Maulana Azad National 55
Institute of Technology,
(May 26th – June 14th , 2014)
Bhopal, MP
4 UG Programme 2015 The LNM Institute Of 55
Information Technology,
(May 18th – June 06th, 2015)
Jaipur, Rajasthan
5 UG Programme 2016 Indian Institute of Space 61
Science and Technology,
(May 18th – June 07th, 2016)
Thiruvananthapuram
(o) On Post-Graduate Level Training Programme: (PGTP) :
Theme: Modelling, Dynamical Systems, PDE, Computations, Measures Theory, Analysis,
Computational PDE with hands on computations. Apart from the feedback taken after completion
of each PGTP (see Appendix- I for more details), special feedback was taken through Google. Some
typical feedback can be seen in Appendix – III. For more detailed feedback, see the link:
https://drive.google.com/copen?id=0B-bLZm3A4-6KVGFZak51MV9aMDg
Manpower trained: Total - 283
Sr. Year of the Programme Institute Participants
No.
1 PG Programme 2012 Indian Institute of 46
Technology Delhi
(May 14th – June 2, 2012)
2 PG Programme 2013 Indian Institute of 44
(May 20th - June 8th , 2013) Technology Madras
3 PG Programme 2014 Indian Institute of 55
Technology Guwahati
(June 2nd – 21st, 2014)
4 PG Programme 2015 Indian Institute of 58
Technology Kanpur
(May 18th – 5th June, 2015 )
5 PG Programme 2016 Indian Institute of 80
Technology Ropar
(May 16th – 4th June, 2016 )
(p). On Advanced Level Training Programme :(ATP) It is mainly meant for Pre Ph.D, and
young researchers. Main topic covered are Distribution Theory, Sobolev Spaces, Elliptic,
Parabolic and hyperbolic PDE (Existence of weak solution, Regularity Theory), Finite element
Methods with hands on computations. Every year, there is a special topic like Option
Derivative, Navies – Stokes Equations, Control & Optimal Control etc. covered. For these
programme, see the feedback in after each such programme Appendix- I and for typical
consolidated feedback, see Appendix- III. For more detailed special feed back, see the link:
https://drive.google.com/ c open?id=0B-bLZm3A4-6KVGFZak51MV9aMDg
Manpower trained: Total - 280
Sr. Year of the Programme Institute Participants
No.
1 Advanced Level Training Programme IIT Bombay 41
2012
(June10th -30th , 2012)
2 Advanced Level Training Programme Department Of Mathematics, 59
2013 Indian Institute Of Science,
(June 24th To July 7th , 2013) and Bangalore.
Cimpa Research School : July 8th to 19th,
2013
3 Advanced Level Training Programme Indian Institute Of Science 61
2014 Education And Research,
May 26th - June13th , 2014 Thiruvananthapuram.
4 Advanced Level Training Programme Birla Institute Of Technology 53
2015 (May 25th –June 14th , 2015) & Science, Pilani – K. K.
Birla Goa Campus
5 Advanced Level Training Programme Birla Institute of Technology 66
2016 (May 25th –June 14th , 2016) & Science, Pilani
Hyderabad Campus
APPENDIX – I
ACTIVITY-1 : TRAINING PROGRAMMES
1.1. Undergraduate Training Programme (UGTP)
The main objective of the UGTP is to introduce participants to new vistas in the areas of
differential equations through modelling and expose them to scientific computing lab sessions
with hands on computing.
1. 2012-2013: (Participated: 45)
Held at IMA, Bhubaneshwar during May 7 – 26, 2012, this programme was coordinated by
M.C. Joshi and Swadhin Pattanayak and Seventy Nine (79) candidates from all over India
applied for this programme, of which forty five (45) were selected based on their academic
records and recommendations. Resource Persons available during this programme were : P.C.
Das and N. Parhi (NISER, Bhubaneshwar); P. Dhanumjaya (BITS, Goa); Raju K. George (IIST,
Trivandrum); M.C. Joshi, Neela Nataraj and Amiya K. Pani (IITB); S. Pattanayak (IMA,
Bhubaneshwar); Amit Apte (TIFR, CAM, Bangalore); B. V. Rathish Kumar, P. Shunmugaraj
(IITK). The graph given below represents feedbacks of the participants out of 10 ranking points
against different topics covered in this Program.
Feedback for UG Level Training Programme 2012
10
8
nt 6
oi
P
g 4
n
ki
an 2
R
0
ANLYS MM ODE-II NA-I CT SST
LA MSFPT ODE-I BVP NA-II SCI
Topics

(Linear Algebra (LA), Analysis (ANSYS), Metric Space and Fixed Point Theorem (MSFT),
Mathematical Modelling (MM), Ordinary Differential Equations – I (ODE I), Ordinary
Differential Equations – II (ODE II), Boundary Value Problem (BVP), Numerical Analysis-I
(NA), Numerical Analysis-II (NA), Control Theory (CT), Scilab (SCI), Scilab Spoken Tutorial
(SST)
2. 2013-2014 : (Participated: 55)
This program was held at Birla Institute of Applied Sciences, Bhimtal, during June 3 – 22, 2013
which was coordinated by Anant Pant (BIAS, Bhimtal), M.C.Joshi (IITB), R P Pant and
Mahesh Joshi (Kumaun University, Nainital). One hundred thirty five (135) candidates from all
over India applied for this program out of which fifty five (55) were selected based on their
academic records and recommendations. Resource Persons: Amiya Kumar Pani (IITB), Mohan
C. Joshi (IITB), S. Baskar (IITB); Raju K. George (IIST, Thiruvananthapuram), A K
Nandkumaran (IISc Bangalore), P. Shunmugraj (IITK), P.S. Datti (TIFR Bangalore), Peeyush
Chandra (IITK), R P Pant (Kumaun University, Nainital), Mahesh Joshi (Kumaun University,
Nainital) and P. Dhanumjaya (BITS-Pilani, Goa). Feedback of the above program is shown
below in the form of a graph :
st)
e Feedback for UG Programme 2013
h
g
hi
e
h 16
g t Topic 1:
n 14
ei Analysis,
b 12
0 Topic 2:
2
& 10
st 8 Linear
e Algebra,
w
o 6
he l 4 Topic 3:
ng t 2
ei
b 0
0
g ( 1 2 3 4 5 6 7 8 9 10 11 12 13
n
ki Topic No.
n
a
R