GIAN Course on

Generalized Iterative Methods for Nonlinear Differential Equations

18 December 2017 - 22 December 2017

Venue: Indian Institute of Technology Indore (IIT Indore)

Overview of the course

Mathematical modeling of real life problems in science, engineering and other branches including social science leads to nonlinear dynamic systems with initial and /or boundary conditions. Quantitative and qualitative properties of these dynamic systems, such as existence, uniqueness play an important role in applying these models to our advantage. The explicit computation of solutions for the nonlinear problem is rarely possible. However, the method of lower and upper solutions is an effective tool to prove the existence results for nonlinear dynamic systems. These lower and upper solutions not only act as bounds for the solution but also assure the interval of existence for initial value problems. Monotone sequences that converge uniformly and monotonically to the minimal and maximal solutions can be generated by effectively associating the method of lower and upper solutions with monotone method. This method scales down the original problem to a computationally easier linear problems. If further uniqueness conditions are satisfied, the minimal and maximal solutions converge to the unique solution of the nonlinear problem. Through this iterative method, the existence of the solution can be confirmed both theoretically and computationally. However, the monotone method is useful when the forcing term is either increasing or can be made increasing by an addition of a linear term. In order to overcome this problem, coupled lower and upper solutions will be introduced. In this case, the monotone method is referred to as generalized monotone method. The generalized monotone method yields monotone sequences which converge to coupled minimal and maximal solutions. The rate of convergence of the iterates for monotone method or generalized monotone method is linear, whereas, the method of quasilinearization or generalized quasilinearization yields quadratic convergence. In this case, we assume a stronger condition on the nonlinear term, namely convexity, concavity or both. As a byproduct, the iterates converge to the unique solution quadratically.  

Objectives of the course

This compact course focuses on introducing the participants to the method of lower and upper solutions (coupled lower and upper solutions) combined with an iterative method for obtaining existence results for a wide range of nonlinear problems. The lectures are outlined to start from the very basic concepts and move towards providing proper training to apply the theory to various real life problems. This course will act as a platform to learn the foundations of dealing with nonlinear problems with applications. In addition, it promotes future collaborative activities between researchers from varied disciplines. Participants will also gain much by carefully applying this seemingly different topic in their own research fields. The technical sessions are given prominence to make sure that it will be fruitful for all the participants.  

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Registration Fee

  • M.Sc/M.Phil/B.Tech/M. Tech. Students: ₹ 1,000/-

  • Ph.D. Student/ Post Doctoral: ₹ 2,000/-

  • Faculty, Scientists, Engineers and delegates from Industry: ₹ 3,000/-

  • Participants from abroad: $200

(The above fee includes lunch, instructional materials for tutorials and assignments, 24 hours internet facility. The participants will be provided with accommodation on payment basis.)

Registration and Accommodation

We request the participants to first register at the GIAN portal given at the following link and then based on their selection, register at the GIAN portal of IIT Indore. (Alternatively, one can request for participation by sending an email to vijesh[at]iiti[dot]ac[dot]in along with a short CV of the applicant. After hearing from the organizers about their selection, they can proceed to register at the GIAN portal of IIT Indore.)


Click here for registration at the GIAN portal.


Click here for registration at the IITI GIAN portal (Please mention NEFT/D.D. Number alongwith the date of transferred amount in the appropriate box.)


Bank Details for Fund Transfer or e-payment:

Name of Account Holder: IIT Indore

Name of Bank: Canara Bank

Branch: IIT Indore, Simrol Campus Branch

Account Number: 1476101027440

Account Type: Saving A/c

Bank IFS Code: CNRB0006223

Bank SWIFT Code: CNRBINBBISG


Credit Card payment option is also available at the link https://epayments.iiti.ac.in/. At this link please fill-up with "GIAN_GIMNDE -- Your Name" for Reference Number and fill-up the amount and then proceed to pay. After the payment is successful, please DO NOT forget to save and send the transaction reference page in PDF by email to "vijesh[at]iiti.ac.in".


Accommodation:

The participants may be provided with hostel accommodation, depending on availability, on payment basis.

The outstation participants who require accommodation should send a request to vijesh[at]iiti[dot]ac[dot]in with the subject `Request for Accommodation'.

The student participants who requested accommodation should fill up the following application form and send their request to hostel[at]iiti[dot]ac[dot]in (with a cc to vijesh[at]iiti[dot]ac[dot]in).

Click here for application form for hostel accommodation.

The rates for accommodation charges are

Institute Hostel accommodation (sharing basis): ₹ 250 per night

Institute Guest House (Executive Suite): ₹ 2, 000 + tax per night (per unit)

Accommodation charges do not include meals expenses

The Teaching Faculty

Professor Aghalaya S Vatsala is a Professor at the University of Louisiana at Lafayette since 1996. After her Master’s degree from Bangalore University in 1968, she completed her doctoral degree from IIT Madras in 1973. She worked as C.S.I.R postdoctoral fellow for two years at IIT, Madras. She served as a faculty at the Regional College of Education, (NCERT) Mysore. She moved to United States in 1978, and worked as a visiting faculty at University of Texas at Arlington, Texas, and Oklahoma State University, Stillwater, Oklahoma. She joined University of Louisiana at Lafayette in 1984, and currently serving as Pennzoil endowed Professor of Mathematics, in the Mathematics department. Professor Aghalaya S. Vatsala is one of the pioneers in developing monotone iterative methods for nonlinear differential equations. Her contributions in the area of ordinary and partial differential equations made her well known in the mathematics world. She has successfully completed various sponsored research projects including two sponsored research projects from U.S. Army. She has successfully organized more than twenty five workshops/conferences including special sessions in second, third and fourth World Congress of Nonlinear Analysts and special session in various American Mathematical Society Annual Conferences.
Aghalaya S. Vatsala has also served as an editorial member for various international journals including Nonlinear Analysis: Hybrid Systems (2009-2011) and Nonlinear Analysis: Theory, Methods & Application Series A & C (till 2011). She is also a member of various international academic bodies including Board of Experts for International Federation of Nonlinear Analysis. She has published more than 125 research papers in reputed international journals. She is also a reviewer for various mathematical databases and a co-author of various research monographs. One of the research monographs entitled ‘Monotone Iterative Technique for Nonlinear Differential Equations’ co-authored with G.S. Ladde and V. Lakshmikantham is cited more than thousand times as per google scholar statistics. She has supervised twelve students for their PhD degree. She has been honored with UL Lafayette foundation distinguished professor award in 2014. Apart from delivering numerous presentations in conferences, she also has motivated students by her talks about career in mathematical sciences as a part of outreach program. For more details, visit:  http://www.ucs.louisiana.edu/~asv5357/  

The Host Faculty

 

Dr. V. Antony Vijesh received his PhD degree from IIT Madras in 2007 and is currently working as an Associate Professor at IIT Indore. His broad area of research is nonlinear analysis and numerical analysis. Presently his group is developing efficient numerical methods for partial differential equations and integro-differential equations using finite difference method and wavelets. The other work aspect of this group is to study the existence and uniqueness of nonlinear differential equations via various iterative schemes and analyze their convergence behavior with systematic error estimates. Their findings have been published in reputed international journals. For more details, visit http://iiti.ac.in/people/~antony/  

Contact Details

Discipline of Mathematics

Indian Institute of Technology Indore

Simrol, Khandwa Road, Indore

Madhya Pradesh, India- 453 552

Tel: +91 731 2438 445

Email: antony[at]iiti[dot]ac[dot]in

Web-page: http://iiti.ac.in/people/~antony/