Swayam Central

# Mathematical Methods for Boundary Value Problems

By Prof.Somnath Bhattacharyya   |   IIT Kharagpur
This course is intended to provide methods to solve linear and nonlinear boundary value problems involving ordinary as well as partial differential equations. The course will start providing mathematical tools based on integral transformation, Fourier series solution and Greens function for obtaining analytic solutions for BVPs. The applicability of the BVP in several practical contexts, e.g. boundary layer flow, transport phenomena and population balance models will be made. Numerical solutions based on the shooting methods will be introduced. Finite difference methods for linear BVP of second-order and higher orders will be discussed. Iterative techniques to solve nonlinear BVP are included in this course. Algorithms for block tri-diagonal system to handle higher order and system of BVPs will be discussed. Computation of elliptic type of PDEs arises in diffusion dominated process will be described. All the methods will be illustrated by working out several examples. This course, apart from being a part of regular undergraduate/ postgraduate mathematics course, will provide a guidance to solve BVPs arise in mathematical modeling of several transport phenomena. Pre-requisite for this course should be the basic knowledge of undergraduate calculus.

INTENDED AUDIENCE: Undergraduates of any Engineering course, Mathematics, Physics and Postgraduate student
of Mathematics/ Mechanical/ Aerospace/Chemical Engineering
PREREQUISITES:          Basic UG course in Mathematics/ Undergraduate Calculus

Learners enrolled: 1996

### SUMMARY

 Course Status : Ongoing Course Type : Core Duration : 4 weeks Start Date : 29 Jul 2019 End Date : 23 Aug 2019 Exam Date : 29 Sep 2019 Category : Mathematics Level : Undergraduate
This is an AICTE approved FDP course

### COURSE LAYOUT

Week 1: Boundary Value Problems ( BVP); Strum-Liouville Problems; Eigen Values, Eigen Functions. Solution of homogeneous/ non-homogeneous BVPs by Eigen function expansion.
Week 2Eigen function expansion techniques for PDEs; Green’s function; Dirichlet Problems;   Maximum Principle.
Week 3Numerical Techniques for BVP: Shooting Method; Finite Difference Method; Block tri-diagonal system of equations; Numerical Methods for Non-linear BVPs
Week 4Finite difference method for PDEs; Stability analysis; Crank-Nicolson Scheme; ADI scheme;  Elliptic type of Partial Differential Equations; Successive-Over-Relaxation Method.

### BOOKS AND REFERENCES

1. Linear Partial Differential Equations for Scientists and Engineers. T. Myint-U and L. Debnath. 4 th Ed. Birkhauser, Boston (2007).
2. Partial Differential Equations with Numerical Methods. Stig Larsson and Vidar Thomée. Springer ( 2009).
3. Numerical Methods for Two-Point Boundary-Value Problems. H. B. Keller, Waltham, Mass., Blaisdell ( 1993).

### INSTRUCTOR BIO

Prof. S. Bhattacharyya is a senior professor in the Dept. Mathematics, IIT Kharagpur. His specialization is Applied Mathematics. He is teaching courses on Integral Transform Techniques, Partial Differential Equations, Numerical solutions of PDEs and other related courses on a regular basis for the B.Tech students at IIT Khargpur for the past 26 years. His research works involve numerical solutions of PDEs and he has published more than 120 research papers in reputed international journals. He has undertaken several sponsored research projects and guided 15 PhD students. Prof. Bhattcahrayya has organized and delivered lectures in several Conferences, AICTE sponsored short term courses and GIAN courses on the topics related to Applied Mathematics. He has received several fellowships for research collaboration in USA, UK and Germany.

### COURSE CERTIFICATE

• The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
• The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
• Date and Time of Exams: 29 September 2019 ,Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
• Registration url: Announcements will be made when the registration form is open for registrations.
• The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
• Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE
• Average assignment score = 25% of average of best 3 assignments out of the total 4 assignments given in the course.
• Exam score = 75% of the proctored certification exam score out of 100
• Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75
• If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.
• Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kharagpur. It will be e-verifiable at nptel.ac.in/noc.
• Only the e-certificate will be made available. Hard copies are being discontinued from July 2019 semester and will not be dispatched