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Courses » Integral Equations,calculus of variations and its applications

Integral Equations,calculus of variations and its applications

ABOUT THE COURSE 

This course is a basic course offered to PG students of Engineering/Science background. It contains Fredholm and Volterra integral equations and their solutions using various methods such as Neumann series, resolvent kernels, Green’s function approach and transform methods. It also contains extrema of functional, the Brachistochrone problem, Euler’s equation, variational derivative and invariance of Euler’s equations. It plays an important role for solving various engineering sciences problems.  Therefore, it has tremendous applications in diverse fields in engineering sciences.

Important For Certification/Credit Transfer:

Weekly Assignments and Discussion Forum can be accessed ONLY by enrolling here

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Note: Content is Free!

All content including discussion forum and assignments, is free


Final Exam (in-person, invigilated, currently conducted in India) is mandatory for Certification and has INR Rs. 1100 as exam fee.


INTENDED AUDIENCE

It is a core as well as elective course for PG students of technical institutions/ universities/colleges.


PRE-REQUISITES

Not Required


INDUSTRIES THAT WILL RECOGNIZE THIS COURSE

Nil

1501 students have enrolled already!!

COURSE INSTRUCTOR


Dr. P. N. Agrawal  is a Professor in the Department of Mathematics, IIT Roorkee. His area of research includes approximation Theory and Complex Analysis. He delivered 13 video lectures on Engineering Mathematics in NPTEL Phase I and recently completed Pedagogy project on Engineering Mathematics jointly with Dr. Uaday Singh in the same Department. Further he has completed online certification course “Mathematical methods and its applications” jointly with Dr. S.K. Gupta of the same department. He taught the course on “Integral equations and calculus of variations” several times to MSc (Industrial Mathematics and Informatics) students. He has supervised nine Ph.D. theses and has published more than 160 research papers in reputed international journals of the world. Currently, he is supervising six research students.



Dr. D. N. Pandey  is an Associate Professor in the Department of Mathematics, IIT Roorkee. Before joining IIT Roorkee he worked as a faculty member in BITS-Pilani Goa campus and LNMIIT Jaipur. His area of expertise includes semigroup theory, functional differential equations of fractional and integral orders. He has already prepared e-notes for course titled “Ordinary Differential Equations and Special Functions” under e- Pathshala funded by UGC. Also, he has published a book titled “Nonlocal Functional Evolution Equations: Integral and fractional orders, LAP LAMBERT Academic Publishing AG Germany”. He has delivered several invited talks at reputed institutions in India and abroad. He has guided three Ph.D. theses and has published more than 60 papers in various international journals of repute. Currently, he is supervising five research students.

COURSE PLAN:

Week 1    
Definition and classification of linear integral equations    
Conversion of IVP into integral equations    
Conversion of BVP into integral equations    
Conversion of integral equationsinto differential equations    
Integro-differential equations

Week 2    
Fredholm integral equation with separable kernel: Theory    
Fredholm integral equation with separable kernel: Examples    
Solution of integral equations by successive substitutions    
Solution of integral equations by successive approximations    
Solution of integral equations by successive approximations: Resolvent kernel

Week 3    
Fredholm integral equations with symmetric kernels:Properties of eigenvalues and eigen functions    
Fredholm integral equations with symmetric kernels:Hilbert Schmidt theory    
Fredholm integral equations with symmetric kernels:Examples    
Construction of Green’s function-I    
Construction of Green’s function-II

Week 4    
Green’s function for self adjoint linear differential equations    
Green’s function for non- homogeneous boundary value problem    
Fredholm alternative theorem-I    
Fredholm alternative theorem-II    
Fredholm method of solutions

Week 5    
Classical Fredholm theory: Fredholm first theorem-I    
Classical Fredholm theory: Fredholm first theorem-II    
Classical Fredholm theory: Fredholm second and third theorem    
Method of successive approximations    
Neumann Series and resolvent kernels-I

Week 6    
Neumann Series and resolvent kernels-II    
Equations with convolution type kernels-I    
Equations with convolution  type kernels-II    
Singular integral equations-I    
Singular integral equations-II

Week 7    
Cauchy type integral equations-I    
Cauchy type integral equations-II    
Cauchy type integral equations-III    
Cauchy type integral equations-IV    
Cauchy type integral equations-V

Week 8    
Solution of integral equations using Fourier transform    
Solution of integral equations using Hilbert- transform-I    
Solution ofintegral equations using Hilbert- transform-II    
Calculus of variations: Introduction    
Calculus of variations: Basic concepts-I

Week 9    
Calculus of variations:Basic concepts-II    
Calculus of variations: Basic concepts and Euler’s equation    
Euler’s equation:Some particular cases    
Euler’s equation:A particular case and Geodesics    
Brachistochrone problem and Euler’s equation-I

Week 10   
Euler’s equation-II    
Functions of several independent variables    
Variational problems in parametric form    
Variational problems of general type    
Variational derivative and invariance of Euler’s equation
 
Week 11    
Invariance of Euler’s equation and isoperimetric problem-I    
Isoperimetri c problem-II    
Variational problem involving a conditional extremum-I    
Variational problem involving a conditional extremum-II    
Variational problems with moving boundaries- I

Week 12    
Variational problems with moving boundaries- II    
Variational problems with moving boundaries- III    
Variational problems with moving boundaries; One sided variation    
Variational problem with a movable boundary for a functional dependent on two functions    
Hamilton’s  principle;Variational   principle   of least action


SUGGESTED READING

1. R. P. Kanwal, Linear Integral Equations, Theory and Techniques, Birkhauser, 1924.
2. M. Krasnov, A. Kiselev, G. Makarenko, Problems and  Exercises  in  Integral  Equations,  Mir  Publishers, Moscow, 1971.
3. W. V. Lovitt, Linear Integral Equations, Dover publications, 2005.
4. I. M. Gelfand,    S.V.   Fomin,    Calculus    of Variations, Dover Publications, 2000.
5. Robert Weinstock, Calculus of Variations with Applications to Physics and Engineering, Dover Publications, 1974.
6. L. Elsgolts,   Differential   Equations   and   the Calculus of Variations, MIR Publishers, 1970.
7. A. S. Gupta,   Calculus   of   Variations   with Applications, PHI Learning, 2015.


CERTIFICATION EXAM :
  • The exam is optional for a fee.
  • Date of Exams : October 28 (Sunday)
  • Time of Exams : Morning session 9am to 12 noon; Afternoon session: 2pm to 5pm
  • Exam for this course will be available in both morning & afternoon sessions.
  • 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.

CERTIFICATION:

  • Final score will be calculated as : 25% assignment score + 75% final exam score
  • 25% assignment score is calculated as 25% of average of  Best 8 out of 12 assignments
  • E-Certificate will be given to those who register and write the exam and score greater than or equal to 40% final score. 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 Roorkee.It will be e-verifiable at nptel.ac.in/noc.