Integral equations, calculus of variations and its applications

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.

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

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

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.