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Course11110710360 lectures

Integral Equations, Calculus of variations and its applications

Prof. P. N. Agrawal and Prof. D. N. Pandey · IIT Roorkee

Start with lecture 1 At NPTEL

Lectures

Unit-1

  1. Definition and classification of linear integral equations Video
  2. Conversion of IVP into integral equations Video
  3. Conversion of BVP into an integral equations Video
  4. Conversion of integral equations into differential equations Video
  5. Integro-differential equations Video

Unit-2

  1. Fredholm integral equation with separable kernel: Theory Video
  2. Fredholm integral equation with separable kernel: Examples Video
  3. Solution of integral equations by successive substitutions Video
  4. Solution of integral equations by successive approximations Video
  5. Solution of integral equations by successive approximations: Resolvent kernel Video

Unit-3

  1. Fredholm integral equations with symmetric kernels: Properties of eigenvalues and eigenfunctions Video
  2. Fredholm integral equations with symmetric kernels: Hilbert Schmidt theory Video
  3. Fredholm integral equations with symmetric kernels: Examples Video
  4. Construction of Green function-I Video
  5. Construction of Green function-II Video

Unit-4

  1. Green function for self adjoint linear differential equations Video
  2. Green function for non-homogeneous boundary value problem Video
  3. Fredholm alternative theorem-I Video
  4. Fredholm alternative theorem-II Video
  5. Fredholm method of solutions Video

Unit-5

  1. Classical Fredholm theory: Fredholm first theorem-I Video
  2. Classical Fredholm theory: Fredholm first theorem-II Video
  3. Classical Fredholm theory: Fredholm second theorem and third theorem Video
  4. Method of successive approximations Video
  5. Neumann series and resolvent kernels-I Video

Unit-6

  1. Neumann series and resolvent kernels-II Video
  2. Equations with convolution type kernels-I Video
  3. Equations with convolution type kernels-II Video
  4. Singular integral equations-I Video
  5. Singular integral equations-II Video

Unit-7

  1. Cauchy type integral equations-I Video
  2. Cauchy type integral equations-II Video
  3. Cauchy type integral equations-III Video
  4. Cauchy type integral equations-IV Video
  5. Cauchy type integral equations-V Video

Unit-8

  1. Solution of integral equations using Fourier transform Video
  2. Solution of integral equations using Hilbert transform-I Video
  3. Solution of integral equations using Hilbert transform-II Video
  4. Calculus of variations: Introduction Video
  5. Calculus of variations: Basic concepts-I Video

Unit-9

  1. Calculus of variations: Basic concepts-II Video
  2. Calculus of variations: Basic concepts and Euler equation Video
  3. Euler equation: Some particular cases Video
  4. Euler equation : A particular case and Geodesics Video
  5. Brachistochrone problem and Euler equation-I Video

Unit-10

  1. Euler's equation-II Video
  2. Functions of several independent variables Video
  3. Variational problems in parametric form Video
  4. Variational problems of general type Video
  5. Variational derivative and invariance of Euler's equation Video

Unit-11

  1. Invariance of Euler's equation and isoperimetric problem-I Video
  2. Isoperimetric problem-II Video
  3. Variational problem involving a conditional extremum-I Video
  4. Variational problem involving a conditional extremum-II Video
  5. Variational problems with moving boundaries-I Video

Unit-12

  1. Variational problems with moving boundaries-II Video
  2. Variational problems with moving boundaries-III Video
  3. Variational problems with moving boundaries; One sided variation Video
  4. Variational problem with a movable boundary for a functional dependent on two functions Video
  5. Hamilton's principle: Variational principle of least action Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Roorkee
Department
Mathematics
Taught by
Prof. P. N. Agrawal, Prof. D. N. Pandey
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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