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Calculus of Variations and Integral Equations

Calculus of Variations and Integral Equations. Instructors: Prof. D. Bahuguna and Dr. Malay Banerjee, Department of Mathematics and Statistics, IIT Kanpur.

Calculus of Variations: Concepts of calculus of variations, Variational problems with the fixed boundaries, Variational problems with moving boundaries, Sufficiency conditions.
Integral Equations: Solutions of integral equations, Volterra integral equations, Fredholm integral equations, Fredholm theory - Hilbert-Schmidt theorem, Fredholm and Volterra integro-differential equation. (from nptel.ac.in)

Lecture 34 - Successive Approximation Method: Iterated Kernels, Neumann Series

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Calculus of Variations

Lecture 01 - Introduction to Calculus of Variations: Brachistochrone Curve, Problem of geodesics, Isoperimetric Problem

Lecture 02 - Piecewise Continuity and Differentiability, Integration by Parts, Line Integral

Lecture 03 - Green's Theorem, Normal Derivatives, Matrices and Determinants

Lecture 04 - Matrices and Determinants, Picard's Theorem, Wronskian, Surface Integral

Lecture 05 - Divergence Theorem

Lecture 06 - Fundamental Concepts of the Calculus of Variations: Brachistochrone Problem, Problem of Geodesics, Isoperimetric Problem

Lecture 07 - The Concept of Variation of a Functional

Lecture 08 - Examples of Applications of the Euler's Equation

Lecture 09 - More Examples of Applications of the Euler's Equation

Lecture 10 - More General Functionals

Lecture 11 - Functionals containing Several Independent Variables

Lecture 12 - Poisson Equation, Functionals of Three Independent Variables, Isoperimetric Problems

Lecture 13 - The Problem of Finding Geodesics

Lecture 14 - The Functionals with Moving Boundary Points

Lecture 15 - Various Cases of the Functionals with Moving Boundary Points

Lecture 16 - Cases of the Discontinuities of the Extremals

Lecture 17 - Sufficient Conditions for an Extremum, Various Notions of Fields of Extremals

Lecture 18 - Jacobi's Condition, The Weierstrass Function

Lecture 19 - Strong and Weak Extrema of Functionals

Lecture 20 - Variational Problems involving Conditional Extremum

Integral Equations

Lecture 21 - Introduction to Integral Equations and Examples

Lecture 22 - Solutions of Integral Equations

Lecture 23 - Conversion of Initial Value Problem to Integral Equation

Lecture 24 - Methods for Solving Volterra Integral Equations: Successive Approximation Method

Lecture 25 - Methods for Solving Volterra Integral Equations: Laplace Transform Method, Series Solution Method

Lecture 26 - Methods for Solving Volterra Integral Equations: Adomian Decomposition Method

Lecture 27 - Methods for Solving Volterra Integral Equations: Method of Successive Substitution

Lecture 28 - The Uniform Convergence of the Resolvent Kernel, Volterra Integral Equation of the First Kind

Lecture 29 - Fredholm Integral Equation

Lecture 30 - Sturm-Liouville Boundary Value Problems, Green's Functions

Lecture 31 - Properties of Green's Functions, Eigenvalues and Eigenfunctions of Sturm-Liouville Boundary Value Problems

Lecture 32 - Sturm-Liouville Boundary Value Problems: Eigenfunction Expansion

Lecture 33 - Methods for Solving Nonhomogeneous Fredholm Integral Equation: Fredholm Alternative Theorem, Adomian Decomposition Method, Successive Approximation Method

Lecture 34 - Successive Approximation Method: Iterated Kernels, Neumann Series; Conversion of Fredholm Integral Equation into a System of Linear Equation

Lecture 35 - Solution of the System of Linear Equation

Lecture 36 - Fredholm Theory to Obtain Resolvent Kernel for Fredholm Integral Equation

Lecture 37 - Hilbert-Schmidt Theorem and its Consequences; Finding the Solution of the Fredholm Integral Equation with Symmetric Kernel

Lecture 38 - Singular Integral Equations

Lecture 39 - Fredholm and Volterra Integro-Differential Equations

Lecture 40 - Nonlinear Integral Equations