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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)

Introduction to Calculus of Variations


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

References
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.