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September 23, 2023

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Description

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)

Course Curriculum

    • Lecture 01 – Introduction to Calculus of Variations Unlimited
    • Lecture 02 – Piecewise Continuity and Differentiability, Integration by Parts, Line Integral Unlimited
    • Lecture 03 – Green’s Theorem, Normal Derivatives, Matrices and Determinants Unlimited
    • Lecture 04 – Matrices and Determinants, Picard’s Theorem, Wronskian, Surface Integral Unlimited
    • Lecture 05 – Divergence Theorem Unlimited
    • Lecture 06 – Fundamental Concepts of the Calculus of Variations Unlimited
    • Lecture 07 – The Concept of Variation of a Functional Unlimited
    • Lecture 08 – Examples of Applications of the Euler’s Equation Unlimited
    • Lecture 09 – More Examples of Applications of the Euler’s Equation Unlimited
    • Lecture 10 – More General Functionals Unlimited
    • Lecture 11 – Functionals containing Several Independent Variables Unlimited
    • Lecture 12 – Poisson Equation, Functionals of Three Independent Variables, Isoperimetric Problems Unlimited
    • Lecture 13 – The Problem of Finding Geodesics Unlimited
    • Lecture 14 – The Functionals with Moving Boundary Points Unlimited
    • Lecture 15 – Various Cases of the Functionals with Moving Boundary Points Unlimited
    • Lecture 16 – Cases of the Discontinuities of the Extremals Unlimited
    • Lecture 17 – Sufficient Conditions for an Extremum, Various Notions of Fields of Extremals Unlimited
    • Lecture 18 – Jacobi’s Condition, The Weierstrass Function Unlimited
    • Lecture 19 – Strong and Weak Extrema of Functionals Unlimited
    • Lecture 20 – Variational Problems involving Conditional Extremum Unlimited
    • Lecture 21 – Introduction to Integral Equations and Examples Unlimited
    • Lecture 22 – Solutions of Integral Equations Unlimited
    • Lecture 23 – Conversion of Initial Value Problem to Integral Equation Unlimited
    • Lecture 24 – Methods for Solving Volterra Integral Equations: Successive Approximation Method Unlimited
    • Lecture 25 – Methods for Solving Volterra Integral Equations: Laplace Transform Method Unlimited
    • Lecture 26 – Methods for Solving Volterra Integral Equations: Adomian Decomposition Method Unlimited
    • Lecture 27 – Methods for Solving Volterra Integral Equations: Method of Successive Substitution Unlimited
    • Lecture 28 – The Uniform Convergence of the Resolvent Kernel, Volterra Integral Equation Unlimited
    • Lecture 29 – Fredholm Integral Equation Unlimited
    • Lecture 30 – Sturm-Liouville Boundary Value Problems, Green’s Functions Unlimited
    • Lecture 31 – Eigenvalues and Eigenfunctions of Sturm-Liouville Boundary Value Problems Unlimited
    • Lecture 32 – Sturm-Liouville Boundary Value Problems: Eigenfunction Expansion Unlimited
    • Lecture 33 – Methods for Solving Nonhomogeneous Fredholm Integral Equation Unlimited
    • Lecture 34 – Successive Approximation Method: Iterated Kernels, Neumann Series Unlimited
    • Lecture 35 – Solution of the System of Linear Equation Unlimited
    • Lecture 36 – Fredholm Theory to Obtain Resolvent Kernel for Fredholm Integral Equation Unlimited
    • Lecture 37 – Hilbert-Schmidt Theorem and its Consequences Unlimited
    • Lecture 38 – Singular Integral Equations Unlimited
    • Lecture 39 – Fredholm and Volterra Integro-Differential Equations Unlimited
    • Lecture 40 – Nonlinear Integral Equations Unlimited

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