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Integral Equations, Calculus of Variations and its Applications

Integral Equations, Calculus of Variations and its Applications. Instructors: Dr. P. N. Agarwal and Dr. D. N. Pandey, Department of Mathematics, IIT Roorkee. This course is a basic course offered to PG students of Engineering/Science background. It contains Fredholm and Volterra integral equations and their solutions using various methods such as Neumann series, resolvent kernels, Green's function approach and transform methods. It also contains extrema of functional, the Brachistochrone problem, Euler equation, variational derivative and invariance of Euler equations. It plays an important role for solving various engineering sciences problems. Therefore, it has tremendous applications in diverse fields in engineering sciences. (from nptel.ac.in)

Lecture 45 - Brachistochrone Problem and Euler Equation

In this lecture we discuss the well known Brachistochrone problem i.e. to find the curve which is traversed by a particle sliding from A to B in the shortest time and the Euler equations in the case of several dependent variables.

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Lecture 01 - Definition and Classification of Linear Integral Equations

Lecture 02 - Conversion of Initial Value Problem into Integral Equations

Lecture 03 - Conversion of Boundary Value Problem into Integral Equations

Lecture 04 - Conversion of Integral Equations into Differential Equations

Lecture 05 - Integro-differential Equations

Lecture 06 - Fredholm Integral Equation with Separable Kernels: Theory

Lecture 07 - Fredholm Integral Equation with Separable Kernels: Examples

Lecture 08 - Solutions of Integral Equations by Successive Substitutions

Lecture 09 - Solution of Integral Equations by Successive Approximations

Lecture 10 - Solutions of Integral Equations by Successive Approximations: Resolvent Kernel

Lecture 11 - Fredholm Integral Equations with Symmetric Kernels: Properties of Eigenvalues and Eigenfunctions

Lecture 12 - Fredholm Integral Equations with Symmetric Kernels: Hilbert Schmidt Theory

Lecture 13 - Fredholm Integral Equations with Symmetric Kernels: Examples

Lecture 14 - Construction of Green's Function

Lecture 15 - Construction of Green's Function (cont.)

Lecture 16 - Green's Function for Self Adjoint Linear Differential Equations

Lecture 17 - Green's Function for Non-homogeneous Boundary Value Problem

Lecture 18 - Fredholm Alternative Theorem

Lecture 19 - Fredholm Alternative Theorem (cont.)

Lecture 20 - Fredholm Method of Solutions

Lecture 21 - Classical Fredholm Theory: Fredholm First Theorem

Lecture 22 - Classical Fredholm Theory: Fredholm First Theorem (cont.)

Lecture 23 - Classical Fredholm Theory: Fredholm Second and Third Theorem

Lecture 24 - Method of Successive Approximations

Lecture 25 - Neumann Series and Resolvent Kernel

Lecture 26 - Neumann Series and Resolvent Kernel (cont.)

Lecture 27 - Equations with Convolution Type Kernels

Lecture 28 - Equations with Convolution Type Kernels (cont.)

Lecture 29 - Singular Integral Equations

Lecture 30 - Singular Integral Equations (cont.)

Lecture 31 - Cauchy Type Integral Equations I

Lecture 32 - Cauchy Type Integral Equations II

Lecture 33 - Cauchy Type Integral Equations III

Lecture 34 - Cauchy Type Integral Equations IV

Lecture 35 - Cauchy Type Integral Equations V

Lecture 36 - Solution of Integral Equations using Fourier Transform

Lecture 37 - Solution of Integral Equations using Hilbert Transforms

Lecture 38 - Solution of Integral Equations using Hilbert Transforms (cont.)

Lecture 39 - Calculus of Variations: Introduction

Lecture 40 - Calculus of Variations: Basic Concepts

Lecture 41 - Calculus of Variations: Basic Concepts (cont.)

Lecture 42 - Calculus of Variations: Basic Concepts and Euler Equations

Lecture 43 - Euler Equations: Some Particular Cases

Lecture 44 - Euler Equation: A Particular Case and Geodesics

Lecture 45 - Brachistochrone Problem and Euler Equation

Lecture 46 - Euler Equation (cont.)

Lecture 47 - Functions of Several Independent Variables

Lecture 48 - Variational Problems in Parametric Form

Lecture 49 - Variational Problems of General Type

Lecture 50 - Variational Derivative and Invariance of Euler Equation

Lecture 51 - Invariance of Euler Equation and Isoperimetric Problem

Lecture 52 - Isoperimetric Problem (cont.)

Lecture 53 - Variational Problem Involving a Conditional Extremum

Lecture 54 - Variational Problem Involving a Conditional Extremum (cont.)

Lecture 55 - Variational Problems with Moving Boundaries

Lecture 56 - Variational Problems with Moving Boundaries (cont.)

Lecture 57 - Variational Problems with Moving Boundaries (cont.)

Lecture 58 - Variational Problem with Moving Boundaries; One Sided Variation

Lecture 59 - Variational Problems with a Movable Boundary for a Functional Dependent on Two Functions

Lecture 60 - Hamilton's Principle: Variational Principle of Least Action