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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 48 - Variational Problems in Parametric Form

In this lecture we discuss the extremal of a functional when the integrand is given in the parametric form.


Go to the Course Home or watch other lectures:

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