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September 23, 2023

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Description

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)

Course Curriculum

  • Lecture 01 – Definition and Classification of Linear Integral Equations Unlimited
  • Lecture 02 – Conversion of Initial Value Problem into Integral Equations Unlimited
  • Lecture 03 – Conversion of Boundary Value Problem into Integral Equations Unlimited
  • Lecture 04 – Conversion of Integral Equations into Differential Equations Unlimited
  • Lecture 05 – Integro-differential Equations Unlimited
  • Lecture 06 – Fredholm Integral Equation with Separable Kernels: Theory Unlimited
  • Lecture 07 – Fredholm Integral Equation with Separable Kernels: Examples Unlimited
  • Lecture 08 – Solutions of Integral Equations by Successive Substitutions Unlimited
  • Lecture 09 – Solution of Integral Equations by Successive Approximations Unlimited
  • Lecture 10 – Solutions of Integral Equations by Successive Approximations: Resolvent Kernel Unlimited
  • Lecture 11 – Fredholm Integral Equations with Symmetric Kernels Unlimited
  • Lecture 12 – Fredholm Integral Equations with Symmetric Kernels: Hilbert Schmidt Theory Unlimited
  • Lecture 13 – Fredholm Integral Equations with Symmetric Kernels: Examples Unlimited
  • Lecture 14 – Construction of Green’s Function Unlimited
  • Lecture 15 – Construction of Green’s Function (cont.) Unlimited
  • Lecture 16 – Green’s Function for Self Adjoint Linear Differential Equations Unlimited
  • Lecture 17 – Green’s Function for Non-homogeneous Boundary Value Problem Unlimited
  • Lecture 18 – Fredholm Alternative Theorem Unlimited
  • Lecture 19 – Fredholm Alternative Theorem (cont.) Unlimited
  • Lecture 20 – Fredholm Method of Solutions Unlimited
  • Lecture 21 – Classical Fredholm Theory: Fredholm First Theorem Unlimited
  • Lecture 22 – Classical Fredholm Theory: Fredholm First Theorem (cont.) Unlimited
  • Lecture 23 – Classical Fredholm Theory: Fredholm Second and Third Theorem Unlimited
  • Lecture 24 – Method of Successive Approximations Unlimited
  • Lecture 25 – Neumann Series and Resolvent Kernel Unlimited
  • Lecture 26 – Neumann Series and Resolvent Kernel (cont.) Unlimited
  • Lecture 27 – Equations with Convolution Type Kernels Unlimited
  • Lecture 28 – Equations with Convolution Type Kernels (cont.) Unlimited
  • Lecture 29 – Singular Integral Equations Unlimited
  • Lecture 30 – Singular Integral Equations (cont.) Unlimited
  • Lecture 31 – Cauchy Type Integral Equations I Unlimited
  • Lecture 32 – Cauchy Type Integral Equations II Unlimited
  • Lecture 33 – Cauchy Type Integral Equations III Unlimited
  • Lecture 34 – Cauchy Type Integral Equations IV Unlimited
  • Lecture 35 – Cauchy Type Integral Equations V Unlimited
  • Lecture 36 – Solution of Integral Equations using Fourier Transform Unlimited
  • Lecture 37 – Solution of Integral Equations using Hilbert Transforms Unlimited
  • Lecture 38 – Solution of Integral Equations using Hilbert Transforms (cont.) Unlimited
  • Lecture 39 – Calculus of Variations: Introduction Unlimited
  • Lecture 40 – Calculus of Variations: Basic Concepts Unlimited
  • Lecture 41 – Calculus of Variations: Basic Concepts (cont.) Unlimited
  • Lecture 42 – Calculus of Variations: Basic Concepts and Euler Equations Unlimited
  • Lecture 43 – Euler Equations: Some Particular Cases Unlimited
  • Lecture 44 – Euler Equation: A Particular Case and Geodesics Unlimited
  • Lecture 45 – Brachistochrone Problem and Euler Equation Unlimited
  • Lecture 46 – Euler Equation (cont.) Unlimited
  • Lecture 47 – Functions of Several Independent Variables Unlimited
  • Lecture 48 – Variational Problems in Parametric Form Unlimited
  • Lecture 49 – Variational Problems of General Type Unlimited
  • Lecture 50 – Variational Derivative and Invariance of Euler Equation Unlimited
  • Lecture 51 – Invariance of Euler Equation and Isoperimetric Problem Unlimited
  • Lecture 52 – Isoperimetric Problem (cont.) Unlimited
  • Lecture 53 – Variational Problem Involving a Conditional Extremum Unlimited
  • Lecture 54 – Variational Problem Involving a Conditional Extremum (cont.) Unlimited
  • Lecture 55 – Variational Problems with Moving Boundaries Unlimited
  • Lecture 56 – Variational Problems with Moving Boundaries (cont.) Unlimited
  • Lecture 57 – Variational Problems with Moving Boundaries (cont.) Unlimited
  • Lecture 58 – Variational Problem with Moving Boundaries; One Sided Variation Unlimited
  • Lecture 59 – Variational Problems with a Movable Boundary for a Functional Dependent on … Unlimited
  • Lecture 60 – Hamilton’s Principle: Variational Principle of Least Action Unlimited

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