# InfoCoBuild

### 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)

 Definition and Classification of Linear Integral Equations

 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

 References 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 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.