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 Understand the methods to reduce Initial value problems associated with linear differential
equations to various integral equations.
 Categorize and solve different integral equations using various techniques.
 Solve the singular integral equations and derivation of Hilbert-Schmidt theorem.
 Know the variational problems, extremum of a functional and necessary conditions for the
extremum of a functional.

THEORY (45 Hours)

UNIT I: (15 Hours)
Integral Equations Definitions of Integral Equations and their classification, Eigen values and
Eigen functions. Reduction to a system of algebraic equations, An Approximate Method.
Fredholm Integral equations of the first kind. Method of Successive Approximations Iterative
Scheme for Volterra and Fredholm Integral equations of the second kind. Conditions of uniform
convergence and uniqueness of series solution. Resolvent kernel and its results. Application of
iterative Scheme to Volterra integral equations of the Second kind. Classical Fredholm Theory
Method of solution of Fredholm equations, Fredholm Theorems
UNIT II: (15 Hours)

Symmetric Kernels Introduction to Complex Hilbert Space, Orthonormal system of functions,
Riesz-Fischer Theorem, Fundamental properties of Eigen values and Eigen functions for
symmetric kernels, Expansion in Eigen function and bilinear form, Hilbert Schmidt Theorem and
some immediate consequences, Solutions of integral equations with symmetric kernels. Singular
Integral Equations The Abel integral equation, Cauchy principal value for integrals, Cauchy-type
integrals, singular integral equation with logarithmic kernel, Hilbert- kernel, solution of Hilbert-
type singular integral equation

UNIT III: (15 Hours)
Calculus of Variations Variational problems, the variation of a functional and its properties,
Extremum of a functional, Necessary condition for an extremum, Euler‘s equation and its
generalization, Variational derivative, General variation of a function and variable end point
problem.

*TUTORIAL (15 Hours (1 Hour per week))

SUGGESTED READINGS:

1. R.P. Kanwal, Linear Integral Equation. Theory and Techniques, Academic Press, New
York, 1971.
2. S. G. Mikhlin, Linear Integral Equations (translated from Russian) Hindustan Book
Agency, 1960
3. J.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice Hall, New Jersy, 1963.
4. M. D. Raisinghania, Integral Equations & Boundary Value Problems, Sultan Chand &
Sons.

Math.425 Operational Research 3+1*

LEARNING OBJECTIVES:

The primary objective of this course is to introduce:
 The history and applications and uses of OR techniques in decision making. The convex
set theory to find the optimal Basic feasible solution of LPP. Modelled the real-world

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