Integral Equations & Calculus of Variations-M401
Paper-I
Course Outcomes
1.Fully understand the properties of geometrical problems
2.Be familiar with variational problems
3.Be familiar isoperimetric problems
4.Be thorough with different types of integral equations
INTEGRAL EQUATIONS:
Unit I
Volterra Integral Equations
Basic concepts
Relationship between Linear di-erential equations
Volterra Integral equations
Resolvent Kernel of Volterra Integral equation.
Di erentiation of some resolvent kernels
Solution of Integral equation by Resolvent Kernel
The method of successive approximations
Convolution type equations
Solution of Integro-di erential equations with the aid of the Laplace Transformation
Volterra integral equation of the 1st kind
Euler integrals
Abel'sproblem
Abel'sintegral equation and its generalizations.
Unit II
Fredholm Integral Equations
Fredholm integral equations of the second kind, Fundamentals
The Method of Fredholm Determinants
Iterated Kernels, constructing the Resolvent Kernel with the aid of Iterated Kernels
Integral equations with Degenerated Kernels.
Hammer steintype equation
Characteristic numbers and Eigen functions and its properties.
Green's function
Construction of Green's function for ordinary differential equations
Special case of Green's function
Using Green's function in the solution of boundary value problem.
CALCULS OFVARIATIONS:
Unit III
Introduction
The Method of Variations in Problems with fixed Boundaries
Definitions of Functionals
Variation and Its properties
Euler's'equation
Fundamental Lemma of Calculus of Variation
The problem of minimum surface of revolution
Minimum Energy Problem
Brachistochrone Problem
Variational problems involving Several functions
Functional dependent on higher order derivatives
Euler Poisson equation.
Unit-IV
Functional dependent on the functions of several independent variables
Euler's equations in two dependentvariables
Variational problems in parametric form
Applications of Calculus of Variation
Hamilton's principle
Lagrange'sEquation
Hamilton'sequations.
TextBooks:
[1] M.KRASNOV,A.KISELEV,G.MAKARENKO,ProblemsandExercisesinIntegralEquations(1971).
[2] S.Swarup,IntegralEquations,(2008).
[3] L.ELSGOLTS,Di erentialEquationsandTheCalculusofVariations,MIRPublishers,MOSCOW.
Elementary Operator Theory-M402
Paper-II
Course Outcomes
1.This course prepares a student to take a second course in operator algebra, Spectral theory etc,.This is highly applicational.
2.Its study open ways to different research areas in this branch particularly in the area of functional analysis broadly, like representation theory, operators on different function spaces etc..
3.The course will be useful for all students who are aiming at writing a master thesis in mathematics (or applied mathematics) with specialization in analysis.
4.Students get an understanding of bounded and unbounded operators
Unit I
Spectral theory in finite dimensional normed spaces
Basic concepts of spectrum
Spectral properties of bounded linear operators
Further properties of resolvent and spectrum.
(Sections7.1,7.2,7.3&7.4of[1]).
Unit II
Compact linear operators on normed spaces
Further properties of compact linear operators
Spectral properties of compact linear operators on normed spaces
Operator equations involving compact linear operators.
(Sections 8.1,8.2,8.3and8.5of[1]).
Unit III
Spectral properties of bounded self adjoint linear operators
Further spectral properties of bounded linear operators
Positive operators
Square root of a positive operator.
(Sections9.1,9.2,9.3and9.4of[1])
Unit-IV
Projection operators
Properties of projection operators
Spectral family
Spectral family of a bounded self adjoint linear operator.
(Sections9.5,9.6,9.7and9.8of[1])
TextBook:
IntroductoryFunctionalAnalysisbyE.Kreyszig,JohnWileyandSons,NewYork,1978.
References:
[1] ElementsofFunctionalAnalysisbyBrownandPage,D.V.N.Comp.
[2] FunctionalAnalysisbyB.V.Limaye,WileyEasternLimited,(2ndEdition).
[3] AHilbertSpaceProblemBookbyP.R.Halmos,D.VanNostrandCompany,Inc.1967.
Analytic Number Theory-M403
Paper-III
Course Outcomes
1.Understand better the distribution of prime numbers
2.Know the basic theory of zeta- and L-functions
3.Understand the proof of Dirichlet's Theorem
4.To investigate the distribution of prime numbers
Unit I
Averages of arithmetical function
The bigoh notation
Asymptotic equality of functions
Euler summation formula
Some asymptotic formulas
The average order of d(n)
The average order of the divisor functions(n)
The average order of (n)
An application to the distribution of lattice points visible throughna the origin
The average order of (n) and (n)
The partial sums of dirichlet product
Applicationsto (n) and (n)
Another identity for the partial sums of a dirichlet product.
(Sections3.1to3.12).
Unit II
Some elementary theorems on the distribution of prime numbers
Introduction chebyshev's functions-
(x) and (x)- Relation connecting (n) and (n)
Some equival ent forms of the prime number theorem
Inequalities for (n) and pn.
(Sections4.1to4.5)
Unit III
Shapiro's Tauberian theorem
Applications of shapiro's theorem
Anasymptoticformulaforthepartialsums P p x1=p
The partial sums of the mobins function
Sel berg Asymptotic formula.
(Sections4.6to4.11except4.10)
Unit-IV
Finite Abelian groups and their character
Construction of sub groups
Characters of finite abelian group
The character group
Theorth ogonality relations for characters Dirichlet characters
Sums involving dirichelet characters the non vanishingof L(1; ) for real non principal .
(Sections6.4to6.10)
TextBook:
TomM.Apostol-AnIntroductiontoAnalyticNumberTheory,Springer.
Integral Transforms-M404(A)
Paper-IV
Course Outcomes
1.Calculate the Laplace transform of standard functions both from the definition and by using tables.
2.Select and use the appropriate shift theorems in finding Laplace and inverse Laplace transforms.
3.Select and combine the necessary Laplace transform techniques to solve second-order ordinary differential equations involving the Dirac delta (or unit impulse).
4.Calculate both real and complex forms of the Fourier series for standard periodic waveforms and convert from real-form Fourier series to complex-form and vice-versa.
Unit I
FOURIERTRANSFORM:
Introduction
Classes of functions
Fourier Series and Fourier Integral Formula
Fourier Trans forms
Fouriers in e and cosine Trans forms
Linearity property-
Change of Scale property
Shifting property
The Modulation theorem
Evaluation of integrals by means of inversion theorems
Fourier Transform of some particular functions
Convolution or Faltungof two integrable functions
Convolution or Faltingor Faltung Theorem for FT
Parseval'srelations
FourierTrans form of the derivative of a function
Fourier Trans form of some more useful functions
Fourier Transforms of Rational Functions
Other important examples concerning derivative of FT
The solution of Integral Equations of Convolution Type
Fourier Trans form of Functions of several variables
Application of Fourier Transform to Boundary ValueProblems.
Unit II
THE LAPLACETRANSFORM:
Introduction-Defnitions
Suficient conditions for existence of Laplace Transform
Linearity property of Laplace Transform
Laplace trans forms of some elementary functions
First shift theorem
Second shift theorem
The change of scale property
Examples
LaplaceTransform of derivatives of a function
Laplace Transform of Integral of a function
Laplace Transform of tnf(t)
Laplace Trans form of f(t)/t
Laplace Transform of a periodic function
The Initial-ValueTheorem
The Final-ValueTheorem of Laplace Transform
Examples
LaplaceTransforms of some special functions
The Convolution of two functions
Applications.
Unit III
THE INVERSELAPLACETRANSFORMANDAPPLICATION
Introduction
Calculation of Laplace inversion of some elementary functions
Method of expansion into partial fractions of the ratio of two
The general evaluation technique of inverse Laplace transform
Application of Laplace Transforms.
Finite Laplace Transforms:
Introduction
Definition of Finite Laplace Transform
Finite Laplace Transform of elementary functions
Operational Properties
The Initial Value and the Final Value Theorem
Applications.
Unit-IV
The MellinTransform:
Introduction-Definition of Mellin Trans form
MellinTransform of derivative of a function
Mellin Trans form of Integral of a function
Mellin Inversion theorem
Convolution theorem of Mellin Transform
Illustrative solved Examples
Solution of Integral equations
Application to Summation of Series
The Generalised MellinTransform
Convolution of generalised Mellin Transform
FiniteMellinTransform
The Z-Transform
Introduction
Transform
Definition
Some Operational Properties of Z-Transform
Applicationof Z-Transforms.
TextBook:
[1] AnIntroductiontoIntegralTransformsbyBaidyanathPatra,CRCPress,TaylorFrancisGroup.
References:
[1] IntegralTransformsbyA.R.VasishtaandR.K.Guptha.
Advanced Operations Research-M405(B)
Paper-V
Course Outcomes
1.To provide a formal quantitative approach to problem solving and an intuition about situations where such an approach is appropriate
2..To introduce some widely used advanced operations research models.
3.Identify and develop operational research models from the verbal description of the real system.
4.Use mathematical software to solve the proposed models.
Unit I
Characteristics of Gametheory
Minimax(Maxmin)criterion and optimal strategy
Saddlepoints
Solutionof Games with saddlepoints
Rectangular Games without saddlepoints
Minimax(Maxmin)principleforMixed strategyGames
Equivalence of RectangularGame and Linearprogrammingproblem
Solution of(m n) Games by Simplex method
Arithmetic method for(2 2) Games
concept of Dominance
Graphical method for (3 3)Games without saddlepoint.
Unit II
Inventory Problems:
Analytical structure of inventory Problem
ABCanalysis
EOQProblems with and without shortage with
(a) Production is instantaneous
(b) finite constantrate
(c) shortage permitted
random models where the demand follows uniformdistribution.
Unit III
Non -Linear programming
un constrained problems of Maxima and Minima
constrained problems of Maxima and Minima
Constraints in the form of Equations
LagrangianMethod
Suficient conditions for Max(Min)ofObjectivefunctionwithsingleequalityconstraint
With more than one equality constraints
Constraints in the form of Inequalities
Formulation of Non-Linearprogrammingproblems
General Non linear programming problem
Canonicalform-GraphicalSolution
Unit-IV
Quadratic programming
Kuhn-TuckerConditions
Non-negativeconstraints
Generalquadraticprogramming problem
Wolfe'smodified simplex method
Beales'sMethod
Simplex methodfor quadratic Program-ming.
TextBooks:
[1] S.D.Sharma,OperationsResearch.
[2] KantiSwarup,P.K.GuptaandManmohan,OperationsResearch.
[3] O.L.Mangasarian,Non-LinearProgramming,McGrawHill,NewDelhi.