Integral Equations & Calculus of Variations-AM401
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 differential equations and Volterra Integral equations
Resolvent Kernel of Volterra Integral equation.
Differentiation of some resolvent kernels
Solution of Integral equation by Resolvent Kernel
The method of successive approximations
Convolution type equations
Solution of Integro-differential equations with the aid of the Laplace Transformation
Volterra integral equation of the 1st kind
Euler integrals
Abel's problem, Abel's integral 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.
Hammerstein type 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.
Finite ElementMethods-AM402
Paper-II
Course Outcomes
1.Implement numerical methods to solve mechanics of solids problems.
2.Formulate and Solve axially loaded bar Problems.
3.Formulate and analyze truss and beam problems.
4.Implement the formulation techniques to solve two-dimensional problems using triangle and quadrilateral elements.
Unit I
Introduction.
Weighted Residual Methods.
Least Square Method.
PartitionMethod.
GalerkinMethod.
Moment Method.
Collocation Method. Variational Methods.
Ritz Method.
Unit II
Finite Elements.
Linesegment Element.
Triangular Element.
Rectangular Elements with examples.
Numerical Integration over Finite Elements.
Unit III
Finite Element Methods.
Ritz Finite Element Method.
Least Square Finite Element Method.
Galerkin Finite ElementMethod.
Boundary Value Problems in Ordinary Differential EquationS.
Assembly of Element Equations.
Boundary Value Problem in PDE's (with Linear triangular element).
Mixed boundary conditions.
Boundary points-Examples.
Unit-IV
Eigen value problemS.
Finite Element Error Analysis.
Approximation Errors.
Various measures of Error.
Convergence of solution.
Accuracy of the solution-Examples.
(5.1to5.4)of[2]
TextBooks:
[1] M.K.Jain,NumericalSolutionofDi erentialEquations.NewAgeInt.(P).Ltd.,NewDelhi.(forUnitsI,
IIandIII)
[2] J.N.Reddy,FiniteElementMethod,McGraw-HillInternationalEdition,EngineeringMechanicsSeries.
(for UnitIV).
Functional Analysis-AM403
Paper-III
Course Outcomes
1.Facility with the main, big theorems of functional analysis.
2.Ability to use duality in various contexts and theoretical results from the course in concrete situations.
3.Capacity to work with families of applications appearing in the course, particularly specific calculations needed in the context of Baire Category.
4.Be able to produce examples and counterexamples illustrating the mathematical concepts presented in the course.
Unit I
Normed Space.
BanachSpaces.
Further properties of normed spaces.
Finitedimensional normed spaces and subspaces
compactness and finite dimension.
linear operators.
Bounded and continuous linear operators.
Unit II
Linear Functionals.
normed spaces of operators.
Dualspace-Inner product space-Hilbert Space.
Fur-ther Properties of Inner product Spaces.
Orthogonal complements and direct sums.
Orthogonal sets and sequences.
Unit III
Series related to Ortho normal Sequences and sets.
Total Orthonormal sets and sequences.
Representation of Functions on Hilbert spaces.
Hilbert-Adjoin tOperator,Self-Adjoint,unitary and normal operators.
Unit-IV
Hahn-BanachTheorem.
Hahn-BanachTheorem for Complex Vector Spaces and Normed Spaces.
Adjoint Operator.
Category Theorem.
Uniform Boundedness Theorem.
Open Mapping Theorem.
Closed Linear Operators, Closed Graph Theorem.
TextBook:
Introductory Functional Analysis with Applications by Erwin Kreyszig, JohnWiley and sons,NewYork.
References:
[1] Functional Analysis by B.V.Limaye 2nd Edition.
[2] Introduction to Topology and Modern Analysis by G.F.Sinmmons.Mc.Graw-Hill International Edition.
Magneto hydro dynamics-AM404(B)
Paper-IV
Course Outcomes
1.describe the derivation of fluid equations, energy equation .
2.describe electromagnetic fields in the energy and momentum fluxes .
3.explain two fluid equations .
4.explain Landau damping .
Unit I
Introduction.
A brief remainder of the laws of electrodynamics.
Governing equations of Electrohy.
dro dynamics.
The electric eldand Lorentz force.
Ohm's law and volumetric Lorentz force.
Ampere'slaw.
Faraday's law in differential form.
reduced form of Max well equations for MHD.
Unit II
Transport equation for imposed magnetic field(B).
an important kinematic equation.
the significance of Faraday's law and Faraday's law in ideal conductors.
Vorticity.
Angular momentum and Biot-Savart Law.
Unit III
Advection and Difusion of Vorticity.
Kelvin's Theorem.
Helm holtz law and helicity.
Prandtl-Batchelor theorem.
Fluid flow in the presence of Lorentz force.
Equations of MHD and dimension less groups.
Maxwell stresses.
Unit-IV
Kinematics of MHD.
Analogy to vorticity.
Difusion of amagnetic field.
Advection in ideal conductors.
Alfven's theorem.
Manetich elicity.
Advection plus difusion.
Azimuthal field generation by differential rotation.
Magnetic reconnection.
TextBook:
P.A.Davidson,"AnIntroductiontomagnetohydrodynamics",CambridgeUniversityPress,2001.
References:
[1] PHRoberts,AnIntroductiontoMagnetohydrodynamics",Longman'sPublishers,1961.
[2] HKMo at,Magnetic eldgenerationinelectricallyconducting
uids,CambridgeUniversiytPress,1978.
[3] RMoreau,Magnetohydrodyanimcs,KluwerAcademyPublishers,1990.
Advanced Operations Research-AM405(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.