Courses Description

Limits, Continuity, Derivatives, The Chain Rule, Applications of derivatives, Mean Value theorem, Extreme Values, Curve Sketching, Integration, Integration by substitution running the chain rule backwards, Applications of integrals, Techniques of integration, Transcendental Functions-Natural Log and Exponential functions, L'Hôpital's Rule, inverse trigonometric function, hyperbolic functions Infinite Series, Complex Numbers

Text Book:1. ANTON, BIVENS & DAVIS, CALCULUS , Laurie Rosatone, (10th Edition) (2012). Reference Books:2. Thomas and George.B., Calculus and Analytical Geometry, Addison-Wesley,(11th Edition). 3. Ronald E Walpole and Raymond H. Myers, Probability and Statistics for Engineers and Scientists, Macmillan publishing Company, New York, 7th edition,(2001).

Conic Sections (hyperbola, parabola, ellipse circles), Vectors in IR (dot product, vector product) lines and planes, geometry of space curves, Multivariable Calculus (double integral, area volume, surface area, Jacobian, parametric equation and polar coordinates, multiple Integrals, Calculus of Vector Fields (line integral, surface integral independence of path, Green's theorem, divergence theorem, Stokes' theorem).

Text Book:1. ANTON, BIVENS & DAVIS, CALCULUS, Laurie Rosatone, (10th Edition) (2012). Reference Books:2. Thomas and George.B., Calculus and Analytical Geometry, Addison-Wesley,(11th Edition). 3. Ronald E Walpole and Raymond H. Myers, Probability and Statistics for Engineers and Scientists, Macmillan publishing Company, New York,7th edition,(2001).

First-order differential equations and their applications, linear differential equations with constant coefficients and their applications, Euler- Cauchy equations, method of variation of parameters, theory of power series method, Legendre's equations, Bessel's Equation. Introduction to system of differential equations. Laplace transform and its applications. Introduction to partial differential equations.

Text Book:Dennis G. Zill, Differential Equations, Prindle, Weber and Schmidt(3rd Edition), (1982). Reference Books:Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley and sons, (9th Edition), (2006). Earl D. Rainville, Phillip E. Bedient, Elementary Differential Equations Prentice Hall, Upper Saddle River, NJ07458 (8th Edition), (1997).

Basic Concepts, Matrix Operations, System of Linear Equations, Linear Independence, Iterative Solutions of Linear System, Introduction to Linear Transformations, Inverse of a Matrix, Matrix Factorization, Introduction to Determinants, Cramer's Rule, Vector Spaces, Eigenvalues and its Applications, Inner Product Spaces, Orthogonality, The Gram- Schmidt Process.

Text Book:1. David C. Lay, Linear Algebra and its Applications, Pearson Education, (3rd Edition), (2004). Reference Books:2. Erwin Kreyszig, Advanced Engineering Mathematics, Jhon Wiley and Sons, (8th Edition), (1982). 3. Howard Anton, Chris Rorres, Elementry Linear Algebra, Jhon Wiley and Sons, (8th Edition), (2000).

Solution of non-linear equations (bisection method, secant method and Newton's method), solution of linear algebraic equations, interpolation, numerical integration, finite difference, numerical solutions of ordinary differential equations (Euler's method, Runge-Kutta, boundary value problems). Computer implementation (Matlab(Retd) and C)

Text Book:1. Numerical Methods using MATLAB, John H. Mathews, Kurtis D. Fink, (4th Edition), (2009). Reference Books:1. Numerical Analysis , Richard L. burden, J. Douglas Faires, 7thEdition, Thomson Books/Cole (2005). 2. Numerical Analysis: A Practical Approach, Melvin J. Maron, Macmillan Company, Inc, New York.(1998) 4. Applied Numerical Analysis, Curtis F. Gerald and Patrick O. Wheatley, 6th Edition, Addison Wesley, Longman (2002).

A quick review of Complex numbers, Analytic function and their properties, elementary functions of Complex variable, Integration of complex functions, Cauchy-Gaursat theorem, Cauchy integral formula, Fundamental theorem of algebra. Series representation of analytic function, Residues and poles, evaluation of improper real integrals, Mappings by Elementary functions, Conformal mappings and their properties, some simple applications of conformal mappings.

Text Book:1. Dennis G. Zill, Patrick D. Shanahan, Complex Variables with Applications, Jones and Bertlett, (2nd Edition), (2003). Reference Books:2. Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons, (8th Edition), (1999). 3. Ruel V. Churchill, James Ward Brown, Complex Variables and Applications, McGraw-HILL, (4th Edition), (1984).

This course provides an elementary introduction to probability and statistics with applications. Topics include: basic probability models; combinatorics; random variables; discrete and continuous probability distributions; statistical estimation and testing; confidence intervals; and an introduction to linear regression.

Text Book:Ronald E Walpole and Raymond H. Myers, Probability and Statistics for Engineers and Scientists, Macmillan publishing Company, New York, 8th edition,(2005). Reference Book:Statistics for Engineers and sciences, William Madenhall and Terry Sincich 4th Edition, Prentice Hall. 3. Advance level Statistics, 3rd Edition, J-Crawshaw and J chambers. 4. Statistics for business and Economics, Paul Newbold, 4th Edition, Pentice Hall.

Ordered sets, supremum and infimum, completeness properties of the real numbers, limits of numerical sequences, limits and continuity, properties of continuous functions on closed bounded intervals, derivatives in one variable, the mean value theorem, sequences of functions, power series, point-wise and uniform convergence, functions of several variables, open and closed sets and convergence of sequences in , limits and continuity in several variables, properties of continuous functions on compact sets, differentiation in n – space, the Taylor series in with applications, the inverse and implicit function theorems.

1. Reference Book: Robert G. Bartle, Donald R. Sherbert. Introduction to Real Analysis, 4th Edition, John Willey, New York (2011) RL. Brabenec, obert. Introduction to Real Analysis, PWS Publishing Company,(1997)

Definition and Examples of Groups, Order of Group, Order of an Element, Abelian Groups, Subgroups, Cyclic Groups. Fundamental Theorem of Cyclic Group , Complexes and Coset Decomposition of Groups, Index of Subgroup in a Group., Lagrange’s Theorem and Applications, Centre of a Group, Normalizer in a Group, Centralizer in a Group, Conjugacy Relation and Congruence Relation in a Group. 16, Subgroup, Cyclic groups, Coset decomposition of a group, Lagrange’s theorem and its consequences, Conjugacy classes, Centralizers and Normalizes, Normal subgroups and Quotient groups, Definition of a homomorphism, Endomorphism and automorphism, Isomorphism theorems and related results,Definitions and basic results of, Cosets, left cosets and right cosets, Doublecosets and related theorems, Cauchy's theorems for abelian groups, Cauchy's theorems for general groups, Sylow's Theorems.

Reference Book: 1. ABSTRACT ALGEBRA by D. S. Dummit and R. M. Foote, 2nd Edition, John Wiley & Sons, Inc. 2. A FIRST COURSE IN ABSTRACT ALGEBRA by J. B. Fraleigh, 7th Edition, Pearson Education

Historical background; Motivation and applications. Index notation and summation convention; Space curves; The tangent vector field; Re parametrization; Arc length; Curvature; Principal normal; Binormal; Torsion; The osculating, the normal and the rectifying planes; The Frenet-Serret Theorem; Spherical images; Sphere curves; Spherical contacts; Fundamental theorem of space curves; Line integrals and Green’s theorem; Local surface theory; Coordinate transformations; The tangent and the normal planes; Parametric curves; The first fundamental form and the metric tensor; Normal and geodesic curvatures; Gauss’s formulae; Christoffel symbols of first and second kinds; Parallel vector fields along a curve and parallelism; The second fundamental form and the Weingarten map; Principal, Gaussian, Mean and Normal curvatures; Dupin indicatrices; Conjugate and asymptotic directions; Isometries and the fundamental theorem of surfaces.

1. Reference Books:Kuhnel, Wolfgang. Differential Geometry: Curves – Surfaces – Manifolds. Student mathematical library, vol. 16. Providence, RI: American Mathematical Society, 2002. Gray, Alfred, Simon Salamon, and Elsa Abbena. Modern Differential Geometry of Curves and Surfaces with Mathematica. Boca Raton, FL: Chapman & Hall/CRC, 2006.Millman, R.S and Parker, G.D. Elements of Differential Geometry (Prentice-Hall Inc., New Jersey, 1977).

Metric Spaces, open sets, closed sets, convergence and continuity in metric Spaces Course, Topological spaces, bases and subspaces, product topology, subspace topology, closed sets and limit points, closure, imterior and boundary, Housdorff spaces, homomorphism, homomorphic spaces, compactness, connectedness, first-countable and second countable spaces, regular and normal spaces.

Reference Book: 1. E. Paul Long, An Introduction to general topology. 2. J. F. Simmons, Introduction to topology modern analysis, McGram Hill, New York

Linear programing, simplex method, duality theory, unconstrained optimization, optimality conditions, one dimensional problems, multi-dimensional problems, method of steepest decent, constrained optimization with equality constraints, optimality conditions, Lagrange multipliers, Hessians and bordered Hessians. Inequality constraints and Kuhn-Tucker theorem.

1. Reference Book: D. Bertsimas and J. N. Tsitsiklis, Introduction to Linear Optimization. D. G. Luenberger, Linear and Nonlinear Programing. Addison-Wesley, 1984. S. P Boyd and L. Vandenberghe, Convex Optimization W. L. Winston, Introduction to Mathematical Programing Duxbury Press, Second Edition, 1995. R. Vanderbei, Linear Programing: Foundations and Extensions Princeton University Press

This course provides a brief review of basic probability concepts and distribution theory. It covers mathematical properties of distributions needed for statistical inference. Measure of central tendency, measure of dispersion, measure of skewness and kurtosis, Sample spaces, events (Borel sets), axioms and laws of probability, independence, conditional probability, Bayes Theorem, discrete and continuous random variables and vectors, distribution functions, densities, Expectation & Functions of Random Variables, Specific Parametric Distributions-Univariate: Binomial, Poisson, Hypergeometric, geometric, negative binomial, exponential, gamma, normal and related families, exponential families, Joint distributions, conditional distributions, independence of random variables, probability inequalities for random variables (Chebyshev, Jensen), Multivariate distributions: trinomial, multinomial, bivariate normal, Random samples, central limit theorem, laws of large numbers, Slutzky's theorem, Normal models, order statistics

Reference Book: 1. Introduction to Mathematical Statistics, Hogg and Craig, Prentice Hall, Upper Saddle RIver 2. Mathematical Statistics with Applications, by D.D. Wackerly, W. Mendenhall and R.L. Scheaffer, Duxbury Press, 6th edition(2002)

Series of numbers and their convergence, series of functions and their convergence, Dabroux upper and lower sums and integrals, Dabroux integrability, Riemann sums and the Riemann integral, Riemann integration in , change of order of variables of integration. Riemann integration in, and, Riemann-Steiltjes integration, Functions of bounded variation, the length of a curve in .

Reference Book: Robert G. Bartle, Donald R. Sherbert. Introduction to Real Analysis, 4th Edition, John Willey, New York (2011) 2. RL. Brabenec, obert. Introduction to Real Analysis, PWS Publishing Company,(1997)

Partial differential equations of the first order. Nonlinear PDEs of first order Applications of 1st order partial differential equations. Partial differential equations of second order, Mathematical modeling of heat, Laplace and wave equations. Classification of 2nd order PDEs Boundary and initial conditions. Reduction to canonical form and the solution of 2nd order PDEs, Technique of separation of variable for the solution of PDEs with special emphasis on Heat, Laplace and wave equations. The method of spherical means, Kirchhoff's formula and Minkowskian geometry, Geometric energy estimates for wave equations. Laplace, Fourier and Hankel transform for the solution of PDEs and their application to boundary value problems.

>Reference Books: 1. Haberman, R., Elementary Applied Partial Differential Equations, Prentice Hall, Inc.New Jersey, 1983.2. Sneddon, I.N., Elements of Partial Differential Equations, McGraw-Hill Book Company, 1987. 3. Zauderer, E., Partial Differential Equations of Applied Mathematics, John Wiley & Sons, Englewood Cliff, New York, 1983.

Metric Spaces, open sets, closed sets, convergence, completeness, normed Spaces,Banach spaces, Bounded and continuous linear operators and functionals, Dual spaces, Finite dimensional spaces, Inner-product space, Hilbert space, orthogonal and orthonormal sets, orthogonal comple-ments, Gram-Schmidt orthogonalization process, representation of functionals, Reizrepresentation theorem, weak and weak* Convergence.

Reference Book: Dover Kreyszig E, Introductory Functional Analysis with Applications, John Wiley,New York. 2. Rudin W, Functional Analysis, 1973, McGraw Hill, New York

Kinematics: Rectilinear motion of particles. Uniform rectilinear motion, uniformly accelerated rectilinear motion. Curvilinear motion of particle, rectangular components of velocity and acceleration. Tangential and normal components. Radial and transverse components. Projectile motion. Kinetics: Work, power, kinetic energy, conservative force fields. Conservation of energy, impulse, torque. Conservation of linear and angular momentum. Non-conservative forces. Simple Harmonic Motion: The simple harmonic oscillator, period, frequency. Resonance and energy. The damped harmonic oscillator, over damped, critically damped and under damped. Motion, forces and vibrations. Central Forces and Planetary Motion: Central force fields, equations of motion, potential energy, orbits. Kepler’s law of planetary motion. Apsides and apsidal angles for nearly circular orbits. Motion in an inverse square field. Planer Motion of Rigid Bodies: Introduction to rigid and elastic bodies, degree of freedom, translations, rotations, instantaneous axis and center of rotation, motion of the center of mass. Euler’s theorem and Chasles’ theorem. Rotation of a rigid body about a fixed axis, moments and products of inertia. Parallel and perpendicular axis theorem. Motion of Rigid Bodies in Three Dimensions: General motion of rigid bodies in space. The momental ellipsoid and equimomental systems. Angular momentum vector and rotational kinetic energy. Principal axes and principal moments of inertia. Determination of 17 principal axes by diagonalizing the inertia matrix. Euler Equations of Motion of a Rigid Body: Force free motion. Free rotation of a rigid body with an axis of symmetry. Free rotation of a rigid body with three different principal moments. The Eulerian angles, angular velocity and kinetic energy in terms of Euler angles. Motion of a spinning top and gyroscopes-steady precession, sleeping top.

1. Reference Book: E. DiBenedetto, Classical Mechanics. Theory and Mathematical Modeling, Birkhauser Boston, 2011. John R. Taylor, Classical Mechanics, University of Colorado, 2005. H. Goldstein, Classical Mechanics, Addison-Wesley Publishing Co., 1980. C. F. Chorlton, Text Book of Dynamics, Ellis Horwood, 1983. M. R. Spiegel, Theoretical Mechanics, 3rd Edition, Addison-Wesley Publishing Company, 2004. G. R. Fowles and G. L. Cassiday, Analytical Mechanics, 7th edition, Thomson Brooks/COLE, USA, 2005.

Constraints, Degree of freedom, General 3-D rigid body dynamics, Holonomic and non-holonomic constraints.Generalized co-ordinates, general equation of dynamics, Lagrange’s equations, conservation laws, ignorable co-ordinates. Lagrange’s equations for holonomic systems, Explicit form of Lagranges equation in terms of tensors. Hamilton’s principle, principle of least action, Hamilton’s equations of motion, Hamilton-Jacobi Method. Poisson Brackets (P.B’s); Poisson’s theorem; Solution of mechanical problems by algebraic technique based on (P.B’s). Small oscillations and normal modes, vibrations of strings, transverse vibrations, normal modes, forced vibrations and damping, reflection and transmission at a discontinuity, Longitudinal vibrations, Rayleigh’s principle.

Reference Books:1. Chorlton, F., Textbook of dynamics, Van Nostrand, 1963. 2. G. Meirovitch. L., Methods of Analytical Dynamics, McGraw-Hill, 1970. 3. Murray R. Spiegel, Theory and problems of theoretical mechanics. SI (metric) edition McGraw-Hill, New York, 1967. 4. D.T. Greenwood, Principles of Dynamics (2nd Edition)

Introduction to Operations Research and real life Phases, Introduction to Linear Programming (LP) with examples, Graphical Solutions to Mathematical Model with Special Cases, Simplex Algorithm and its different cases, Big M Method and Two Phase Method, Sensitivity Analysis/ Post Optimality Analysis, Duality and its Economic Interpretation, Dual Simplex Method, Scheduling and Blending Problems, The Transportation Problems, The Transshipment Problems, The Assignment Problems, Integer Programming, Network Models, Inventory Models, Dynamic Programming and Queuing Theory.

Reference Book: 1. Hamdy A. Taha, Operations Research - An Introduction, (Macmillan Publishing Company Inc., New York, 1987).

Functions of Random Variables: Distribution function technique; Transformation technique: One variable, several variables; Moment-generating function technique Sampling Distributions: The distribution of the mean; The distribution of the mean: Finite populations; The Chi-Square distribution; The t distribution; The F distribution Regression and Correlation: Linear regression; The methods of least squares; Normal regression analysis; Normal correlation analysis; Multiple linear regression; Multiple linear regression (matrix notation)

Reference Books: 1. J. E. Freund, Mathematical Statistics, (Prentice-Hall Inc). 2. Hogg and Craig, Introduction to Mathematical Statistics, (Collier Macmillan). 3. Mood, Greyill and Boes, Introduction to the Theory of Statistics, (McGraw Hill). 4. R. E. Walpole, Introduction to Statistics, latest edition, (Macmillan Publishing Company London) 5. M. R. Spiegel, L. J. Stephens, Statistics, (McGraw Hill Book Company)

Proportional and first-order logic, Equivalence, inference and method of proof, Mathematical induction, diagonalization principle, Basic counting, Recursion, Graphs, Trees, Spanning Trees Set operations, relations, functions, Boolean algebra, Truth tables and minimization of Boolean expressions, Applications to computer science and engineering.

Reference Books:1. Discrete Math and Applications, Rosen, (latest edition) 2. Discrete Math and Applications, Susanna S. Epp

Eulerian approach, Lagrangian description, Properties of fluids, Transport properties, Kinematic properties, thermodynamics properties, Boundary conditions for viscous flows and heat conducting flows problems, Conservation of mass (equation of continuity), conservation of momentum (equations of Navier-Stokes equations ), conservation of energy (energy equations), Dimensionalization and dimensionless parameters in viscous flow, Vorticity transport equation, Stream function, Steady flow, unsteady flow, creeping flow and boundary layer flow, Couette flows, Poiseuille flow, Couette Poiseuille flow between parallel plates, Stokes first problem, Stokes second problem, Unsteady flow between two infinite plates, Asymptotic suction flows: uniform suction on a plane, flow between parallel plates with top suction and bottom injection

Reference Book: 1. Frank M. White, Viscous Fluid Flow, Second Edition, McGRAW-HILL, Inc. 2. Hermann Schlichting,Boundary-layer Theory ,Seventh Edition, McGraw-Hill Series in MechanicalEngineering. 3. G.K. Batchelor, An introduction to fluid dynamics,Cambridge University Press.

Integral equation formulation of boundary value problems, classification of integral equations, method of successive approximation, Hilbert-Schmidt theory, Schmidt’s solution of non- homogeneous integral equations, Fredholm theory, case of multiple roots of characteristic equation, degenerate kernels. Introduction to Wiener-Hopf technique.

Reference Book: 1. B. Noble., Methods based on the Wiener-Hopf technique, Pergamon Press, 1958. 2. Abdul J. Jerri., Introduction to integral equations with applications, Marcel Dekker Inc. New York, 1985.

Measure Spaces with measure, measurable function, idea of − σ fields, outer measure, Lebesgue measure, measurable sets, complete measure spaces, measurable functions, Egorov’s theorem, Lebesgue Integration, comparison between Riemann integration and Lebesgue integration, L2-space, the Riesz-Fischer theorem.

Reference Book: 1. Holmos PR, Measure Theory, van Nostrand, New York. 2. D. L. Cohn, , Measure Theory, Birkhauser 1980.

Rings and Fields, Definitions and Examples of subrings, Ideals, homomorphisms and related theorems. Properties of subrings and Ideals, product of Ideals, Intersection and Sum of ideals. Homomorphism theorems. Prime ideals, Principal ideals, Maximal ideals, regular rings, Internal and External direct sums of rings. Polynomial rings, Matrix rings and their properties. Unique factorization domain, factorization theory. Principal ideal domains and Euclidean domains, arithmetic in Euclidean domains.

Reference Books: 1. Fraleigh, J.A., A First Course in Abstract Algebra, Addision Wesley Publishing Company, 1982 Herstein, I.N., Topies in Algebra, John Wiley & Sons 1975 Lang, S., Algebra, Addison Wesley 1965 Hartley, B., and Hawkes, T.O., Ring, Modules and Linear Algebra, Chapman and Hall, 1980.

Definition and examples of manifolds; Differential maps; Submanifolds; Tangents; Coordinate vector fields; Tangent spaces; Dual spaces; Multilinear functions; Algebra of tensors; Vector fields; Tensor fields; Integral curves; Flows; Lie derivatives; Brackets; Differential forms; Introduction to integration theory on manifolds; Riemannian and semi-Riemannian metrics; Flat spaces; Affine connections; Parallel translations; Covariant differentiation of tensor fields; Curvature and torsion tensors; Connection of a semi-Riemannian tensor; Killing equations and Killing vector fields; Geodesics; Sectional curvature.

Reference Books: Kuhnel, Wolfgang. Differential Geometry: Curves – Surfaces – Manifolds. Student mathematical library, vol. 16. Providence, RI: American Mathematical Society, 2002. Gray, Alfred, Simon Salamon, and Elsa Abbena. Modern Differential Geometry of Curves and Surfaces with Mathematica. Boca Raton, FL: Chapman & Hall/CRC, 2006.A.Pressley Elementary Differential Geometry, 2nd Edition, Springer (2012) Millman, R.S and Parker, G.D. Elements of Differential Geometry (Prentice-Hall Inc., New Jersey, 1977).

Some Theorems from the Theory of Differential Equations; Initial Value Problems for First Order Ordinary Differential Equations and for Systems of First Order Ordinary Differential Equations Linear Difference; Equations with Constant Co-efficient , The General Linear Multistep Methods; Convergence; Order and Error Constant; Local and Global Truncation Error; Consistency and Numerical Stability; General Methods for Finding Intervals of Absolute and Relative Stability; Predictor-Corrector Methods; Derivation of Classical Runge-Kutta Methods; Runge-Kutta Methods of order Greater Than Four; Error Estimates and Error Bounds for Runge Kutta Methods; Comparison with P Predictor-Corrector Methods; Implicit Runge-Kutta Methods.

Reference Book: 1. Greenspan, Numerical solutions of ODE’s for classical Relativistic and Nanosystems, 2006. 2. C. E. Froberg, Numerical mathematics, The Benjamin Cummings Pub. Com. Inc., 1985. 3. G. M. Phillips, P. J. Taylor, Theory and Applications of Numerical Analysis, Academic Press, 1973. 4. W. E. Pre et al., Numerical Recipes, Cambridge University Press, 1986. 5. M. K. Jain, Numerical Solution of Differential Equations, Wiley Eastern Ltd. 6. W. E. Milne, Numerical Solution of Differential Equations, Dover Pub. Inc., N.Y.

Review of basic complex analysis, Maximum modulus principle, Schwarz lemma, Phragmen-Lindel of theorems, zeros of analytic functions, Jesen‘s formula, Weirestrass factorization theorem, gamma function, Stiring‘s approximation.

Reference Books: 1. S. Lang, complex analysis, Springer- Verlag. 2. Ahlfors, complex analysis.

Brief review of complex functions and complex integrations, class of analytic functions, Riemann Mapping Theorem, class of univalent functions, class of convex univalent and starlike univalent functions, close-to-convex functions, Coefficient bounds, growth and distortion results of functions contained in subclasses of the class of univalent functions, class of Caratheodary functions with positive real part and related classes, functions with bounded radius rotation, functions with bounded boundary rotation, differential subordination, Hadamard product (or convolution), some linear differential and integral operators.

Similarity solution, Berman problem, Plane stagnation flow, axisymmetric stagnation flow, flow near an infinite rotating disk, Jeffery Hammel flow in a wedge shaped region and it solution for small wedge angle, Stokes solution for an immersed sphere, Derivation of boundnry-Iayer equations for two-dimensional flow, The laminar boundary layer equations, The approximate method due to the von Karman and K. Pohlhausen for two dimensional flows, Blasius problem of flat plate flow, Falker-Skan wedge flows, Heat transfer for Falker-Skan flows, two dimensional steady free convection, viscous flows over a stretching sheet, thin film flows.

Reference Books: 1. Frank M. White, Viscous Fluid Flow, Second Edition, McGRAW-HILL, Inc. 2. Hermann Schlichting,Boundary-layer Theory ,Seventh Edition, McGraw-Hill Series in MechanicalEngineering. 3. G.K. Batchelor, An introduction to fluid dynamics,Cambridge University Press.

Introduction, functional, Fundamental theorem of variational calculus, Euler-Lagrange Equation, Brachistochrone problem, variation of end point, Isoperimetric problems, Dido problem, Geodesics problems, Euler-Lagrange equation with Several variables.

Reference Books: Curant, R. and D. Hilbert: Methods of Mathematical Physics, Vol I. Interscience Press, 1953. Elsgolc, L.E.: Calculus of Variations, Pergamon Press Ltd., 1962. Weinstock, Robert: Calculus of Variations with Applications to Physics and Engineering, Dover, 1974.

Gradient method, subgradient method, proximal gradient method, Nesterov's acceleration technique, alternating direction method of multi-pliers, coordinate descent method, and stochastic/randomized algorithms. Applications of these optimization methods for solving problems in contemporary applications arising from big data analytics, machine learning, statistics, signal processing

Reference Books: 1. J. Nocedal and S.J. Wright, Numerical Optimization, Springer, 2006 2. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer, 2003

History and overview of cryptography, One time pad and stream ciphers, Block ciphers, Block cipher abstractions: PRPs and PRFs, Attacks on block ciphers, Message integrity: definition and applications, Collision resistant hashing, Authenticated encryption: security against active attacks, Arithmetic modulo primes, Cryptography using arithmetic modulo primes, Public key, Arithmetic modulo composites encryption, Digital signatures: definitions and applications, More signature schemes, Identification protocols, Authenticated key exchange and SSL/TLS session setup, Zero knowledge protocols

1. Reference Books: J. Katz and Y. Lindell, Introduction to Modern Cryotography. A. Menezes, P. Van Oorschot, S. Vanstone, Handbook of Applied Cryptography

Quotient topology, homotopy of paths, the fundamental group, covering spaces, fundamental group of the circle, retraction and fixed points, Borsuk-Ulam theorem, deformation retracts and homotopy type, fundamentalbgroups of various surfaces, direct sums of Abelian groups, free products of groups, free groups, Seifert-van Kampen theorem and applications.

Reference Books: 1. Allen Hatcher, Algebraic Topology, 2001. 2. Tom Dieck, Tammo, Algebraic topology, European Mathematical Society (2010).

Introductory mathematical concepts, Deformation and strain, Stress, Plane theory of elasticity (Cartesian coordinates), Plane theory of elasticity (polar coordinates), Three-dimensional elasticity theory, Prismatic bars subjected to end loads, General solutions of elasticity,

1. Reference Books: Arthur P. Boresi, Ken P. Chong, James D. Lee. “Elasticity in Engineering Mechanics”, (latest edition) John Wiley &Sons, Inc., 2011,

The course will emphasize the solution of real life problems using the finite element method underscoring the importance of the choice of the proper mathematical model, discretization techniques and element selection criteria. Introduction to finite element analysis, Direct stiffness approach: Spring elements, Bar and truss elements, Introduction to differential equations and strong formulation, Principle of minimum potential energy and weak formulation, Finite element formulation of linear elastostatics, The constant strain triangle, The quadrilateral element, Practical considerations in FEM modeling, Convergence of analysis results, Higher order elements, Isoparametric formulation.

1. Reference Books: A First Course in the Finite Element Method Author: Daryl Logan Year Published: 2011 Edition: Fifth Publisher: Cengage Learning.

This course gives a broad overview of Monte Carlo methods. These methods include many important tools for students interested in applied probability,finance and statistics,Pseudo-random numbers and quasi-random numbers,Generating random variables ,Generating stochastic processes ,Markov Chain Monte Carlo ,Variance reduction,Gradient estimation, Stochastic optimization.

Reference Books: Glasserman, P. Monte Carlo Methods in Financial Engineering. Springer, 2004. 2. Kroese, D. P., T. Taimre and Z. Botev. Handbook of Monte Carlo Methods. Wiley, 2011. 3. Robert, C. P. and G. Casella. Monte Carlo Statistical Methods, Second Edition. Springer, 2005.

Definition and properties of Gamma function, Beta function, Incomplete Gamma function, Definition and generating function of Legendre polynomials. Recurrence relation and Legendre polynomials, Recurrence relation and Legendre differential Equations, Rodrigue’s formula, Hermite polynomials, Bessel functions An integral form of Bessel functions and orthogonality, Differential equation solvable with Bessel functions.

Reference Books: L. C. Andrews, Special functions for Engineers and Applied Mathematics, McMillan Publishing Company. N.W. Lebedev, Special functions and their applications, Dover Publishing Inc., 1972 E.D. Rainvill, Special Functions, McGraw Hill, 1992

Measurements and units, Basic Quantities ,International system of units, British system of units and their inter-conversion, force and motion, friction and drag force, viscosity, newton’s law of motion, newton’s law of universal gravitation, conservation laws, thermal properties, Newton’s law of cooling, electric and magnetic properties, charge, current, capacitor, resistor, inductor, Ohm’s law, Coulomb’s law, Faraday’s laws, Elastic Properties; Stress, Strain, Hooks law, Torsion

1. Reference Book: Fundamentals of Physics, Extended Edition, David Halliday, Robert Resnick, Jearl Walker

This course is an advanced introduction to applications and theory of numerical methods for solution of differential equations, especially of physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods. Topics include finite differences, spectral methods, well-posedness and stability, boundary and nonlinear instabilities. The course assumes familiarity with basic (numerical) linear algebra and will involve a certain amount of programming in MATLABTM or any programming language of your choice.

Reference Books: 1. R. L. Burden, J. D. Fairs; An Introduction to Numerical Analysis, 1993.2. G. D. Smith, Numerical Solutions of P.D.Es, 1999. 3. J. H. Wilkinson, The Algebraic Eigenvlaue Problems, 1965. 4. U. Asher et al., Numerical solution of Boundary Value Problems in ODE’s, 1986. 5. Numerical Mathematics, Matheus Grasselli, Dmitry Pelmvovsky, Jonez Batlett, 2009

Pre requisite: Undergraduate Engineering Mathematics Applied calculus, application of numerical linear algebra, vector calculus, Ordinary and Partial Differential Equations; solution techniques with exact and approximate methods. Coupled equations. Difference equations. Integro-differential equations and solution techniques. Integral equations (Volterra and Fredholm type).Exact solutions for infinite and semi- infinite systems, Optimization of multivariable calculus, quadratures, Laplace and Fourier transformations.

This course surveys the basic concepts of computer modeling in science and engineering using discrete particle systems and continuum fields. It covers techniques and software for statistical sampling, simulation, data analysis and visualization, and uses statistical, quantum chemical, molecular dynamics, Monte Carlo, mesoscale and continuum methods to study fundamental physical phenomena encountered in the fields of computational physics, chemistry, mechanics, materials science, biology, and applied mathematics. Applications are drawn from a range of disciplines to build a broad-based understanding of complex structures and interactions in problems where simulation is on equal footing with theory and experiment. A term project allows development of individual interests. Students are mentored by a coordinated team of participating faculty from across the Institute.

A presentation of the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic and hyperbolic partial differential equations and integral equations central to a wide variety of applications in science, engineering, and other fields. Topics include: Mathematical Formulations; Finite Difference and Finite Volume Discretizations; Finite Element Discretizations; Boundary Element Discretizations; Direct and Iterative Solution Methods.

1. Reference Books: The Finite Element Method, P.E. Lewis & J.P. Ward

MA 700 Computational Fluid Dynamics

Pre-requisite: Undergraduate course on Fluid Mechanics/Dynamics Numerical solution of 1D, 2D and 3D Transport equations, Euler's equations, Burger's equations, Incompressible Navier-Stokes equations, compressible Navier-Stokes equations, advanced numerical techniques, multi-grid methods, and spectral methods.

1. Reference Book: Computational Fluid Dynamics, K.A. Hoffmann, S.T Chang EES, USA

This course is about modeling multi-domain engineering systems at a level of detail suitable for design and control system implementation. It also describes Network representation, state-space models, Multiport energy storage and dissipation, Legendre transforms Nonlinear mechanics, transformation theory, Lagrangian and Hamiltonian forms, and Control- relevant properties. The application examples may include electro- mechanical transducers, mechanisms, electronics, fluid and thermal systems, compressible flow, chemical processes, diffusion, and wave transmission.

Normed Spaces, Approximation by Algebraic Polynomials ,Exercises on Approximation by Polynomials, Approximation by Trigonometric Polynomials, Exercises on Trigonometric Polynomials, Characterization of Best Approximation, Exercises on Chebyshev Polynomials, Simple Application of Chebyshev Polynomials, Lagrange Interpolation, Orthogonal Polynomials, Gaussian Quadrature

1. Reference Book: Approximation Theory and Methods, M.J.J.D. Powell, 1982

This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions.

2. Reference Books: Linear Partial Differential Equations for Scientists and Engineers, Myint-U, Tyn, and Lokenath Debnath, 4th Edition, 2006.

Modeling and conservation laws, Scaling and non-dimensionalisation, Simple boundary layer theory, Nonlinear oscillations, Multiple scale methods, Ordinary differential equations: hysteresis and stability, SturmLiouville systems, Integral equations and eigenfunctions, Partial differential equations: shocks, similarity solutions, Calculus of Variations; optimal control.

Reference Books: 1. Strang, Gillbert,; ‘ Computational Science and Engineering’, Welleslet-Cambridge Press, 2007. 2. D. L. Powers, Boundary Value Problems and Partial Differential Equations, Academic Press, 2005. 3. W. E. Boyce, Elementary Differential Equations, John Wiley and Sons, 2005. 4. M. L. Krasnov, G. I. Makarenko and A. I. Kiselev, Problems and Exercises in the Calculus of Variations, Imported Publications, Inc., 1985. 5. A. D. Snider, Partial Differential Equations: Sources and Solutions, Prentice Hall Inc., 1999.

Systems of partial differential equations, characteristics, Weak solutions, Riemann's function, Maximum principles, comparison methods, well- posed problems, and Green's functions for the heat equation and for Laplace's equation, Fredholm alternative and Green's functions for non-self-adjoint problems, application of delta functions, Further development of hyperbolic equations, Cauchy-Kovalevskaya theorem, Riemann invariants, shocks and weak solutions, causality.

Linear Algebra is a central and widely applicable part of Mathematics. It is estimated that many (if not most) computers in the world are computing with matrix algorithms at any moment in time whether these be embedded in visualization software in a computer game or calculating prices for some financial option. This course builds on elementary linear algebra and in it we derive, describe and analyze a number of widely used constructive methods (algorithms) for various problems involving matrices.

Lagrangian and Eulerian descriptions of motion, analysis of strain, Balance laws of continuum mechanics, Frame-indifference, Constitutive equations for a nonlinear elastic material. Linear elasticity as a linearization of nonlinear elasticity. Incompressibility and models of rubber. Exact solutions for incompressible materials, phase transformations, shape-memory effect.

Strong solutions, questions of existence and uniqueness, diffusion processes, Cameron Martin formula, weak solution and martingale problem. Some selected applications chosen from topics including option pricing and stochastic filtering.

1. Reference Book: Stochastic Differential Equations, An introduction with Applications, B. Oksendal, 6th Edition.

Convection, stability, boundary layers, parameterized convection, Rotating flows, atmosphere and oceans, Waves, geostrophy, boilers, condensers, fluidised beds. Flow régimes. Homogeneous, drift-flux, two-fluid models. Ill-posedness, waves, density wave oscillations. Coatings and foams. Gravity flows, Droplet dynamics, contact lines, Drying and wetting, Foam drainage.

Reference Books: 1. Frank M. White, Viscous Fluid Flow, Second Edition, McGRAW-HILL, Inc. 2. G.K. Batchelor, An introduction to fluid dynamics,Cambridge University Press.

Introduction to signal processing, Signal formation and sampling in MRI and FMRI, Representation of a signal in the frequency domain (Fourier methods), Compression (wavelets), Filter operations for attenuation of a signal, Extracting information from a signal without a model (PCA,ICA), De-noising, Applications in speech and biomedical signal processing, Applications in learning how the brain functions (FMRI).

1. Reference Book: Mathematical Methods and Algorithms for Signal Processing, Todd K. Moon, Wyn C. Stirling, Prentice Hall, 2008.

The course starts with some background to the geology, geophysics and engineering involved in the recovery of oil from deep inside the Earth. Flow through porous media - balance laws, Darcy's law, analytical and numerical methods, Grid generation and geometric modeling - surface modeling, structured and unstructured grids, stochastic sampling, groundwater modeling, the study of subsurface pollution and remediation, and even to problems outside the geosciences.

1. Reference Books: The Theory of Critical Phenomena, J. J. Binney, N. J. Dowrick, A. J. Fisher and M. E. J. Newman, 1992. Fundamentals of Numerical Reservoir Simulation, D. W. Peaceman, 1977.

Equations of inviscid compressible flow including flow relative to rotating axes, Models for linear wave propagation including Stokes' waves, Inertial waves, Rossby waves and simple solutions, Theories for Linear waves: Fourier Series, Fourier integrals, method of stationary phase, dispersion and group velocity. Flow past thin wings, simple wave flows applied to one-dimensional unsteady gas flow and shallow water theory, Shock Waves: weak solutions, Rankine Hugoniot relations, oblique shocks, bores and hydraulic jumps.

1. Reference Book: Waves and Compressible Flow, H. Ockendon, J. R. Ockenden, 2004.

This is an advanced course in statistical analysis which highlights the significance of planning, collection, analysis and interpretation of complex data. Students are taught to devise research problems and hypothesis and prepare the necessary tests to avoid Type I or Type II errors. The course also utilizes extensive use of univariate, bivariate and discriminant analysis. The last few units stress on deriving useful information from the tests in order to help students make smart decisions.

1. Reference Books: Introduction to the theory of statistics, Mood A. M., Graybill F. A. Boes D. C. 1998. Introduction to statistical inference, Kiefer, Jack C. Statistical Inference, George Casella, Roger L. Berger

It covers a vast variety of mathematical concepts including linear equations and functions, quadratic and polynomial functions, exponential and logarithmic functions, matrix algebra, set theory, permutations and combinations, probability, random variables, frequency distribution time series, measures of central tendency, measures of dispersion, estimation, forecasting and index numbers.

Reference Books: Frank S. Budnick, Applied Mathematics for Business, Economics and Social sciences, McGraw Hill, New York, 4th edition, (1993). 2. Levin Rubin, Statistics for Management, Printice Hall International, 7th edition(1998)

Introduction to symmetry methods, Symmetry methods for ODEs and PDEs manually and by using computer packages, Invariants, canonical variables using the symmetries of differential equations, solution of ODEs and PDEs using classical methods, Linearization of the ODE and system of ODEs using point symmetries, Computation of the first integrals for ODEs and conservation laws for PDEs using suitable methods, Exact solutions of ODEs and PDEs via first integrals and conservation laws.

Reference Books: CRC handbook of Lie group analysis of differential equations, Volume 1: Symmetries, exact solutions and conservation laws by Nail H. Ibragimov (CRC Press) 2. Applications of Lie groups to differential equations by Peter J Olver (Springer - Verlag) 3. Differential equations: Their solution using symmetries by Hans Stephani (Cambridge University Press)

Brief review of complex functions and complex integrations, class of analytic functions, Riemann Mapping Theorem, class of univalent functions, class of convex univalent and starlike univalent functions, close-to-convex functions, Coefficient bounds, growth and distortion results of functions contained in subclasses of the class of univalent functions, class of Caratheodary functions with positive real part and related classes, functions with bounded radius rotation, functions with bounded boundary rotation, differential subordination, Hadamard product (or convolution), some linear differential and integral operators.

1. Reference Books: L. V. Ahlfors, Conformal Invariants. Topics in Geometric Function Theory, McGraw–Hill, New York, 1973. J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1993. Goodman, A.W., Univalent functions, Vol. I & II, polygonal publishing house, Washington, New Jersey (1983). Duren, P. L., Univalent functions, Grundlehren der Math. Wissenchaften, Springer-Verlag, NewYork-Berlin (1983).

Vector spaces, Normed Spaces, Inner Product spaces, Completeness, Hilbert Spaces, Linear and bounded operators, Orthoprojections, Isometric and unitary operators, Continuous functions of self-adjoint operators, Coercivity, Elliptic forms, Regularity, closed operators, adjoint and eigenfunction expansions, Spectral Integrals, Differential Operators in .

Reference Books: 1. Karl E. Gustafson, Introduction to Partial Differential Equations and Hibbert Space Methods 1997. 2. R. A. Kennedy, P. Sadeghi, Hilbert Space Methods in Signal Processing, 2013.

Basic concepts, Existence and uniqueness results. Fixed-points formulation, Wiener-Hopf equations, Equivalence between variational inequalities and Wiener-Hopf equations, Iterative methods, Auxiliary principle techniques, Dynamical systems, Sensitivity analysis, convergence analysis, numerical solutions of obstacle problems, variational inclusions, resolvent equations, applications . Convex sets, convex hull, their properties, separation theorems, hyperplane, Best approximation theorem and its applications, Farkas and Gordan Theorems, Extreme points and Polyhedral. Convex functions, Basic Definitions, properties, various generalizations, differentiable convex functions, subgradient, characterization and applications in linear and nonlinear optimization, complementarity problems and its equivalent formulations.

Reference Books: D. Kinderlehrer and G. Stampacchia, An introduction to Variational Inequalities and their Applications, SIAM Publishing Co., Philadelphia, PA, 2000.M.Aslam Noor, Some Developments in general variational inequalities, Applied Mathematics and Computation, Vol. 152, 2004, pages 199-277.

Optimization by equality constraints, Direct substitution method and Lagrange multiplier method, necessary and sufficient conditions for an equality constrained optimum with bounded independent variables. Inequality constraints and Lagrange multipliers. Kuhn-Tucker Theorem. Multidimensional optimization by Gradient method. Convex and concave programming. Calculus of variation and Euler Lagrange equations. Functionals depending on several independent variables. Variational problems in parametric form. Generalized mathematical formulation of dynamics programming. Non-linear continuous models. Dynamics programming and variational calculus. Control theory.

This course will focus on model development, simulation & implementation of models, and visualization of results using MATLAB tools. After the completion of this course students should be able to build mathematical models of physical systems from the first principles and to use software tools. Different types of mathematical models-deterministic, stochastic, static, dynamic and transient models etc. Some examples of mathematical models from environmental, biological sciences and technological domains. Modeling process, identification of involved processes and sub-processes, mathematical formulations of individual (sub) processes, Solution Algorithm development-Linear, nonlinear Algebraic equations, Computer implementation of the model-Using Symbolic computation. Use of computer model for analysis, control and design of the system, Evaluation of what-if scenarios with varied data, Numerical Simulation-Introduction, structure of Continuous and Discrete Simulation models, Special purpose simulation languages. Application of simulation models. Case Studies for different domains. Use of Commercial software for simulating results.

1. Reference Books: Kai Veltn, Mathematical Modeling and Simulation, Introduction for Scientists and Engineers, Wiley 2009. MATLAB user’s manual by Mathworks Inc. Simulink user’s manual by Mathworks Inc.

The General Linear Multistep Methods, Convergence; Order and Error Constant, Local and Global Truncation Error, Consistency and Numerical Stability, Attainable order of Stable Methods. Problems in Applying Linear Multistep Methods, General Methods for Finding Intervals of Absolute and Relative Stability, Predictor- Corrector Methods, Order and Convergence of the General Explicit One-Step Method, Derivation of Classical Runge-Kutta Methods, Runge-Kutta Methods of order Greater Than Four, Error Estimates and Error Bounds for Runge Kutta Methods;, Comparison with P Predictor-Corrector Methods, Implicit Runge-Kutta Methods. Finite difference formulas and discretization methods, Multi-level schemes, Convergence, stability and consistency, The local truncation error.

Reference Books: 1. Greenspan, Numerical solutions of ODE’s for classical Relativistic and Nano systems, 2006. 2. C. E. Froberg, Numerical mathematics, The Benjamin Cummings Pub. Com. Inc., 1985. 3. M. K. Jain, Numerical Solution of Differential Equations, Wiley Eastern Ltd. 4. W. E. Milne, Numerical Solution of Differential Equations, Dover Pub. Inc., N.Y. 5. LeVeque, Randall J. Finite Difference methods for Ordinary and Partial Differential Equations. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2007.

Introduction of Classical Algebra , Modern Algebra, Binary Operations, Algebraic Structures, Boolean Algebras, Groups, Quotient Groups, Symmetry Groups in Three Dimensions, Monoids and Machines, Monoids and Semigroups, Finite-State Machines, Quotient Monoids and the Monoid of a Machine, Latin Squares, Orthogonal Latin Squares, Finite Geometries, Magic Squares, Geometrical Constructions, Constructible Numbers, Duplicating a Cube, Trisecting an Angle, Squaring the Circle, Constructing Regular Polygons, Error-Correcting Codes.

1. Reference Book: William J. Gilbert, Modern Algebra with Applications, John Wiley & Sons, Inc., publication, 2nd edition, 2000.

Approximation of derivatives through Taylor series, truncation error, order of convergence, Parabolic Partial Differential Equations, Explicit methods, Implicit methods , Numerical solutions of elliptic, parabolic and hyperbolic PDEs, Numerical Solution to System of linear and nonlinear equations, Condition number and spectral properties of a matrix, Newton and fixed point methods, applications in MATLAB.

1. Reference Books: Applied Numerical Methods with software, S. Nakamura, Prentice-Hall 1991 Computational Fluid Dynamics by Klaus A. Hoffmann EES, 2000 Numerical Analysis by R. L. Burden 5th Edition 2000

Prerequisite: Multivariable calculus and linear algebra Linear, integer, nonlinear and dynamic programming, classical optimization problems, network theory. CGU

Reference Books: B. E. Gillett, Introduction to Operations Research, (Tata McGraw Hill Publishing Company Ltd., New Delhi). F. S. Hillier and G. J. Liebraman, Operations Research, (CBS Publishers and Distributors, New Delhi, 1974). 3. C. M. Harvey, Operations Research, (North Holland, New Delhi, 1979).

Difference equations, Dimensional analysis, Expansions, Approximate solutions of linear differential equations, order symbols, Asymptotic series, Quadratic and cubic algebraic equations and its solutions by perturbation method, Regular perturbation, Singular perturbation, Boundary layer, The method of matched asymptotic expansion, equations with large parameter, , Solution of partial differential equations by perturbation methods, Asymptotic expansion of integrals Laplace’s method, Watson’s Lemma, Riemann-Lebesgue lemma.

Reference Books: 1. A.Nayfeh, Perturbation methods. 2. I.Stakgold, Boundary Value Problems of Mathematical Physics. 3. B.Noble, Methods based on the Wiener-Hopf technique for the solution of Partial Differential Equations.

Numerical Solution to System of linear equations, Finite Difference method , Taylor series, finite difference equations, Finite difference approximations of derivatives and mixed partial derivatives, Parabolic Partial Differential Equations, Explicit methods, Implicit methods , Elliptic Partial Differential Equations, The Jacobi Iteration Method, The Point Gauss-Seidel Iteration Method, The Line Gauss-Seidel Iteration Method, Point Successive Over-Relaxation Method (PSOR), Line Successive Over-Relaxation Method (LSOR) , The Alternating Direction Implicit Method (ADI), Hyperbolic Partial Differential Equations, Lax method, Mid-point Leap frog, Lax-Wendroff methods, MacCormack method, upwind schemes, modified R-K methods, TVD schemes, Uniform Grid, Staggered Grid (Marker and Cell method), Numerical Algorithms (Applications), Multi-Grid Methods.

2. Reference Books: Applied Numerical Methods with software, S. Nakamura, Prentice-Hall 1991 Computational Fluid Dynamics by Klaus A. Hoffmann EES, 2000 Numerical Analysis by R. L. Burden 5th Edition 2000

Stochastic models of inventory, reliability, queuing, sequencing, and transportation. Applications of these models to problems arising in industry, government, and business.

Reference Books: B. E. Gillett, Introduction to Operations Research, (Tata McGraw Hill Publishing Company Ltd., New Delhi). F. S. Hillier and G. J. Liebraman, Operations Research, (CBS Publishers and Distributors, New Delhi, 1974). 3. C. M. Harvey, Operations Research, (North Holland, New Delhi, 1979).

Equations of electrodynamics, Equations of Fluid Dynamics, Ohm's law equations of magneto hydro dynamics. Motion of a viscous electrically conducting fluid with linear current flow, steady state motion along a magnetic field, wave motion of an ideal fluid. Magneto-sonic waves. Alfve's waves, damping and excitation of MHD waves, characteristics lines and surfaces. Kinds of simple waves, distortion of the profile of a simple wave, discontinuities, simple and shock waves in relativistic magneto hydro dynamics, stability and structure of shock waves, discontinuities in various quantities, piston problem, oblique shock waves.

Reference Books: P. A. Davidson, An introduction to Magnetohydrodynamics, Cambridge texts in Applied Mathematics, 2001. 2. J. P. Freidberg, Ideal Magnetohydrodynamics, 1987. 3. J. P. Goedbloed and S. Poedts, Principles of Magnetohydrodynamics, 2004.

Prerequisites: A graduate course in probability theory, a basic course in numerical methods, and facility in programming a computer using a language such as FORTRAN, C++. This is an advanced course in which stochastically- motivated mathematical methods are applied to problems of various kinds (e.g. radiation transport, semiconductor, geological and financial modeling, or statistical mechanics) that can be solved by simulations carried out on a computer. Problems studied in this way include the most naturally formulated as integral equations over relatively high dimensional phase spaces, as well as those in which estimates of integrals of functions of a large number of variables are sought. This should be regarded as an advanced course in the applications of probability theory to numerical analysis.

Reference Books: 1. Ross, S. M. Simulation, Fifth Edition. Academic Press, 2012. 2. Rubinstein, R. and D. P. Kroese. Simulation and the Monte Carlo Method. Wiley, 2007

Prerequisites: linear algebra, multivariate calculus, and (preferably) experience in programming in Mat lab. This course will survey widely used methods for continuous optimization, focusing on both theoretical foundations and implementation as numerical software. Topics include linear programming (optimization of a linear function subject to linear constraints), line search and trust region methods for unconstrained optimization, and a selection of approaches (including active-set, sequential quadratic programming, and interior methods) for constrained optimization.

Reference Books: S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004 (has electronic version from the author's website) 2. D. Bertsekas, Nonlinear Programming, Athena Scienti_c, 1999 3. J. Nocedal and S.J. Wright, Numerical Optimization, Springer, 2006 4. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer, 2003

This graduate level course will focus on finite element and finite volume based CFD codes. The topics to be addressed include convection dominated transport problems with standard finite element approximations, analysis of accuracy and stability issues, Lax-Wendroff and Taylor-Galerkin methods for unsteady convective transport, streamline diffusion, edge-oriented stabilization, jump-stabilization, Non-linear high resolution schemes base on flux correction. Practical implementations using softwares ANSYS and Featflow. The students are expected to solve a number of one and two dimensional

1. Reference Book: 1. S. Turek, Efficient Solvers for incompressible Flow problems, Springer 1999. 2. D. Kuzman, A guide to Numerical Methods for transport Equation, Friedrich-Alexender- Universitat Frlangen-Nurenberg, 2010. R. Glowinski, Numerical Methods for Fluids, 2003.

Transforms covered will include: Fourier, Laplace, Hilbert, Hankel, Mellin, Radon, and Z. The course will be relevant to mathematicians and engineers working in communications, signal and image processing, continuous and digital filters, wave propagation in fluids and solids, etc. Problems using the techniques covered in this course.

Reference Books: 1. Lovitt, W.V., Linear integral equations, Dover Publications 1950. 2. Smith, F., Integral equations, Cambridge University Press. 3. Tricomi, F.G., Integral equations, Interscience, 1957.

This graduate level course is concerned with numerical simulation of turbulence in single-phase flows and multiphase/multicomponent systems. The underlying mathematical models are derived from the fundamental equations of fluid mechanics using suitable averaging procedure. The topics include a self-contained introduction to Reynolds Averaged Navier-Stokes Equations (RANS) models, Large Eddy Simulations (LES),

3. Reference Books: 1. C. D. Wilcox, Turbulence Modeling for CFD. DWC Industries, 1998. 2. B.Mohammadi and O. Pironneau, Analysis of the k-epsilon turbulence model. John Wiley & Sons, 1994. D.A. Drew and S. L. Passman, Theory of Multicomponent Fluids, Springer,1998.

Subordination, Hypergeometric functions, Integral operators, second order differential subordination and applications, first order linear differential subordination and applications, special differential subordinations, higher order differential subordinations, applications of differential subordination in other fields.

1. Reference Book: Miller, S. S. & Mocanu, P. T., Differential Subordination Theory and Applications. Marcel Dekker Inc., New York, Basel (2000).

Introduction of convolution product. Basic and algebraic properties of Hadamard Product, Hadamard Product and Some Linear differential and integral operators, convolution on some subclasses of the class of univalent functions, convolution on functions with bounded boundary, bounded radius and bounded Mocanu variations, some problems on convolution and at some of the relations between the convolution and the subordination, Applications of convolution in geometric function theory and other related fields.

1. Reference Book: Stephan Ruscheweyh, Convolutions in Geometric Function Theory. Sem. Math. Sup., vol. 83. Presses University Montreal, Montreal (1982).

Equations of dynamic and its various forms. Equations of Langrange and Euler. Jacobi's elliptic functions and the qualitative and quantitative solutions of the problem of Euler and Poisson. The Problems of Langrange and Poisson. Dynamical system. Equations of Hamilton and Appell. Hamilton-Jacobi theorem. Separable systems. Holder's variational principle and its consequences.

Reference Books: 1. D. T. Greenwood, Classical Dynamics (Dover, 1997). 2. F. Chorlton, Chorlton Text Book of Dynamics (Ellis Horwood, 1983). 3. H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics (Addison-Wesley Publishing Co., 2003).

Number Theory is often considered one of the most beautiful and elegant topics in mathematics. We will study properties concerning the integers, such as divisibility, congruences, and prime numbers. More advanced topics include encryption, quadratic reciprocity, and Diophantine approximation. Finally we will introduce elliptic curves and see how these curves relate to the proof of Fermat's last theorem.

Reference Books: 1. D.M. Burton, Elementary Number Theory, McGraw-Hill, 2007. 2. S.B. Malik , Basic Number Theory, Vikas Publishing house, 1995. 3. K.H. Rosen, Elementary Number Theory and its Applications, 5th edition, Addison- Wesley, 2005. 4. I. Niven, H.S. Zuckerman, H.L. Montgomery, An Introduction to the theory of Numbers, John Wiley and Sons, 1991. 5. A. Adler, J.E. Coury, The Theory of Numbers, Jones and Bartlett Publishers, 1995.