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MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR. DEPARTMENT OF MATHEMATICS

SYLLABUS

MAT 101 Mathematics-I 3L-1T 4 Credit

Matrices: Rank and inverse of matrix by elementary transformations, consistency of linear

system of equations and their solution, Eigen values and Eigen vectors, Cayley- Hamilton

theorem (statement only) & its applications, diagonalization of matrices.

Differential Calculus : Curvature , concavity, convexity and points of inflexion, asymptotes,

partial differentiation, Euler’s theorem on homogeneous functions, total differentiation,

approximate calculation, curve tracing (Cartesian and five polar curves-Folium of Descartes,

Limacon, Cardioids, Lemniscates of Bernoulli and Equiangular spiral).

Integral Calculus: Improper Integrals, area and length of curves. Surface area and volume of

solid of revolution, multiple integrals, change of order of integration. (Cartesian form).

Vector Calculus: Differentiation and integration of vector functions of scalar variables, scalar

and vector fields, gradient, directional derivative, divergence, curl. Line integral, surface integral

and volume integral. Green’s, Gauss’s and Stokes’s theorems (statement only) and their simple

applications.

Text Books:

1. . R.K.Jain & S. R. K. Iyengar, Advanced Engineering Mathematics, Narosa.

2. Thomas & Finney, Advanced Calculus and Geometry, Addison-Wesley Pub. Co.

Reference Books : 1. D. W. Jordan & P. Smith, Mathematical Techniques, OXFORD. 2. Peter V. O’Neil, Advanced Engineering Mathematics, Cengage Learning, New Dehli. 3. B.V.Ramana, Higher Engineering Mathematics, McGraw – Hill.

MAT 101 A Mathematics-I 3L 1T 4 Credits

Differential Calculus: (Cartesian form) Asymptotes, curvature, concavity, convexity and points

of inflexion, curve tracing, partial differentiation, Euler’s theorem on homogeneous functions.

Integral Calculus: (Cartesian form) Area and length of curves, surface area and volume of solid

of revolution, double integrals, change of order of integration.

Matrix: Rank and inverse of a matrix by elementary transformation, consistency of linear

system of equations and their solutions, Eigen values, Eigen vectors, Cayley- Hamilton theorem

(statement only) & its applications.

Coordinate Geometry of Three Dimensions: Equation of a sphere, plane section of a sphere,

tangent plane, orthogonality of spheres, definition and equation of right circular cone and right

circular cylinder.

Text Books:

1. R.K.Jain & S R K Iyengar, Advanced Engineering Mathematics, Narosa.

2. Thomas & Finney Advanceed Calculus and Geometry, Addison-Wesley Pub. Co. Reference Books:

1. Shanti Narayan, A Text book of Matrices, S.Chand.

2. Shanti Narayan, Differential Calculus, S.Chand.

3. Shanti Narayan, Integral Calculus, S.Chand.

4. B.V.Ramana, Higher Engineering Mathematics, McGraw – Hill.

MAT-102 Mathematics-II 3L-1T 4 credits --------------------------------------------------------------------------------------------------------------------

Differential Equations: Differential equations of first order and first degree - linear form,

reducible to linear form, exact form, reducible to exact form, Picard’s theorem (statement only).

Linear Differential Equations with Constant Coefficients: Differential equations of second

and higher order with constant coefficients.

Second Order Ordinary Differential Equations with Variables Coefficients: Homogeneous,

exact form, reducible to exact form, change of dependent variable (normal form), change of

independent variable, method of variation of parameters.

Series Solution: Sequence, power series, radius of convergence, solution in series of second

order LDE with variable coefficient (C.F. only). Regular singular points and extended power

series( Frobenius Method).

Fourier Series : Fourier series, half range series, change of intervals, harmonic analysis.

Partial Differential Equations: Formulation and classification of linear and quasi- linear partial

differential equation of the first order; Lagrange’s method for linear partial differential equation

of the first order, solution by separation of variables method, Wave and Diffusion equation in

one dimension.

Text Books:

1. Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley.


Reference Books:

1. Peter V. O’Neil, Advanced Engineering Mathematics, Cengage Learning, New Delhi.

2. M Ray, A Text Book On Differential Equations, Students Friends & Co., Agra-2.

3. Robert C. Mcowen, Partial Differential Equation, Pearson Education.

4. George F. Simmons & S.G. Krantz, Differential Equation, Tata McGraw – Hill.

5. R.K.Jain & S R K Iyengar, Advanced Engineering Mathematics, Narosa, New Delhi.

6 T. Amaranath , An Elementary Course in Partial Differential Equations, Narosa, New Delhi.

7. S.G.Deo and V. Raghavendra: Ordinary Differential Equations, Tata McGraw Hill Pub. Co.,

New Delhi.

MAT-401 Mathematics III 3L-0T 3 credits

Laplace transforms: Laplace transforms of elementary functions, Laplace inverse

transformations, Heavisides’ unit step function, Dirac delta function, shifting theorem,

transforms of derivatives and integrals, convolution theorem, Solution of ordinary differential

equation with constant coefficients and partial differential equations with special reference to

heat equation, wave equation and Poisson Equation

Fourier transforms: Fourier transforms, inverse Fourier transform, Fourier sine and cosine

transforms, Fourier integral formula.

Solution of second order PDE: By separation of variables method : Wave equation and

Diffusion equation in two dimension, Laplace equation in two & three dimensions, Poisson

equation(Cartesian coordinates).

Z- Transforms: Linearity, Z -Transform of elementary functions, shifting theorem, initial and

final value theorems, Convolution theorem, inversion of Z-Transforms, solution of difference

equations by Z- Transforms.

Text Books:

1. Larry S. Andrews, Integral Transforms for Engineers, Bhimsen K. Shivamoggi.

2. Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley.

Books Recommended:

1. H.K.Dass, Advanced Engineering Mathematics, S.Chand.

2. .R.K.Jain and S R K Iyengar, Advanced Engineering Mathematics, Narosa .

3. Larry S. Andrews, Integral Transforms for Engineers, Bhimsen K. Shivamoggi.


5. T. Amarnath, Partial Differential Equation and its application, Narosa .

6. Sankara Rao, Introduction to Partial Differential Equations, Prentice Hall of India.

MAT-402 Complex Analysis 3L-0T 3 credits

Analytic Functions; Functions of complex variable, limits and continuity, differentiability,

Cauchy – Riemann equations, analytic function, harmonic functions, Milne’s Thompson’s

method, conjugate functions.

Conformal Mappings: Mappings or transformations, conformal mapping, necessary and

sufficient conditions for w=f(z) to represent conformal mapping, linear, bilinear and some

important transformations, cross ratio, Schwarz – Christoffel transformations.

Complex Integration: Line integral, Cauchy fundamental theorem, Cauchy-Goursat theorem,

Cauchy integral formula, Cauchy derivative formula, Morera’s theorem.

Expansion of analytic function: Expansion of analytic function as power series, Taylor and

Laurent series, zeros and poles, isolated singularities.

Calculus of Residues: Residue at simple pole, residue at a pole of order greater than unity, the

Cauchy’s residue theorem, evaluation of definite Integrals.

Text Books:

1. E.Kreyszig, Advanced Engineering Mathematics, John Wiley.

Reference Books:

1. S.Ponnusamy, Foundation of Complex Analysis, Narosa Publisher.

2. J.H. Mathews and R.W. Howell, Complex Analysis for Mathematics and Engineering, Narosa

Publisher.

MAT-403 Abstract Algebra 3L -0T 3 credits Group Theory : Groups, semi groups and monoids, cyclic semi graphs and sub monoids,

subgroups and cosets, Congruence relations on Semi groups, Factor groups and homomorphisms,

Morphisms normal sub groups. Structure of cyclic groups, permutation groups, dihedral groups,

Sylow theorems(statement only).

Rings: Rings, subrings, morphism of rings, ideal and quotient rings, Euclidean domains,

commutative rings, integral domains, noncommutative examples, structure of noncommutative

rings, ideal, theory of commutative rings

Field Theory: Integral domains and fields, polynomial representation of binary number.

Text Books:

1. Michael Artin, Algebra, Pearson Education.

2. R.K. Sharma ,S.K. Shah and G. Shankar, Algebra I, Pearson.

References Books:

1. John Fraleigh, First Course in Abstract Algebra, Pearson Education.

2. John A. Beachy and William D. Blair, Abstract Algebra, Second Edition, Waveland Press.

3. John A. Beachy, Abstract Algebra II, Cambridge University Press, London Mathematical

Society Student Texts #47, 1999.

MAT -404 Numerical Methods 3L 3 credits

Error Analysis: Representation of numbers in computers and their accuracy, floating point

arithmetic, concept of zero, errors in computations, types of errors, propagation of errors,

computational methods for error estimation, general error formulae, approximations of functions

and series.

Roots of Algebraic and Transcendental Equations: Bisection method, Regula-falsi method,

iteration method, Newton-Raphson method.

Solution of Simultaneous Algebraic Equations: Gauss elimination method, Gauss Jordan

method, decomposition method, Jacobi and Gauss-Seidel iteration methods.

Interpolation Finite Differences: Newton’s forward and backward differences interpolation

formulae, relations between forward and backward operators, Lagrange’s interpolation formula,

numerical differentiation using Newton’s forward and backward differences interpolation

formulae.

Numerical Integration: Trapezoidal rule, Simpson’s one-third rule, Gaussian quadrature

formula.

Ordinary Differential Equations: Taylor’s series method, Picard’s method, Euler’s and

modified Euler’s methods, Runge-Kutta fourth order method, Milne’s Predictor-Corrector

method.

Text Books:

1. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering

Computation, Wiley Eastern Limited.

2. J N Sharma, Numerical methods for Engineers and Scientists, 2nd edition Narosa Publishing

House New Delhi.

Books Recommended:

1. S.S. Sastry, Introductory Methods of Numerical Analysis, Printice Hall of India.

2. G.D. Smith, Numerical Solutions to Partial Differential Equations, Brunel Univ. Clarendon

Press.

3.V.N. Vedamurthy and N.Ch.S.N. Iyengar, Numerical Methods,Vikas Publishers.

4. B.S.Grewal, Numerical Methods in Engineering and Science, Khanna Publishers.

MAT-405 Probability and Statistics 3L-0T 3 credits

Probability Theorem: Axiomatic definition, properties of probability, conditional probability,

independence, Baye’s theorem.

Discrete Distributions: Probability distribution functions and cumulative distribution functions,

Mean and variance, moment-generating functions, marginal and conditional probability

distributions, binomial and Poisson distribution.

Continuous Distributions: Probability density functions and cumulative distribution functions,

mean and variance, moment generating functions, marginal and conditional probability

distributions, some specific continuous distributions, normal and exponential distributions.

Functions of Random Variables: Distribution function technique, transformation technique,

moment-generating function technique.

Sampling Distributions: Chi-Square, t- test and F- test , Law of large numbers , central limit

theorem, estimation of parameter and testing of hypothesis.

Text Books:

1. Hogg, R.V. & Craig, A.T., Introduction to Mathematical Statistics, 5th Ed., Prentice-Hall,

Inc., Englewood Cliffs, N.J., 1995.

2. Mood, A.M., Graybill, F.A. and Boes, D.C., Introduction to the Theory of Statistics, 3rd Ed.

McGraw Hill, Inc., New York, 1974.

Reference Books:

1. DeGroot, Morris H., and Mark J. Schervish. Probability and Statistics. 3rd ed. Boston, MA:

Addison-Wesley, 2002. ISBN: 0201524880.

2. Freund, W.J., Mathematical Statistics, 5th Ed., Prentice-Hall, Inc., Englewood Cliffs, N.J.,

1994.

3. Hoel, P.G., Mathematical Statistics, 5th Ed., John Wiley & Sons, Inc., NewYork, 1984.

MAT-406 Operation Research 3L-0T 3 credits 1. Introduction of Operations Research: The history, nature and significance of operations

research (OR), models and modeling in OR, Applications and scope of OR, general methods of

solving the problems in OR.

2. Linear programming: Introduction, general structure of linear programming (LP) models,

methods of solving: graphical method, simplex method. Duality in LP, sensitivity analysis.

3. Transportation Problem: Mathematical statement of transportation problem , methods of

finding Basic Feasible Solution, test of optimality, MODI’S method for optimal solution,

variation in transportation problem.

4. Quadratic programming: Wolfe’s and Beale’s method.

5. Network Analysis: Project planning and control with PERT-CPM

Text Books:

1. S.D. Sharma, Operations Research, Kedarnath Publications.

Reference Books: 1. S.S. Rao, Engineering Optimization Theory & Practice.

2. J.K. Sharma, Operations Research Problems and Solutions.

3. Winston, Operations Research.

4. H.A. Taha, Operations Research.

5. P.K. Gupta & D.S. Hira, Operations Research.

MAT-407 Information Theory and Coding 3L-1T 4 credits

Mathematical Foundation of Information Theory in communication system. Measures of

Information- Self information, Shannon’s Entropy, joint and conditional entropies, mutual

information and their properties.

Discrete Memory less channels: Classification of channels, calculation of channel capacity.

Source Coding, and Channels Coding. Unique decipherable Codes, condition of Instantaneous

codes, Average codeword length, Kraft Inequality. Shannon’s Noiseless Coding Theorem.

Construction of codes: Shannon Fano, Shannon Binary and Huffman codes. Higher Extension

Codes. Decoding scheme- the ideal observer decision scheme .Error Correcting Codes:

Minimum distance principle. Relation between distance and error correcting properties of codes,

The Hamming bound. Construction of Linear block codes, Parity check Coding and syndrome

decoding.

Text /References

1. Information theory and Reliable Communication by R.G.Gallager

2. Information Theory by Robert Ash

3. An Introduction to Information Theory by F. M. Reza

4. Error correcting codes by W.W. Peterson and E. J. Weldon

5. The theory of Information and Coding; Student Edition ( Encyclopedia of Mathematics and its

Applications ) S.J. Mceliece

MAT-408 Linear Algebra and theory of Matrices 3L 3 Credits Fields and linear equations.Vector spaces, sub spaces, linear combinations, spanning sets, basis

and dimensions, linear transformations. Rank and nullity of linear transformation.

Representation of transformations by matrices. duality and transpose of a linear transformation.

Linear functional, dual space.

Eigen values and Eigen Vectors, characterstics polynomials, minimal polynomials. Cayley

Hamilton’s theorem, triangularization, diagonalization. Inner product spaces. Orthogonality,

Gram – Schmidt orthonormalization. Orthogonal projections. Linear functions and adjoints.

Unitary and normal operators. Spectral theorem for normal operators.

Bilinear forms, symmetric and skew symmetric bilinear forms, bilinear forms and vectors, matrix

of a bilinear form.

Books Recommended:

1. Linear Algebra, K. Hoffman and R. Kunze, PHI Learning, 2009.

2.Linear Algebra, S. Lang, Springer India, 2005.

3.Linear Algebra, M. Artin, Pearson education.

4. Linear Algebra, Herstein, Wiley India Pvt. Ltd., 2006.

New Syllabus

M.Sc. (APPLIED MATHEMATICS )

I Semester

Course number

Course Name Subject area code

L T P C CWS MTE PRE ETE Total

MAT-611 Abstract Algebra DC 3 1 - 4 10 30 - 60 100 MAT-612 Real Analysis DC 3 1 - 4 10 30 - 60 100 MAT-613 Statistics & Probability

Theory DC 3 1 - 4 10 30 - 60 100

MAT-614 Ordinary Differential Equation

DC 3 1 - 4 10 30 - 60 100

MAT-615 Operations Research DC 3 1 - 4 10 30 - 60 100 HST-603 Comprehensive English

dynamics of communication (for tech. eng. Communication skills)

IDC 2 - - 2 - - - - -

Total 17 5 - 22 50 150 - 300 500 II Semester

Course number



MAT-621 Linear Algebra and Theory of Matrices

DC 3 1 - 4 10 30 - 60 100

MAT-622 Complex Analysis DC 3 1 - 4 10 30 - 60 100 MAT-623 Computer Language

and Computer Lab I DC 2 0 4 4 10 30 20 40 100

MAT-624 Partial Differential Equations

DC 3 1 - 4 10 30 - 60 100

MAT-625 Discrete Mathematical Structure

DC 3 1 - 4 10 30 - 60 100

Total 14 4 4 20 50 150 20 280 500

III Semester

IV Semester

Course number



MAT-631 Fluid Mechanics DC 3 1 - 4 10 30 - 60 100 MAT-632 Numerical Analysis DC 3 1 0 4 10 30 20 40 100 MAT-633 Integral Transforms DC 3 1 - 4 10 30 - 60 100 MAT-634 Information .Theory and

Coding DC 3 1 - 4 10 30 - 60 100

Elective I DE 3 1 - 4 10 30 - 60 100 MAP-635 Computer lab-II PRJ 0 0 4 2 - - - - - Total 15 5 4 22 50 150 20 280 500

Course number



MAT-641 Functional Analysis DC 3 1 - 4 10 30 - 60 100 MAT-642 Integral Equations and

Calculation of Variations

DC 3 1 - 4 10 30 - 60 100

MAT-643 Mathematical Modelling

DC 3 1 - 4 10 30 20 40 100

Elective II DC 3 1 - 4 10 30 - 60 100

MAT-644 Project PRJ - - - 10 - - - 100 100 Total 12 4 - 26

40 120 20 320 500

Grand Total of all semesters

58 18 8 90 190 570 60 1180 2000

M. Sc. Electives I to III

S. No. Code Course Name L T P C 1. MAT-651 Number Theory 3 1 - 4 2. MAT-652 Applied Stochastic Processes 3 1 - 4 3. MAT-653 Advanced matrix theory 3 1 - 4 4. MAT-654 Special Function 3 1 - 4 5. MAT-655 Combinatorics & Graph Theory 3 1 - 4 6. MAT-656 Fractional Calculus & its application 3 1 - 4 7. MAT-657 Computational Fluid Dynamics 3 1 - 4 8. MAT-658 Computer programming in Fortran 3 0 2 4 9. MAT-659 Numerical optimization technique 3 1 - 4 10. MAT-660 Polynomials 3 1 - 4 11. MAT-661 Queuing Theory and Applications 3 1 - 4 12. MAT-662 Network Flows Algorithms 3 1 - 4 13. MAT-663 Combinatorial Optimization 3 1 - 4 14 MAT-664 Topology 3 1 - 4 15 MAT-665 Cryptography 3 1 - 4 16 MAT-666 Error Correcting Codes 3 1 - 4

MAT-611 Abstract Algebra 4 Credits( 3L+1T+0P)

Groups: Normal subgroups, quotient groups, homomorphism and isomorphism theorems of

groups. Maximal subgroups. Composition series of a group. Direct product. Cauchy’s and

Sylow’s theorems for finite groups.

Rings: Subrings and Ideals. Principal and Maximal Ideals. Quotient rings, isomorphism of rings,

characteristic of a ring. Imbedding of a ring into another ring. Polynomial rings. Irreducible

polynomials. Division algorithm for polynomials over a field. Euclidean algorithm. Euclidean

algorithm for polynomial over a field. Euclidean rings. Properties of Euclidean rings. Unique

factorization domains.

Fields: Simple and algebraic field extensions. Splitting fields and normal extensions. Finite

fields and applications.

Books Recommended:

1. Algebra I, II &III, R.K Sharma et al., Pearson.

2. Algebra, S. Lang, Springer.

3. Algebra, M. Artin, PHI Learning.

4. Topics in Algebra, I.N. Herstein, Wiley India Pvt. Ltd.

MAT-612 Real Analysis 4Credits (3L+1T+0P)

Metric spaces: Definition and examples, convergent sequences. Cauchy sequences. Cauchy’s

general principle of convergence.

Series : Convergence tests. Absolute and conditional convergence. Addition, Multiplication and

rearrangements.

Weierstrass’ theorem, continuity of functions in metric spaces. Discontinuities. Monotonic

functions. Derivative of a real function. Mean value theorems. Continuity of derivatives, Taylor’

theorem. Compactness, Connectedness.

Sequences and series of functions : Cauchy criterion for uniform convergence. Abel’s and

Dirichlet's tests for uniform convergence, Weierstrass Approximation theorem.

Power series. Uniqueness theorem for power series. Fourier series. Bessel’s inequality.

Localization theorem. Parseval's theorem.

Differential calculus of functions of several variables: limit continuity and differentiability,

application to maxima-minima.

Integral calculus of functions of several variables : Jacobians, invertible function , implicit

functions, Riemann integration and Riemann-Stieltjes Integrals.

Books Recommended:

1. Principles of Mathematical Analysis by W.Rudin, Mc Graw Hill, Singapore, 1976.

2. Mathematical Analysis by T.M.Apostol, Narosa Publishing House, 1985.

3. Theory of Functions of a Real Variable, Volume I, I. P. Natanson, Frederick Pub. Co., 1964.

4. Real Analysis by H.L. Royden, McMillan Publication Co. Inc. New York.

MAT-613 Statistics & Probability Theory 4 Credits ( 3L+1T +0P)

Formal concepts: Sample space, outcomes and events, random variable Probability, conditional

probability, expected value, moment generating function, specific discrete and continuous

distributions, e.g. Binomial, Poisson, Geometric, Pascal, Hypergeometric, Uniform, Exponential,

Weibull, Beta, Gamma, Erlang, Normal and student’s ‘t’ distribution. Chi square and F

distribution. Law of large numbers and central limit theorem, sampling distributions, point and

interval estimation, testing of hypothesis, large and small samples Chi-Square test as a test of

goodness of fit.

Books Recommended:

1. A first course in Probability, Sheldon M. Ross, Pearson, 2006.

2. Probability & Statistics for Engineers, Richard A. Johnson, PHI Learning, 2011.

3. An Introduction to Probability theory and its Application, W. FELLER, Wiley, 2008.

MAT-614 Ordinary Differential Equations: 4 Credits (3L+1T+0P)

Ordinary differential equations: System of Simultaneous Linear Differential Equations with

constant and variable coefficients. Pfaffian Equation, Solution in Series, Bessel’s, and Legendre

polynomial, Linear Difference Equations.

Existence and Uniqueness of Initial Value Problems: Picard's and Peano's Theorems.

Boundary Value Problems for Second Order Equations: Green's function, Sturm comparison

theorems and oscillations, Eigen Value Problems and Sturm Liouville Problems, Stability of

Linear and Non Linear Systems.

Books Recommended:

1. Ordinary Differential Equations, Dev Raghvendra et al., Tata McGraw Hill.

2. Elements of Ordinary Differential Equations and Special Functions, A Chakrabarti, Wiley,

1990.

3. Text book of Ordinary Differential Equation, C.R.Mondal, PHI Learning, 2008.

MAT-615 Operations Research 4 Credits (3L+1T+0P)

Introduction to Linear Programming Problem: Statement of Linear Programming Problem,

Transportation problem, Assignment Problem.

Dynamic Programming: Introduction, Solution of Linear Programming Problem using

Dynamic Programming.

Non-linear programming

Unconstrained optimization via iterative methods: Direct search methods (Univariate method),

Gradient methods (steepest descent (Cauchy’s) method).

Constrained optimization: Lagrange multipliers, Kuhn Tucker conditions.

Quadratic programming: Wolfe’s and Beale’s method.

Network Analysis: Project planning and control with PERT-CPM

Books Recommended:

1. Engineering Optimization Theory & Practice, S.S. Rao, John Wiley and Sons, 2009.

2. Operations Research, S.D. Sharma, Kedar Nath Publ.

3. Operations Research Problems and Solutions, J.K. Sharma, Macmillan Publishers India

Ltd., 2008.

4. Operations Research, Winston, Duxbury Press.

5. Operations Research, Hamdy A. Taha, Macmillan, 1982.

6. Operations Research- P.K. Gupta & D.S. Hira

MAT-621 Linear Algebra and theory of matrices 4 Credits( 3L+1T+0P)

Fields and linear equations.Vector Spaces, Sub Spaces, Linear combinations, spanning sets,

Basis and Dimensions, Linear Transformations. Rank and Nullity of linear Transformation.

Representation of transformations by matrices. Duality and transpose of a linear transformation.

Linear Functionals. Dual Space.

Eigen values and Eigen Vectors. Characterstics Polynomials, minimal polynomials. Cayley

Hamilton’s theorem, Triangularization, Diagonalization. Inner Product Spaces, Orthogonality,

Gram – Schmidt Orthonormalization. Orthogonal Projections. Linear functions and adjoints.

Unitary and normal operators. Spectral theorem for normal operators.

Bilinear forms, symmetric and skew symmetric bilinear forms, Bilinear forms and vectors,

matrix of a bilinear form.

Books Recommended:

1. Linear Algebra, K. Hoffman and R. Kunze, PHI Learning, 2009.

2.Linear Algebra, S. Lang, Springer India, 2005.

3.Linear Algebra, M. Artin, Pearson Education.

4. Linear Algebra, Herstein, Wiley India Pvt. Ltd., 2006.

MAT-622 Complex Analysis 4 Credits ( 3L+1T+0P)

Complex Functions, Limits, Continuity and differentiability of functions of a complex variable,

analytic functions, harmonic conjugates, the Cauchy-Riemann equations. Complex integration,

Proofs of Cauchy’s integral theorem, Cauchy’s Integral formula and derivatives of analytic

functions , Contour Integration, Liouville’s theorem, maximum modulus principle, argument

principle, Rouche’s theorem. Power Series, Taylor’s theorem, Zeros of Analytic functions,

Laurent’s Theorem. Classification of singularities, poles, residue theorem and its applications.

Contour Integration Conformal and bilinear mappings. Analytic continuation.

Books Recommended:

1.Complex Variables & Applications – Ruel V. Churchill, Jame Sward Brown, Tata McGraw

Hill Eduction, 2009.

2.Functions of One Complex Variable, John B. Conway, Narosa Distributors Pvt. Ltd., 1973.

3.Theory of functions- E.C.Titchmarsh

4.Theory of functions of a complex variable, Shanti Narayan, S. Chand Publishers, 2005.

MAT-623Computer Language and Computer Lab-I 4 Credits( 2L+0T+4P)

Programming in ‘C’: Need of Programming Languages , Flowcharts and algorithm development,

data types constants, variables, declarations, operators and expressions, operator precedence and

associativity, input and output operations, formatting, decision making, branching and looping,

array and character strings, built-in and user defined functions, the scope and lifetime of

variables, structures and unions, pointers, pointer arithmetic/expressions, pointers and arrays,

pointers and structures, dereferencing file handling, command line arguments, defining macros,

preprocessor directives simple use of dynamic memory allocation: malloc and calloc functions.

Books recommended:

1. Programming in C, Balagurusamy, Tata McGraw hill, 2011.

2. The C programming language, Kerminghan and Ritchie, PHI Learning, 2012.

MAT-624 Partial Differential Equations 4 Credits( 3L+1T+0P)

Formulation and classification of Linear and Quasi- Linear Partial Differential Equation of the

First Order, Lagrange’s Method for Linear Partial Differential Equation of the First Order ,

Pfaffian Equation, Cauchy Problems, Complete Integrals of Non Linear Equations of First

Order, Four Standard Forms, Charpit’s Method, Monge’s Method. Linear Equations with

Constant Coefficients, Classification of Second Order Linear PDE and Reduction to Canonical

Forms, Laplace, Poisson’s and Helmholtz, Wave and Diffusion Equations in Various Coordinate

Systems and their Solutions Under Different Initial and Boundary Conditions. Green’s

Functions and Properties. Existence Theorem by Perron’s Method. Heat Equation, Maximum

Principle. Uniqueness of Solutions via Energy Method. Uniqueness of Solutions of IVPs for

Heat Conduction Equation.

Books Recommended:

1. Introduction to Partial Differential Equations, K.Sankara Rao, PHI learning Pvt. Ltd.

2. Partial Differential Equations, P Prasad and R Ravindran, New Age International, 2011.

3. T. Amaranath , An Elementary Course in Partial Differential Equations, Narosa, New Delhi.

4. I. N. Sneddon, Elements of Partial Differential Equation McGraw – Hill, New York.

5. J N Sharma and K Singh, Partial Differential Equations for engineers and scientists, 2nd

Edition, Narosa, New Delhi.

MAT-625: Discrete Mathematical Structures 4 Credits ( 3L+1T+0P)

Formal Logic-Statements, Symbolic Representation and Tautologies, Quantifiers, Predicates and

Validity, Propositional Logic, Lattices: Partially ordered sets and Lattices, Hasse Diagrams,

lattices as algebraic systems sub-lattices, direct product and Homomorphisms, Complete lattices,

Modular lattices, distributed lattices, the complemented lattices, convex sub lattices, Congruence

relations in lattices.

Graphs: Complete graphs, regular graphs, bipartite graphs, Vertex degree, subgraphs, paths and

cycles, the matrix representation of graphs, fusion, trees and connectivity, bridges, spanning

trees, chromatic number, connector problems, shortest path problems, cut vertices and

connectivity. Polya’s theory of enumeration and its application.

Books Recommended:

1. Elements of Discrete Mathematics by C. L. Liu, McGraw-Hill Book Co.

2. Discrete mathematical structures by Kolman, Busby and Ross, 4th edition. PHI, 2002.

3. Discrete Mathematics with Graph Theory by Goodaire and Parmenter, Pearson

edition.2nd edition.

4. Graph Theory with Applications to Engineering and Computer Sciences by N. Deo, PHI

learning, 2009.

Old Syllabus

III Semester

IV Semester

Course number



MAT-631 Operation Research DC 3 1 - 4 10 30 - 60 100 MAT-632 Numerical Analysis

&Computer Lab II DC 2 0 4 4 10 30 20 40 100

MAT-633 Integral Transforms DC 3 1 - 4 10 30 - 60 100 Elective I DE 3 1 - 4 10 30 - 60 100 Elective II DE 3 1 - 4 10 30 - 60 100 MAT-634 Project PRJ - - - 4 - - - - - Total 14 4 4 24 50 150 20 280 500

Course number



MAT-641

Functional analysis DC 3 1 - 4 10 30 - 60 100 MAT- 642

Integral Equations DC 3 1 - 4 10 30 - 60 100 MAT-643

Computer Lab III DC 1 - 6 4 10 30 20 40 100 Elective III DC 3 1 - 4 10 30 - 60 100 MAT-644

Project (ctd.) PRJ - - - 8 - - - 100 100

Department of Mathematics

M. Sc. Electives I to III

S.No. Code Course Name L T P C 1. MAT-651 Number Theory 3 1 - 4 2. MAT-652 Applied Stochastic Processes 3 1 - 4 3. MAT-653 Advanced matrix theory 3 1 - 4 4. MAT-654 Information Theory & Coding 3 1 - 4 5. MAT-655 Combinatorics & Graph Theory 3 1 - 4 6. MAT-656 Fractional Calculus & its application 3 1 - 4 7. MAT-657 Fluid mechanics 3 1 - 4 8. MAT-658 Computer programming in Fortran 3 0 2 4 9. MAT-659 Numerical optimization technique 3 1 - 4

MAT-631Operations Research 4Credits( 3L+1T+0P) Unconstrained optimization using calculus (Taylor’s theorem, convex functions, coercive functions) Unconstrained optimization via iterative methods (Newton’s methods, gradiant/conjugate base method, Quasi-Newton method). Constrained optimization, Penalty methods, Lagrange multipliers, Kuhn Tucker conditions, linear programing (simplex methods, dual simplex method, duality thoery). Modeling for optimization.

Total 10 3 6 24

40 120 20 320 500 Grand Total of all

semesters 55 17 16 94 285 475 60 1180 2000

Books recommended: 1.Non-linear Programming - O.L. Mangasrian. 2.Linear Programming – G. Hadley 3.Fundamental of Queuing theory by Gross & Moris. 4.PERT and CPM Principles and applications – L.S.Srinath MAT-632 Numerical Analysis and computer lab II 5Credits( 3L+1T+4P) The solution of linear equations:- Gauss elimination method, Gauss – Jordan method, Crout’s method, Jacobi’s method, Gauss – Seidel method. The solution of non linear equations:- Regular Falsi method, Newton – Raphson method, Secant method. Interpolation:- Newton’s forward and backward differences, Stirling’s Central Difference formula,Numerical differentiation and Integration(Trepezoidal rule, Simpson’s 1/3 rule and Simpson’s 3/8 rule) , Lagrange’s formula for unequal intervals. Numerical Solution of First ordinary differential equations:- Picard’s method, Euler’s method, Modified Euler’s method, Runge Kutta fourth order method, Milne’s predictor corrector method, Finite difference method for ordinary differential equations. Computer programming in C and C++ in above methods. Books recommended: 1.Elementary Numerical Analysis – Atkinson, K. 2.Numerical Analysis – David Kincaid & W. Cheney. 3.Numerical Analysis – F. Scheid. 4.Numerical Methods for Scientific and Engineering Computation – M.K. Jain, S.R.K Iyengar and R.K. Jain. MAT-633 Integral Transforms 4 Credits( 3L+1T+0P)

Fourier Transforms: Fourier integral Theorem, Fourier Transform, Fourier Cosine Transform, Fourier Sine Transform, Transforms of Derivatives, Fourier transforms of simple Functions, Fourier transforms of Rational Functions, Convolution Integral, Parseval’s Theorem for Cosine and Sine Transforms, Inversion Theorem, , Solution of Partial Differential Equations by means of Fourier Transforms. first order and second order Laplace and Diffusion equations. Hankel Transform: Elementary properties, Inversion theorem, transform of derivatives of functions, transform of elementary functions, Parseval relation, relation between Fourier and Hankel transform, use of Hankel Transform in the solution of Partial differential equations, Dual integral equations and mixed boundary value problems. Laplace Transform: Definition, Transform of some elementary functions, rules of manipulation of Laplace Transform, Transform of Derivatives, relation involving Integrals, the error function, Transform of Bessel functions, Periodic functions, convolution of two functions, Inverse Laplace Transform of simple function , Tauberian Theorems, Solution of Differential Equations- Initial value problems for linear equations with constant coefficients, two-point boundary value problem for a linear equation with constant coefficients, linear differential equation with variable coefficients, simultaneous differential equations with constant coefficients, , Solution of diffusion and wave equation in one dimension and Laplace equation in two dimensions. Books Recommended: 1.The use of Integral Transforms – Ian N. Sneddon 2.Fourier Transforms – Ian N. Sneddon MAT-641 Functonal analysis 4Credits( 3L+1T+0P) Normal Spaces, continuity of a linear mapping. Banach spaces, Linear Transformations and functionals and Normed bounded linear transformation, dual spaces, Hahn – Banach theorem. Hilbert Spaces. Orthonormal sets, Bessel’s Inequality, Parseval’s relation, Riesz Representation theorem, Relationship between Banach Spaces,Hilbert Spaces. Adjoint operators in Hilbert Spaces, Self adjoint operators, positive operators, Projection Operators and orthogonal projections in B&H spaces, Fixed point theorems and their applications, Best approximations in Hilbert Spaces. Gatebux and Frechat Derivatives. Solution of

boundary value problems. Optimization problems. Applications to Integral and differential equations. Books Recommended: 1.Functional Analysis-B V Limaye 2.Functional Analysis- Brown Page MAT-642 Integral Equations 4 Credits( 3L+1T+0P) Definition and classification, conversion of initial and boundary value problems to an integral equation, Eigen-Values and eigen functions. Solutions of homogeneous and general Fredholm integral equations of second kind with separable kernels. Solution of Fredholm and Volterra integral equestions of second kind by methods of successive substitutions and successive approximations, Resolvent kernel and its results. Integral equations with symmetric kernels: Complex Hilbert space, Orthogonal system of functions, fundamental properties of eigen values and eigen functions for symmetric kernels, expansion in eigen-functions and bilinear forms, Hilbert-Schmidt theorm. Solution of Fredholm integral equations of second kind by using Hilbert-Schmidt theorem. Fredholm theorems. Solution of Volterra integral equations with convolution type kernels by Laplace transform. Books recommended: 1.Integral Equations – W. V. Lovitte 2.Linear Integral Equations – R.P.Kanwal 3.Linear Integral Equations – S.G. Mikhlin MAT-643 Computer Lab-III 4Credits(1L+0T+6P)

To develop in introductory depth knowledge in mathematical software like MATHEMATICA, MATLAB. MAT-651Number Theory 4 Credits( 3L+1T+0P) To introduce students to the basic concepts in the theory of numbers, amalgamating classical results with modern techniques using algebraic and analytic concepts. Congruences: Some elementary properties and theorems, linear and systems of linear congurences. Chinese Remainder Theorem. Quadratic congruences. Quadratic Reciprocity Law, Primitive roots. Some elementary arithmetical functions and their average order, Mobius Inversion formula, integer partitions, simple continued fractions, definite and indefinite binary quadratic forms ,some diophantine equations. Books recommended : 1. Number Theory- Shanti Narayan MAT-652 Applied Stochastic Processes 4Credits(3L+1T+0P) Definition and classificaton of general stochastic processes, Examples. Markov chains, Transition Probability Matrices, classificaton of states, Recurrence, examples. Basic Limit theorems of markov chains, Renewal Equation (Discrete case), Absorption probabilities. Random walk and queueing examples. Continous time Markov chains, Pure Birth Processes, Poisson Processes, Birth and Death Processes, Differential Equation of Birth and Death Processes, Examples. Renewal processes, Renewal equations and Elementary Renewal theorem. Brownian

motion, Continuity of paths and the Maximum variables, Variations and Extensions. Books recommended : 1. Elements of Applied Stochastic Processes- V.N. Bhat,. 2., Modeling and Analysis of Stochastic Systems- V.G. Kulkarni 3. Stochastic Models in Queueing Theory- J. Medhi. 4. Probability, Stochastic Processes, and Queuing Theory The Mathematics of Computer Performance Modelling - R. Nelson, 5. Stochastic Processes, 2nd ed.- S. Ross, MAT-653Advanced Matrix Theory 4 Credits( 3L+1T+0P) Quadratic forms and congruence of Matrices:- Quadratic forms, Quadratic forms as a product of matrices, Matrices as representative of linear transformation, the set of quadratic forms over F, congruence of quadratic forms and matrices. Congruence transformation of symmetric matrix. Elementary congruent transformations, congruent reduction of a symmetric matrix, congruence of skew symmetric matrices. Quadratic forms in the real field:- Reduction in the real field, classification of real quadratic forms in n-variables, definite, semi definite and indefinite real quadratic forms. Quadratic characteristics properties of definite, semi definite forms, gram matrices, case of complex field, reduction in the complex field. Hermitian matrices and forms:- Hermitian matrices and forms, linear transformation of Hermitian form, conjunctive transformation of a matrix, conjunctive reduction of Hermitian matrix, types of Hermitian forms, conjunctive reduction of a Hermitian matrices. Characteristic roots and characteristic vectors of matrices:- Characteristic roots and characteristic vectors of a square matrix, Nature of the characteristic roots of special types of matrices, relation between algebraic and geometric multiplicities of characteristic roots, mutual relation between characteristic vectors corresponding to different characteristic roots. Books recommended: 1.Matrix Methods for Engineers and Scientists – S. Barnett 2.Elementary Matrix Theory – Eves, Howard. 3.Matrix Theory– Shanti Narayan

MAT-654 Information Theory and Coding 4Credits(3L+1T+0P) Mathematical Theory of Foundation Of Information Theory in Communication system. Measures of Information- Self information, Mutual Information, Average Information,entropy and its properties. Source Model and Coding, channels Model and Coding. Problems of unique decipherable Codes, condition of Instantaneous codes, Code word length, Kraft Inequality. Noiseless Coding Theorem. Construction of codes: Shannon Fano, Shannon Binary and Huffman codes. Discrete Memory less channels: Classification of channels, calculation of channel capacity. Decoding scheme- the ideal observer. The fundamental theorem of Information theory. Error Correcting Codes: Minimum distance principle. Relation between distance and error correcting properties of codes, The Hamming bound. Parity check Coding. Bounds on the error correcting ability of Parity Check Codes. Linear block codes, systematic linear codes& optimum coding for Binary symmetric channel, The Generator & parity check matrices, Syndrome decoding & Symmetric channels, Hamming codes, Books Recommended: 1. Information Theory - Robert B Ash. 2. Introduction to Information Theory- F. M Reza 3. Introduction to Coding & Information Theory- Steven Romann. 4. Error correcting codes - W.W. Peterson and E. J. Weldon. MAT-655 Combinatorics & Graph Theory 4Credits(3L+1T+0P) Permutation and combinations. Pigeon hole principle, Inclusion and Exclusion Principles, Sequences and selections, Proofs, Induction Graphs: Paths, Cycles, Trees, Coloring. Trees, Spanning Trees, Graph Searching (DFS, BFS), Shortest Paths. Bipartite Graphs and Matching problems. Counting on Trees and Graphs. Hamiltonian and Eulerian Paths.

Groups: Cosets and Lagrange Theorem, Cyclic Groups etc.. Permutation Groups, Orbits and Stabilizers. Generating Functions. Symmetry and Counting: Polya Theory. Books Recommended: 1.Discrete Mathematics- Normal L. Biggs . 2. Discrete Structures, Logic and Computatibility- J. Hein. 3. Elements of Discrete Mathematics-C.L.Liu MAT-656 Fractional Calculus & its application 4Credits( 3L+1T+0P) The Reimann Liouville Fractional Calculus: Fractional Integrals of some functions namely binomial function exponential, the hyperbolic and trigonometric functions, Bessel’s functions, Hypergeometric function and the Fox’s H-function. Drichlet’s Formula, Derivatives of the Fractional Integral and the Fractional Integral of Dervative’s. Laplace Transform of the Fractional integral, Leibniz’s Formula for Fractional Integrals. Derivatives, Lebniz’s Formula of Fractional Derivatives. The Weyl Fractional Calculus – Definition of Weyl Fractional Integral Weyl Fractional Derivatives, A Leibniz Formula for Weyl Fractional Integral and simple applications. Fractional Differential Equations: Introduction, Laplace Transform, Linearly Independent Solutions, Solutions of the Homogeneous Equations, Solution of the Nonhomogeneous Fractional Differential Equations, Reduction of Fraction Differential Equations to ordinary differential equations. Semi Differential equations. Books Recommended: 1.The Fractional Calculus – K.B. Oldham & J.Spanier. 2.The Introduction to the Fractional Calculus & Fractional Differential Equation – K.S. Miller & B.Ross. MAT-657 Fluid mechanics 4Credits( 3L+1T+0P) Theory of very slow motion: Stokes’ equations, Stokes’ flow, Ossen equations, Oseen flow and Lubrication theory. Theory of Laminar Boundary Layers: Two-dimensional boundary layer equations for flow over a plane wall, the boundary

layer on a flat plate(Blasius- Töpfer solution), similar solutions of boundary layer equations, boundary flow: past a wedge and along the wall of a convergent channel, two-dimensional boundary layer equations for flow over a curved surface, separation of boundary layer, Blasius series solution, Görtler new series method, the spread of a jet, flow in the wake of a body, Prandtl-Mises transformation, boundary layer equations for flow past a body of revolution(axially symmetrical boundary layers), Mangler’s transformation. Integral methods for the approximate solutions of laminar boundary layer equations: Kármán momentum integral equation, Kármán-Pohlhausen method and its applications, Walz-Thwaites method, Energy-integral equation, boundary layer control. Books recommended: 1.Viscous Flow Theory Vol.1 ‘Laminar flow’ – S.I. Pai. 2.Boundary Layer Theory – H. Schlichting. MAT-658Computer Programming in Fortran 4Credits( 3L+0T+2P) The flow chart concept, Fortran IV programming. Integer and real operations, control statements, Do statements, Arrays, Input and output operations, formats, subroutines and function subprograms, Logic and complex operations, real operation in double precision, some programs. Books Recommended: 1.A Guide to Fortran IV programming – McCracken, D.D. MAT-659 Numerical optimization technique 4Credits( 3L+1T+0P) Revised Simplex method for LPP, bounded variable problem. Integer Programming: Gomory’s algorithm for all integer programming problem, branch and bound technique. Quadratic forms; concave and convex functioning and multiplier. Lagrauge function and multiplier.

Quadratic programming; Wolfe’s method, Beal’s method. Duality in quadratic programming. Dynamic programming; Principle of optimality due to Bellman, solution of an LPP by dynamic programming. Queuing models. Network Analysis’s: Project planning and control with PERT-CPM. Books recommended: 1.Non-linear Programming - O.L. Mangasian. 2.Linear Programming – G. Madley 3.Fundamental of Queuing theory by Gross & Moris. 4.PERT and CPM Principles and applications – L.S.Srinath

Malaviya National Institute of Technology, Jaipur Department of Mathematics

Comprehensive Course

Title of the Course:-Random Variables & Stochastic Process Course Code :-MAT-701 Course Credits :-04 Concept of a Random Variable, Discrete & Continuous Random Variables and their Event Space, Statistical Averages, Computation of Mean time to Failure, Moment Generating Functions. Bernoulli, Binomial, Negative Binomial, Poisson, Normal, Cauchy, Rectangular, Exponential, Geometric, Hyper-Geometric, WeibuIl, Eralang Distributions, Moments & M.G.F. for above distributions. Two dimensional random variables, joint probability mass Function Joint Probability , Density Functions, Joint Probability Distribution Functions, Marginal Probability Distribution, Conditional Probability Distribution & Conditional Expectation Function involving more than one random variables. Introduction to Stochastic Processes, Classification of Stochastic Processes, Analytical Representation of a Stochastic Process, Autocorrelation Function & its hopelties. The Bernoulli Process, the Poisson Process, Pure Birth, Pure death & Birth-Death Prorecesses. Introduction to Markov Chains, Discrete & Continuous Parameter Markov chains, Computation of n -Step Transition Probabilities; Higher Transition Probabilites & Chapman Kolmogorov Equations, State Classification & Limiting Distribution, Irreducible Finite Chains with Aperiodic States, Queuing Models with general arrival or Service Patterns, Discrete Parameter Birth Death Process, Finite Markov Chains with Absorbing States. Books recommended : 1. Elements of Applied Stochastic Processes- V.N. Bhat,. 2., Modeling and Analysis of Stochastic Systems- V.G. Kulkarni 3. Stochastic Models in Queueing Theory- J. Medhi. 4. Probability, Stochastic Processes, and Queuing Theory The Mathematics of Computer Performance Modelling - R. Nelson, 5. Stochastic Processes, 2nd ed.- S. Ross,



Title of the Course:-Information Theory Course Code :-MAT-702 Course Credits :-04

Entropy as a measure of uncertainty and information Shannon's entropy and entropies of order Algebraic properties and possible interpretations, analytical properties and inequality, joint and conditional entropies. Mutual information. Csiszar’s f-divergence measures and their properties Noiseless coding, unique decipherability, condtion of existence of instantaneous codes, its extension to uniquely decipherable codes, Noiseless coding theorem, construction of optimal codes. Discrete memoryless channels. Models for communication channel, channel capacity. Classification of channels, Calculation of channel capacity, decoding schemes. Fundamental Theorems, Exponential error bound weak converse of Fundamental theorm, Extension of definition of entropies to continuous memory less channels. and properties. Error correcting codes-Minimum distance, principle and error correcting properties. Hamming bounds, parity check coding. Upper and Lower bounds of parity check codes. Books Recommended: 1. Information Theory - Robert B Ash. 2. Introduction to Information Theory- F. M Reza 3. Introduction to Coding & Information Theory- Steven Romann. 4. Error correcting codes - W.W. Peterson and E. J. Weldon.



Title of the Course:-Probability & Stochastic Process Course Code :-MAT-703 Course Credits :-04 Sample Space, Events, Algebra of Events, Classical, Statistical and Axiomatic Definitions of Probability, Conditional Probability, Independent Events, Theorem of Total Probability, Baye's Theorem. Bernoulli, Binomial, Negative Binomial, Poisson, Normal, Cauchy, Rectangular, Exponential, Geometric, Hyper-Geometric, Weibull, Erlang Distributions, Moments & M.G.F. for above distributions. Introduction to Stochastic Processes, Classification of Stochastic Processes, Analytical Represenatation of a Stochastic Process, Autocorrelation Function & its Properties the Bernoulli Process, the Poisson Process, Pure Birth, Pure death & Birth-Death Process, Mathematical Models For M/M/1, M/M/1/N, M/M/S, M/M/S/N queues. Introduction to Markov Chains, Discrete Parameters Markov Chains, Computation of n -Step Transition Probabilities, Higher Transition Probabilities & the Chapman Kolmogorov Equations, State Classification & Limiting Distribution, Irreducible Finite Chains with Aperiodic States, M/G/1 queuing Model, Discrete Parameter Birth Death Process, Finite Markov Chains with Absorbing States.

Books Recommended: 1. A first course in Probability-Sheldon Ross. 2. Probability & Statistics for Engineers- Richard A. Johnson. 3. Stochastic Models in Queueing Theory- J. Medhi. 4. An Introduction to Probability theory and its Application- W.FELLER. 5.Fundamental of Queuing theory by Gross & Moris.



Title of the Course:-Special Functions and Fractional Calculus Course Code :-MAT-704 Course Credits :-04

1. Historical Survay of the development of special functions. The Fox H-Function-Definition, special

cases, Asymptotic expansions, simple transformation formulas and elementary properties, Mellin transform Laplace transform, Multiplication formulas.Simple Integrals involving the H-function.

2. The H-Function of two variables Defination, special cases, elementary properties, Asymptotic

behavior, Derivatives, Contiguous relations, Finite summation formulas and Generating Relations for the H-function of Two Variables.

3. Fractional Calculus- Historical Survay, Defination of the Riemann-Liouville Fractional Integral,

Fractional Integrals of Some functions namely,binomial. Function exponential, The hyperbolic and trigonometric functions, Bessel's functions, Hypergeometric function and the Fox’s H-function. Dirichlet's Formula, Derivatives of the Fractional Integral and the Fractional Integral of Derivative’s, Laplace Transform of the Fractional Integrals, Fractiona1 Derivatives, Laplace Transform of the Fractional Derivatives, Leibniz's Formula for Fractional Derivatives.

4. The Weyl Fractional Calculus- Definition of Wayl Fractional Integral, Weyl Fractional Derivative,

A Leibniz Formula for Weyl Fractional Integral and simple applications. . Books Recommended :- 1. The Fractional Calculus, K.B.Oldham and J. Spanier Academic Press. New York, 1974 2. An Introduction To The Fractional calculus and Fractional Differential Equations, K S.Miller and

B. Ross" John Wiley & Sons. New York 1993. 3. The H-function with Applications. in Statistics and other Disciplines A.M. Mathai and R.K.

Saxena Wiley Eastern, New Delhi, 1978. 4. The H-functions of One and Two Variables with Applications H.M. Srivastava, K.C.Gupta and.

S.P.Goyal, South Asian Publisher, New Delhi, 1982.



Title of the Course:-Generalized Hypergeometric function Course Code :-MAT-705 Course Credits :-04 1 Generalized Hypergeometric function: Definition, Convergence of the series for pFq, Differential

equation and its solution. The pFq with uniqe argument, Saalschutz theorem ,Whipples, Theorem and Dixon Theorem. Contour Integral representation for pFq, Euler type integrals involving pFq.

2 Meijers G-FUNCTION: Definition, elementary properties multiplication formulas, Derivatives,

Recurrence Relations, Mellin Transform and Laplace Transforms of the G Function. 3. H-function of one variable: Definition, identities. special cases differential formulas, recurrence

and Contiguous function relations. Finite and infinite series, Simple finite and infinite integrals involving H-function.

Books Recommended :- 1. Special Functions, ED Rainville, Reprinted by Chelsea Publ. Co. Bronx, NewYork (1971). 2. Higher Transcendental Function, Volume 1,2,3, Harry Bateman, McGRAW-HILL BOOK

Company, Inc. 1953. 3. The H-functions of One and Two Variables with Applications H.M. Srivastava, K.C.Gupta and.

S.P.Goyal, South Asian Publisher, New Delhi, 1982. 4. On the G-function, I-VIII, CS Meijer, Nederl. Akad. Wetensch. Proc. 1946.



Title of the Course:-Advanced Application of Integral Transforms Course Code :-MAT-706 Course Credits :-04

1. A general intergral transform, the H-function transform: Defination special cases, inversion formula and uniqueness theorem, certain general theorems, their special cases and applications Fourier Kernels Symmetrical and Unsymmetrical, self-reciprocal functions.

2. Two dimensional Integral Transforms- definations and elementary properties, A two

dimensional Integral Transform Whose Kernel is H-function of two variables. The inversion formula, properties and their applications.

3. Generalized Hankel Transform and its applications to the solution of dual integral equations.

Books recommended :-

1. The H-Function with Applications in Statistics and other Disciplines A.M. Mathai and R.K.

Saxena Wiley Eastern, New Delhi 1978.

2. The H-functions of one and two Variables with Applications H.M. Srivastava, K.C. Gupta and S.P.Goyal, South Asian Publisher, New Delhi 1982.

3. Mixed boundary Value Problems in Potential Theory. I.N.Sneddon North-Holland publishing

Co.A.John Wiley & Sons New York 1966.

4. Integral Transforms and Their Applications, Lokenath Debnath and Dambaru Bhatta, Chapman and Hall/CRC; 2 edition (11 October 2006).



Title of the Course:- Complex Analysis Course Code :-MAT-707 Course Credits :-04

Complex valued functions, limits, continuity, differentiability, Cauchy Reimann Equations, analytic functions, construction of an analytic function, confonnal mappings.

Complex integration: complex line integrals, Cauchy theorem, Cauchy's integral formula,

Liouville's theorem, Poisson's Integral formula, Morera's theorem, Taylor's and Laurent Series. Singularities, Branch points, Meromorphic functions. and entire functions, residues and

applications in evaluating real integrals, Rouche's theorem, Fundamental theorem of Algebra. Books recommended :- 1. Foundation of complex Analysis, S. Ponnusamy, Alpha Science International Ltd., 2005. 2. Complex Variable and applications, James Ward Brown and Ruel V. Churchill, McGraw-Hill, 2008. 3. Complex Variable: Theory and Application, H.S. Kasana, PHI Learning Private Ltd., 2005.



Title of the Course:- OPERATIONS RESEARCH Course Code :-MAT-708 Course Credits :-04 1. Introduction to Operations Research- The History, Nature & Significance of Operations

Research, Models & Modelling in Operations Research & General methods of solving these Models, Applications & Scope of Operations Research.

2. Linear Programming - Introduction, General Structure of a Linear Programming model, General

Guidelines on Linear Programming model formulation, Graphical Method, Simplex Method, Duality & Sensitivity Analysis, Integer Linear Programming, Dynamic. linear programming.

3. Queuing Theory - Analysis of queues with Poisson arrival & exponentially distributed service

times. Single channel queue with infinite customer population, Multi-channel queue with infinite customer population, Multi- channel queue with finite customer population

4. Transportation & Assignment Problems - Mathematical model of Transportation Problem, Methods of finding Initial B .F .S, Test for Optimality, Variations in Transportation Problcm, Mathematical Statemcnt of an Assignment problem, Solution Methods for an Assignmcnt problem, Variations of an Assignmcnt problem.

Malaviya National Institute of Technology, Jaipur

Department of Mathematics


Title of the Course:- Polynomials Course Code :-MAT-709 Course Credits :-04

Legendre Polynomials Hermite polynomials Laguerre Polynonuals Jacobi PolynomiaIs Orthogonal polynomials The general class of polynomials

1. A Treatise on Generating Functions, HM Srivastava and HL Manocha, Ellis Hortwood Ltd. 1984. 2. Orthogonal Polynomial, Gabor Szego, American Mathematical society, 1939. 3. Higher Transcendental Function, Volume 1,2,3, Harry Bateman, McGRAW-HILL BOOK

Company, Inc. 1953.



Title of the Course:- Integral Equations Course Code :-MAT-710 Course Credits :-04

Linear Integral Equations of the first and second kind of fredholm and volterra. types; Solution.

by successive substitutions and successive approximations. solutions of equations with separable Kernels.

The Fredholm alternative, Hilbert schmidt theory of Symmetric Kernels.

1. Integral Equations, F. G.Tricomi, Dover Publication,USA 1897.

2. Introduction to Integral Equations with Applications, A. Jerri, John wiley & sons,1999.



Title of the Course:- Viscous Fluid Dynamics Course Code :-MAT-711 Course Credits :-04

Basic concepts, Fundamental equations of' the flow of viscous fluids:- Equation of state

,equation of continuity - conservation of mass, Equation of motion (Navier- Stokes equations) --

conservation of momentum, Equation of energy -conservation of energy, Dimensional analysis,

Exact solution of the Navier- stokes equations :- Steady incompressible flow with constant

fluid properties, Steady incompressible flow with variable viscosity, Unsteady incompressible

flow with constant fluid properties, Steady compressible flow, Steady incompressible flow with

fluid suction/injection on the boundaries, Theory of very slow motion:- Stokes equations,

Stokes flow, Oseen equations, Oseen flow, Lubrication theory.

Books recommended: 1.Viscous Flow Theory Vol.1 ‘Laminar flow’ – S.I. Pai. 2.Boundary Layer Theory – H. Schlichting.



Title of the Course:- Fluid Mechanics Course Code :-MAT-712 Course Credits :-04

Ideal and Real Fluids, Pressure, Density, Viscosity, Description of Fluid motion, Lagrangian

method, Eulerian method, Steady and unsteady flows, Uniform and nonuniform flows, One

dimensional, two dimensional and ax symmetric flows, Line of flows, Stream surface, Stream

tube, Streak lines, Local and Material derivative. Equation of continuity. Euler's equation of

motion along a stream line, Equation of motion of an inviscid fluid, conservative field of force,

Integral of Euler's equation, Bernoulli’s equation and its applications, flow from a tank through

a small orifice, Cauchy's integral, Symmetric forms or the equation of continuity, Impulsive

motion of a fluid, Energy equation. Dimensional Analysis, Buckingham’s pi theorem, Variable

in fluid mechanics, Procedures of dimensional Analysis, similitude, Important dimension less

parameter (Reynolds's no., Mech no., Prandtl no. etc.)

Books recommended: 1.Viscous Flow Theory Vol.1 ‘Laminar flow’ – S.I. Pai. 2.Boundary Layer Theory – H. Schlichting.



Title of the Course:- MAGNETOFLUIDDYNAMICS OF VISCOUSFLUIDS Course Code :-MAT-713 Course Credits :-04 Exact solutions of the MHD Equations: MHO flow between parallel plates, MHD flow in a tube of rectangular cross-section, MHD pipe flow, MHD flow in an annular channel, MHD flow between two rotating coaxial cylinders, MHD flow near a stagnation point, MHD flow due to a plane wall suddenly set in motion. MHD boundary layer flow: Two-dimensional MHD boundary layer equations for flow over a plane surface for fluids of large electrical conductivity, MHD boundary layer flow past a flat plate in an aligned magnetic field, Two-dimensional thermal boundary layer equation for MHD flow over a plane surface, Heat transfer in MHD boundary layer flow past a flat plate in an aligned magnetic field, Two dimensional MHD boundary layer equations for flow over a plane surface for fluids of very small electrical conductivity, MHD boundary layer flow past a flat plate in a transverse magnetic field, MHD plane free jet flow, MHD plane wall jet flow, MHD curved wall jet flow, MHD circular free jet flow. Unsteady MHD boundary layer flow: MHD boundary layer flow due to impulsive motion of a plane wall, MHD boundary layer flow due to an accelerated flat plate, MHD boundary layer growth on a body placed symmetricalto the flow, MHD boundary layer growth in a rotating flow, Heat mass and momentum transfer in unsteady MHD free convection flow on an accelerated vertical plate, Unsteady MHD boundary layer flow past a flat plate in an aligned magnetic field for fluids of large electrical conductivity. MFD boundary layer flow: Two-dimensional MFD boundary layer equations for flow over a plane surface (fluids with large electrical conductivity), Similarity solutions for MFD steady boundary layer flow in an aligned magnetic field, Two-dimensional MFO boundary layer equations for flow over a plane surface (fluids with very small electrical conductivity), Similarity solutions for MFD steady boundary layer flow in a transverse magnetic field fixed relative to the fluid, Magnetogasdynamics plane free jet flow.



Title of the Course:- Bicomplex Analysis Course Code :-MAT-714 Course Credits :-04

Course Description Bicomplex numbers, Algebra and calculus of bicomplex numbers, Idempotent representation, Tholomorphicity, elementary bicomplex functions, integration, harmonic analysis, bicomplex manifolds, applicationsin quantum theory Scope & Objective The mathematical concept of bicomplex numbers (quaternions) is introduced in electromagnetics, and is directly applied to the derivation of analytical solutions of Maxwell's equations. It is demonstrated that, with the assistance of a bicomplex vector field, a novel entity combining both the electric and the magnetic fields, the number of unknown quantities is practically reduced by half, whereas the Helmholtz equation is no longer necessary in the development of the final solution. The most important advantage of the technique is revealed in the analysis of electromagnetic propagation through inhomogeneous media, where the coefficients of the (second order) Helmholtz equation are variable, causing severe complications to the solution procedure. Unlike conventional methods, bicomplex algebra invokes merely first order differential equations, solvable even when their coefficients vary, and hence enables the extraction of several closed form solutions, not easily derivable via standard analytical techniques. Text Books and References 1. Stefan Ronn, Bicomplex algebra and function theory, Arxive. 2. G. Baley Price (1991) An introduction to multicomplex spaces and functions, Marcel Dekker ISBN 0-

8247-8345-X. 3. Paul Baird and John C. Wood, Harmonic morphisms and bicomplex manifolds. 4. Irene Sabadini, Michael Shapiro, Frank Sommen(2009), Hypercomplex Analysis, Springer. 5. Clyde Davenport (1991) A Hypercomplex Calculus with Applications to Special Relativity, ISBN 0-

9623837-0-8 6. www.3dfractals.com

http://www.3dfractals.com/



Title of the Course:- GEOMETRIC FUNCTION THEORY Course Code :-MAT-715 Course Credits :-04 Course Description Conformal map, Riemann mapping theorem, fixed point theorem, Riemann surfaces, Schwarz Christoffel transformations for simply connected and multiply connected regions, Applications to flow problems, geometric function theory, harmonic functions, convexity, starlikeness, classes of functions, coefficient estimates, fractional calculus, Bergman spaces, applications to solution of partial differential equations Scope & Objective The geometric approach provides a new way to view the subject of complex variables. It is the source of tantalizing new questions. It also provides a vast array of powerful new weapons to use on traditional problems. Any number of problems about mappings and conformality are rendered transparent by way of geometric language. Text Books and References 1. RKrantz, Steven (2006). Geometric Function Theory: Explorations in Complex Analysis. Springer.

ISBN 0-8176-4339-7. 2. Graham I., Kohr G., Geometricfunction theory in one and higher dimensions, Marcel Dekker

Inc., New York,2003. 3. Nehari Z., Conformal Mapping, Dover publications, New York,1952. 4. Serge Lang, Complex analysis, Springer Verlag, NewYork,4thed. 1999. 5. Duren P.L.,Univalent functions, Springer Verlag, NewYork,1935. 6. HilleE., Analyticfunctiontheory (Vol.II),Chelsea Publications, 2ndEd., 1987.

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