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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
2. B.V.Ramana, Higher Engineering Mathematics, McGraw – Hill.
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT 103
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-401
Mathematics III
Laplace transforms:
3L-0T
3 credits
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.
4. B.V.Ramana, Higher Engineering Mathematics, McGraw – Hill.
5. T. Amarnath, Partial Differential Equation and its application, Narosa .
6. Sankara Rao, Introduction to Partial Differential Equations, Prentice Hall of India.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
M.Sc. (APPLIED MATHEMATICS)
I Semester
Course
number
Course Name
MAT-611
MAT-612
MAT-613
Abstract Algebra
Real Analysis
Statistics & Probability
Theory
Ordinary Differential
DC
Equation
Operations Research
DC
Comprehensive English
IDC
dynamics of communication
(for tech. eng.
Communication skills)
Total
MAT-614
MAT-617
HST-603
Subject
area
code
DC
DC
DC
L
T P C
CWS
MTE
PRE
ETE
Total
3
3
3
1 1 1 -
4
4
4
10
10
10
30
30
30
-
60
60
60
100
100
100
3
1 -
4
10
30
-
60
100
3
2
1 - -
4
2
10
-
30
-
-
60
-
100
-
17
5 -
22
50
150
-
300
500
II Semester
Course
number
Course Name
Subject
area
code
L
T P C
CWS
MTE
PRE
ETE
Total
MAT-622
Linear Algebra and Theory
of Matrices
Complex Analysis
Computer Language
and Computer Lab I
Partial Differential
Equations
Discrete Mathematical
Structure
Total
DC
3
1 -
4
10
30
-
60
100
DC
DC
3
2
1 0 4
4
4
10
10
30
30
20
60
40
100
100
DC
3
1 -
4
10
30
-
60
100
DC
3
1 -
4
10
30
-
60
100
14
4 4
20
50
150
20
280
500
MAT-621
MAT-626
MAT-623
MAT-627
III Semester
L T P C
CWS
MTE
PRE
ETE
Total
Fluid mechanics
Subject
area
code
DC
3
1
-
4
MAT-637
Numerical Analysis
DC
3
1
0
4
MAT-633
Integral Transforms
DC
3
1
-
4
MAT-638
Topology
DC
3
1
-
4
Elective I
DE
3
1
-
4
Computer lab-II
P
0
0
4
2
15 5
4
22
10
10
10
10
10
50
30
30
30
30
30
20
170
-
60
60
60
60
60
30
330
100
100
100
100
100
50
550
Course
Code
Course Name
MAT-636
MAP-635
Total
IV Semester
Course
Code
Course Name
Subject
area
code
L T P C
CWS
MTE
PRE
ETE
Total
MAT-641
MAT-642
MAT-645
Functional Analysis
Integral Equations
Information and
Coding Theory
Elective II
DC
DC
DC
3
3
3
1
1
1
-
4
4
4
10
10
10
30
30
30
-
60
60
60
100
100
100
DC
3
1
-
4
10
30
-
60
100
Project
Total
DC
12 4
-
10 10
30 40
30
150
-
60
340
100
500
90 190
620
20
1250 2050
MAD-646
Grand Total of all
semesters
58 18 8
M. Sc. Electives I to III
S.No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11
12
Course
Code
MAT-651
MAT-652
MAT-653
MAT-667
MAT-655
MAT-656
MAT-668
MAT-659
MAT-661
MAT-664
MAT-665
MAT-666
Course Name
L
T
P
C
Number Theory
Applied Stochastic Processes
Advanced matrix theory
Special Function
Combinatorics & Graph Theory
Fractional Calculus & its application
Computational Fluid Dynamics
Numerical optimization technique
Queueing Theory and Applications
Optimization Algorithms for Networks
Analytic function theory
Mathematical Modelling
3
3
3
3
3
3
3
3
3
3
3
3
1
1
1
1
1
1
1
1
1
1
1
1
-
4
4
4
4
4
4
4
4
4
4
4
4
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-617 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
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-622 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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-621 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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-626Computer 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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-623 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, 2 nd
Edition, Narosa, New Delhi.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-627: 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.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-636
Fluid Mechanics
4 Credits (3L+1T+0P)
Basic concepts:- Fluid, Continuum Hypothesis, Viscosity, Rate of strain quadric, Stress at a
point, Stress Quadric, Relation between Stress and rate of strain Components(Stokes Law of
Friction),Thermal Conductivity.
Fundamental equation of the flow of viscous fluid:-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:-Dynamical
Similarity(Reynolds Law),Inspection Analysis, Dimensional Analysis, Buckingham π-Theorem.
Physical Importance of Non-Dimensional Parameters.
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 of the boundaries.
Books recommended:
1. S.I. Pai.,Viscous Flow Theory ‘Laminar flow’, Vol.1,Van Nostrand Co., Ny,(1956).
2. H. Schlichting, Boundary Layer Theory, 7th Edition, McGraw-Hill (1979).
3. J. L. Bansal, Viscous Flow Theory, Jaipur Publishing House (2008)
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-637
Numerical Analysis
4Credits (3L+1T+0P)
Errors: Floating-point approximation of a number, Loss of significance and error propagation,
Stability in numerical computation.
Linear Systems: Gaussian elimination with pivoting strategy, LU factorization, Residual
corrector method, Solution by iteration (Jacobi and Gauss-Seidal with convergence analysis),
Matrix norms and error in approximate solution, Eigenvalue problem (Power method),
Gershgorin’s theorem (without proof). Nonlinear Equations: Bisection method, Fixed-point
iteration method, Secant method, Newton's method, Rate of convergences, Solution of a system
of nonlinear equations.
Interpolation by Polynomials: Lagrange interpolation, Newton interpolation and divided
differences, Error of the interpolating polynomials
Differentiation and Integration: Difference formulae, Some basic rules of integration.
Differential Equations: Euler method, Runge- Kutta methods, Multi-step methods, PredictorCorrector methods . Finite difference method for ordinary and Partial differential equations.
Books Recommended:1. K. E. Atkinson, An Introduction to Numerical Analysis, 2nd Edition, Wiley-India, 1989.
2. S. D. Conte and C. de Boor, Elementary Numerical Analysis - An Algorithmic Approach, 3rd
Edition, McGraw-Hill, 1981.
3. R. L. Burden and J. D. Faires, Numerical Analysis, 7th Edition, Thomson, 2001.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-633
Integral Transforms
4 Credits (3L+1T+0P)
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.
Fourier Series and Fourier Transforms: Orthogonal set of functions, Fourier series, Fourier sine
and cosine series, Half range expansions, 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.
Books Recommended:
1. Ian N. Sneddon , The use of Integral Transforms ,McGraw Hill; Second Printing edition
,1972.
2. Ian N. Sneddon, Fourier Transforms , Dover Publications,2010 .
3. Loknath Debnath, Integral Transforms and their applications ,Chapman and Hall/CRC; 2
edition ,2006.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-638
Topology
4 Credits (3L+1T+0P)
Topological Spaces: open sets, closed sets, neighbourhoods, bases, sub bases, limit points,
closures, interiors, continuous functions, homeomorphisms. Examples of topological spaces:
subspace topology, product topology, metric topology, order topology. Quotient Topology:
Construction of cylinder, cone, Moebius band, torus, etc. Connectedness and Compactness:
Connected spaces, Connected subspaces of the real line, Components and local connectedness,
Compact spaces, Heine-Borel Theorem, Local -compactness. Separation Axioms: Hausdorff
spaces, Regularity, Complete Regularity, Normality, Urysohn Lemma, Tychonoff embedding
and Urysohn Metrization Theorem, Tietze Extension Theorem. Tychnoff Theorem, One point
Compactification. Complete metric spaces and function spaces, Characterization of compact
metric spaces, equicontinuity, Ascoli-Arzela Theorem, Baire Category Theorem. Applications:
space filling curve, nowhere differentiable continuous function. Optional Topics: Topological
Groups and orbit spaces, Paracompactness and partition of unity, Stone-Cech Compactification,
Nets and filters.
Books Recommended:
1. M. A. Armstrong, Basic Topology, Springer (India), 2004.
2. K. D. Joshi, Introduction to General Topology, New Age International, 2000.
3. J. L. Kelley, General Topology, Van Nostrand, 1955.
4. J. R. Munkres, Topology, 2nd Edition, Pearson Education (India), 2001.
5. G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1963.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAP-635
Computer Lab II
2 Credit (0L-4P)
Exposure to MATLAB and computational experiments based on the syllabus of Numerical Analysis
(MAT- 632.).
Text Books:
Amos Gilat, MATLAB: An Introduction with Applications, 4th Edi.,Wiley, (2011).
Reference Books:
1. Rudra Pratap, Getting Started with MATLAB: A Quick Introduction for Scientists and
Engineers, Oxford, (2010).
2. Brian R. Hunt, Ronald L. Lipsman, Jonathan M. Rosenberg, A Guide to MATLAB: For
Beginners and Experienced Users,3rd Edi., Cambridge,(2014).
3. William J.Palm III, A Concise Introduction to MATLAB, McGraw-Hill Higher Education,
(2008)
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-641
Functional 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 Banach &Hilbert 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, Wiley Publications
2. Functional Analysis- Brown Page
3. Introductory functional analysis with applications, Kreyszig, Wiley publications
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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 equations 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 theorem. 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. W. V. Lovitte ,Linear Integral Equations, over Publications; Reissue edition , (2005).
2. R. P. Kanwal ,Linear Integral Equations, Birkhäuser; 2nd edition , (1996).
3 S.G. Mikhlin, Linear Integral Equations , Routledge, (1961).
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-645 Information Theory and Coding
4 Credits (3L+1T+0P)
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 code word 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. R. G. Gallager, Information Theory and Reliable Communication. Wiley, 1968, ISBN13: 978-0471290483
2. Robert Ash, Information theory, Dover Publications 1990 (first published 1965)
3. R.J.McEliece , The Theory of Information and Coding, Cambridge University Press, Jul2004
4. S. Lin and D. J. Costello, Error Control Coding, 2nd ed. Pearson Prentice Hall, 2004,
ISBN-13: 978-0130426727.
5. T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. WileyInterscience, 2006. ISBN-13: 978-0471241959.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-651
Number 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. Shanti Narayan, Number Theory, Chand,Delhi,(1966).
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-652
Applied Stochastic Processes
4Credits (3L+1T+0P)
Definition and classification of general stochastic processes, Examples. Markov chains,
Transition Probability Matrices, classification of states, Recurrence, examples. Basic Limit
theorems of markov chains, Renewal Equation (Discrete case), Absorption probabilities.
Random walk and queueing examples. Continuous 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. V.N. Bhat , G.N.Miller, Elements of Applied Stochastic Processes, 3rd Edi., Wiley, New
York, (2002).
2.V.G. Kulkarni, Modeling and Analysis of Stochastic Systems,2nd Edi.,Chapman and
Hall,(1996).
3. J. Medhi ,Stochastic Models in Queueing Theory,Academic Press, Amsterdam,(2003)
4. R. Nelson ,Probability, Stochastic Processes, and Queuing Theory
The Mathematics of Computer Performance Modelling ,Springer-Verlag, New York,(1995).
6. S. Ross ,Stochastic Processes, 2nd ed.,Wiley,New York,(1996).
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-653
Advanced 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. S. Barnett ,Matrix Methods for Engineers and Scientists,McGraw-Hill,London,(1979).
2. Eves, Howard ,Elementary Matrix Theory,Dover Publication,(1980).
3. Shanti Narayan and P.K.Mittal,Text book on Matrix Theory,S.Chand,(1953).
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-667
Special Functions
4 Credits (3L+1T+0P)
The Gamma and Beta Functions: Eulers’ integral forГ(z), the beta function, factorial function ,
Legendre’s duplication formula, Gauss’s multiplication theorem, summation formula due to
Euler, behaviour of log Г(z) for large |z |
The Hypergeometric function: An integral representation. Its differential equation and solutions.
, F(a,b,c;1) as a function of the parameters, evaluation of F(a,b,c;1), contiguous function
relations, the hypergeometric differential equation, logarithmic solutions of the hypergeometric
equation, F(a,b,c;z) as a function of its parameters, Elementary series manipulations, simple
transformations, relation between functions of Г(z) and, Г(1-z) quadratric transformations,
theorem due to Kummer, additional properties
The Confluent Hypergeometric function: Basic properties of 1F1, Kummer’s first formula.
Kummer’s second formula,
Generalized Hypergeometric Series: The function pFq, the exponential and binomial functions,
differential equation, contiguous function relations, integral representation pFq, with unit
argument, Saalshutz’ theorem, Whipple’s theorem, Dixon’s theorem, Contour integrals of
Barnes’ type.
Bessel Functions: Definition, Differential equation, differential recurrence relations, pure
recurrence relation, generating function, Bessel’s Integral, index half an odd integer, modified
Bessel functions
Introduction to Legendre function, Meijer G-function and some basic properties.
Books Recommended:
1. Earl. D. Ranvillie, Special Functions , Macmillan, 1960.
2.L.C. Andrews ,Special Functions of Mathematics for Engineers, SPIE Press, 1992.
3. Gabor Szego, Orthogonal Polynomials, American mathematical society, 1939.
4. L.J. Slater,Generalized Hypergeometric Functions , Cambridge University Press; Reissue
edition ,2008.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-655
Combinatory & 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. Normal L. Biggs ,Discrete Mathematics,2nd Edi.,OUP Oxford,(2002).
2. J. Hein ,Discrete Structures, Logic and Computatibility,Jones & Bartlett Pub., 3rd Edi.,(2009).
3. C.L.Liu ,Elements of Discrete Mathematics,3rd Edi., McGraw-Hill,(2008).
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-656
Fractional Calculus and Applications
4 Credits (3L+1T+0P)
The Riemann Liouville Fractional Calculus: Fractional Integrals of some functions namely
binomial function, exponential, the hyperbolic and trigonometric functions, Bessel’s functions,
Hyper-geometric function and the Fox’s H-function. Dirichlet’s Formula, Derivatives of the
Fractional Integral and the Fractional Integral of Derivatives. Laplace Transform of the
Fractional integral, Leibniz’s Formula for Fractional Integrals. Derivatives, Leibniz’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 Non-homogeneous
Fractional Differential Equations, Reduction of Fractional Differential Equations to ordinary
differential equations. Semi Differential equations.
Books Recommended:
1. K.B. Oldham & J. Spanier, The Fractional Calculus: Theory and Applications of
Differentiation and Integration to Arbitrary Order, Dover Publications Inc, 2006.
2.K.S. Miller & B.Ross. ,An Introduction to the Fractional Calculus and Fractional Differential
Equations Hardcover , Wiley-Blackwell, 1993.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-668
Computational Fluid Dynamics
4Credits (3L+1T+0P)
Basic Fluid Dynamics, Numerical Solution of some Fluid Dynamical problems, Local similar
solution of boundary layer equations, Transonic Relaxation methods, Small perturbation
equations, Transonic small Perturbation equations Line relaxation Techniques Time dependent
methods, Finite element method.
Books Recommended:
1. T K Bose ,Numerical Fluid Dynamics, Narosa Publishing House,(1997).
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
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. Lagrange 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. O.L. Mangasarian ,Non-linear Programming,SIAM,(1987).
2. G. Hadley ,Linear Programming,Addison-Wesley Pub.Co.,(1962).
3. Gross & Moris ,Fundamental of Queuing theory,4th Edi.,Wiley,(2008).
4. L.S.Srinath ,PERT and CPM Principles and applications,East-West Press Private
Limited,(1971).
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-661
Queueing Theory and Applications
4 Credits (3L+1T+0P)
Review of probability, random variables, distributions, generating functions; Poisson, Markov,
renewal and semi-Markov processes; Characteristics of queueing systems, Markovian and nonMarkovian queueing systems, embedded Markov chain applications to M/G/1, G/M/1 and
related queueing systems; Networks of queues, open and closed queueing networks; Queues with
vacations, priority queues, queues with modulated arrival process, discrete time queues,
introduction to matrix-geometric methods, applications in manufacturing, computer and
communication networks.
Books Recommended:
1. D. Gross and C. Harris, Introduction to Queueing Theory, 3rd Edition, Wiley, 1998 (WSE
Edition, 2004)
2. L. Kleinrock, Queueing Systems, Vol. 1: Theory, John Wiley, 1975.
3. J. Medhi, Stochastic Models in Queueing Theory, 2nd Edition, Academic Press, 2003
(Elsevier India Edition, 2006).
4. J.A. Buzacott and J.G. Shanthikumar, Stochastic Models of Manufacturing Systems, Prentice
Hall, 1992.
5. R. B. Cooper, Introduction to Queueing Theory, 2nd Edition, North-Holland, 1981.
6. L. Kleinrock, Queueing Systems, Vol. 2: Computer Applications, John Wiley, 1976.
7. R. Nelson, Probability, Stochastic Processes, and Queueing Theory: The Mathematics of
Computer Performance Modelling, Springer, 1995.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-664 OPTIMIZATION ALGORITHMS FOR NETWORKS
4 Credits (3L+1T+0P)
Introduction to graphs and networks: Introduction, concepts and definitions, linear programming.
Tree algorithms: Spanning tree algorithms.
Path algorithms: Shortest path algorithm, all shortest path algorithms, other shortest path
algorithms.
Postman problem: Postman problem for undirected graphs, directed graphs and mixed graphs.
Travelling salesman problem: Salesman problem, existence of Hamiltonian circuits, lower bound
and solution techniques.
Books Recommended:
Edward Minieka ,Optimization Algorithms for Networks and Graphs,2nd Edi.,(1992).
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-665
Analytic function theory
4 Credits (3L+1T+0P)
Analytic functions, complex integral calculus, Families of analytic functions: Convex, Starlike,
Spirallike, Harmonic functions, Univalent functions, Entire and meromorphic functions,
Geometric properties of functions, Conformal mapping on simply connected domains, Mapping
properties of special functions, Riemann mapping theorem, Schwarz-Christoffel transformations,
Potential function, Laplace equation and solution.
Text and Reference Books
1. Graham I., Kohr G., Geometric Function Theory in One and Higher Dimensions, Marcel
Dekker Inc., New York, 2003.
2. Nehari Z., Conformal Mapping, Dover publications, New York, 1952.
3. Serge Lang, Complex Analysis, Springer Verlag, NewYork, 4th ed. 1999.
4. Duren P.L., Univalent Functions, Springer Verlag, NewYork, 1935.
5. Hille E., Analytic Function Theory (Vol. II), Chelsea Publications, 2nd Ed., 1987.
6. Conway J.B., Functions of One Complex Variable, Springer Verlag, NewYork, 2nd Ed.,
1978.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-666
Mathematical Modelling
4 Credits (3L+1T+0P)
What is a model? What is Mathematical modelling? Role of mathematics in
problem solving;
Transformation from real world problem to real world
model and then to Mathematical Model; some illustrations of real world
problems;
Mathematical
formulation,
Dimensional
Analysis,
Scaling,
Validation,
Simulation,
Some
case
studies
with
analysis
(such
as
exponential
growth
and
decay
models,
population
models,
Traffic
flow
models, Optimization models) Projects.
Books:
1. D N P Murthy, N W Page, E Y Rodin, Mathematical Modelling , Pergamon Press,(1990).
2 -Principles of Mathematical Modelling (2004)-Clive L. Dyne, Elsevier Publication
3 Mathematical Modelling –A case study approach (2005), AMS publication-R Illner, C Sean
Bohun, S McCollum, T van Roode
4. Mathematical Modelling- J. N. Kapur
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.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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-FunctionDefinition, 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.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
Title of the Course:-Advanced Application of Integral Transforms
Course Code
:-MAT-706
Course Credits
:-04
1. A general integral transform, the H-function transform: Definition 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- definitions 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).
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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.
Hill,
Complex Variable and applications, James Ward Brown and Ruel V. Churchill, McGraw2008.
3.
Complex Variable: Theory and Application, H.S. Kasana, PHI Learning Private Ltd.,
2005.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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
Comprehensive Course
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.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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.
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
Title of the Course:- MAGNETOFLUIDDYNAMICS OF VISCOUSFLUIDS
Course Code
:-MAT-713
Course Credits
:-04
Exact solutions of the MHD equations: MHD 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 symmetrical to 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 MFD 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.
Reference Book:
1.
Bansal J. L., Magnetofluiddynamics of Viscous Fluids, JPH Jaipur (1994)
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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.
2.
3.
4.
5.
6.
Stefan Ronn, Bicomplex algebra and function theory, Arxive.
G. Baley Price (1991) An introduction to multicomplex spaces and functions, Marcel
Dekker ISBN 0-8247-8345-X.
Paul Baird and John C. Wood, Harmonic morphisms and bicomplex manifolds.
Irene Sabadini, Michael Shapiro, Frank Sommen(2009), Hypercomplex Analysis,
Springer.
Clyde Davenport (1991) A Hypercomplex Calculus with Applications to Special
Relativity, ISBN 0-9623837-0-8
www.3dfractals.com
Malaviya National Institute of Technology, Jaipur
Department of Mathematics
Comprehensive Course
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.
2.
3.
4.
5.
6.
RKrantz, Steven (2006). Geometric Function Theory: Explorations in Complex Analysis.
Springer. ISBN 0-8176-4339-7.
Graham I., Kohr G., Geometricfunction theory in one and higher dimensions, Marcel
Dekker
Inc., New York,2003.
Nehari Z., Conformal Mapping, Dover publications, New York,1952.
Serge Lang, Complex analysis, Springer Verlag, NewYork,4thed. 1999.
Duren P.L.,Univalent functions, Springer Verlag, NewYork,1935.
HilleE., Analyticfunctiontheory (Vol.II),Chelsea Publications, 2ndEd., 1987.
MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MA 510 Simulation And Modeling
3L-0T 3 credits
Definition of a system, System concepts, type of system, continuous & discrete systems,
modeling process verification & validation.
Introduction of Probability Distributions and random processes, Central limit theorem.
Estimation of mean and variance, Confidence interval, Hypothesis testing, Normal distribution,
t-test, ANOVA- an Introduction
Markov chains: CTMC and DTMC
Queuing models: Basic queuing models. Little’s Theorem and network of queues.
Introduction, classification of simulation models, advantages and disadvantages of simulation.
Concept of simulation time and real time. Discrete system simulation. Monte Carlo method,
Random number generators.
Simulation of inventory systems
Introduction to simulation environment and software tools.
Text/References:
1. Principles of Operations Research, Wagner, PhI.
1. Simulation modeling and analysis, Law and Kelton, McGraw Hill.
2. Probability and Statistics with Reliability, Queuing and Computer Science Application,
Kishore S Trivedi, Wiley.
3. System simulation, Gorden G., Prentice Hall of India.