# Syllabus for M. Phil. Course

Mechanics

Oct 24, 2013 (5 years and 3 months ago)

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Department of Mathematics

Chittagong University of Engineering & Technology (CUET)

Chittagong
-

Syllabus for M. Phil
.

Course

Course Code Course Title Credit

Math 6
000

Thesis

30
.00

Math 6101

Special Functions and I
ntegral Transforms
-
I

3.00

Math 6102

Special Functions and
Integral Transforms
-
II

3.00

Math 6103

Qualitative Theory of Differential Equations

3.0
0

Math 6104

Partial
Differential Equ
ations (PDEs)

3.00

M
ath 6201

Fluid Dynamics
-
I

3.00

Ma
th 6202

Fluid Dynamics
-
II

3.00

Math 6301

Similarity Analysis

3.00

Math 6302

Perturbation and Approximation Theory

3.00

Math 6401

Optimization Techniques
-
I

3.00

Math 6402

Optimiza
tion Techniques
-
II

3.00

Math 6501

-
I

3.00

Math 6502

-
II

3.00

Math 6
503

Quantum Field Theory
-
I

3.00

Math 65
04

Quantum Field Theory
-
II

3.00

Math 6505

Mathematical Cosmology

3.00

Math 65
06

Classical Theory of Fields

3.00

Math 6507

Geometrical Methods in Mathematical Physics

3.00

Math 6
601

3.00

M
ath 6602

Graph Theory
-

I

3.00

Math 6603

Graph Theory
-
II

3.00

Math 6701

-
I

3.00

Math 6702

-
II

3.00

Math 6703

Computational Numerical Methods

3.00

Math 6801

Group Theory

3.00

Math 6802

Rings and Modulus

3.00

2

Course Code:

Math 6101

Course Title
:
Special Functions and Integral

Transforms
-
I

Credit Hours
: 3.00

Gamma and Beta Functions: Properties of Gamma function, Continuity and
convergence of gamma and beta functions, integral form of

. Asymptotic
Representation of Gamma function for Large
.

Elliptic Integral and Elliptic Functions: Reduction of elliptic integrals to standard
form, properties of Elliptic function, addition formulae, periods of elliptic function.

The probability integral and related functions; application to t
he theory of heat
conduction and to the theory of vibration. Generating function of the Hermite and
Laguerre polynomials, recurrence relations, the differential equation and the integral
equation satisfied by the polynomials. Integral representations, orth
ogonality and
Laguerre polynomials, Hypergoemetric functions its linear and quadratic
transformations. The confluent hypergeometric function, its integral and asymptotic
representation. Representation of various functions in terms of Hypergeometric and
the

confluent Hypergeometric functions. Hermite functions. Matheus functions and
the Dirac Delta functions. The minkuiski Temple. Theory of generalized function.
Schwartz’s theory of distribution.

Reference
s
:

1)

Artin; L , Holt, Rinehast and winston: New Yo
rk.

2)

Bell
-

W.W, Van: Special functions for scientists and Engineers.

Nostr and Co. Ltd

3)

Lebedev, N.N: Special functins and Their Applications.

Prentice
-

Hall, Englewood, Cliffs,N.j.

4)

McLachlan: Theory and applicatons of Mathien functins.

Dover Puplication,

Springer Verlag, Berlin.

5)

Slater, L.J: Confluent Hypergeometric Functins

Combridge
University

Press, London.

6)

P.M.K. Morse and H. Feshbach: Methods of Theoretical Physics

7)

R. Curant and D. HIlbert: Methods of Theoretical Physics

3

Cours
e Code
: Math 6102

Course Title:

Special Function and Integral Transforms
-
II

Credit Hours
:

3.00

Green’s function and its applications. Green’s Function and Second Order Differential
Operators, Generalized Green’s Function, Green’s Identities, Fourier int
egral theorem
and Fourier
transforms

Multiple Fourier transforms. Fourier transforms of radially
symmetric functions. The solutions of integral equations of convolution type, Use of
Fourier transforms in solving Laplace’s equation, diffusion equations and
wave
equations. The double Laplace transform, the interated Laplace transform, the
Stieltjes transform and the Hankel transform. The perseval relation for Hankel
transform and the relation between Fourier and Hankel
transform
. Use of Hankel
transforms in s
olving partial differential equations.

Reference
s
:

1)

Sneddon, I
-
N: Special Functions of Mathematical physics and chemistry 2
nd

2)

Sneddon, I. N: Fourier transforms Mc Graw
-
Hill Book Co. I nc. New York.

3)

Sneddon, I.N: The use of

Integral transforms Tata
-

Mc Graw
-
Hill Co. Ltd.

4)

P.M.K. Morse and H. Feshbach: Methods of Theoretical Physics

5)

R. Curant and D. HIlbert: Methods of Theoretical Physics

6)

G.F. Roach: Green’s Functions: Cambridge University Press: London.

4

Course Code:
M
ath 6103

Course Title
: Qualitative Theory of Differential Equation

Credit Hours
: 3.00

Geometrical methods for first order linear and nonlinear differential equations. Nature
and stability of the critical points of second order linear and nonlinear system
s,
Canonical forms of Second
-

Order Linear Equations with constant co
-
efficients, An
initial

Value Problem; Characteristics. Stability types of stability, stability by
Liapunov functions and theorems. Limit cycle and periodic solutions. Lotka
-
voltera
popu
lation models. Complexity and Stability, Epidemic models and dynamics of
infections diseases, continuous growth models, Delay models, Periodic fluctuations.
Population models with perturbations. Bifurcation theory and chaotic system.

Reference
s
:

1)

Soloman

Lefschetz: Differential equations, Geometric Theory.

2)

H.I. Freedman: Deterministic Mathematical methods in population Ecology

3)

V.I Arnold: Geometrical Methods in the theory of ODE.

4)

J. Cronin: Differential Equation Introduction and Qualitative theory

5)

S.N
. Chow and J. K. Hale: Methods of Bifurcation
theory.

6)

Shepley L. Ross: Differential Equations, Jahn Wiley & Sons.

7)

M. Haze winkle, R. Jurkovich and J
-
H.P. Paelinck: Bifurcation Analysis.

8)

Robert C. Hiborn: Chaos and Nonlinear Dynamics, Oxford
University Pre
ss.

9)

D.W. Jordan and P. Smith: Nonlinear O. D. Equations.

5

Course Code:

Math 6104

Course Title
:

Partial Differential Equations (PDEs)

Credit Hours
:

3.00

Classificat
ion of PDE (parabolic, elliptic

and hyperbolic
), Existence, uniquene
ss and
represention of solutions for the PDE (wave equation & heat equation). Cauchy,
Dirichlet and Neumann boundary
-
value problems for the Laplace and Poisson
equation. Potential theory in two and higher dimensional domains, initial and
boundary value pro
blems of heat equation and wave equation, Maximum principle of
parabolic equation; Sturm
-
Liouville systems, boundary and eigenvalue problems,
method of eigenfunction expansions. The sturm
-

Liouville equation, Green’s Function
and Generalized functions.

Reference
s
:

1.

Prashad, P. and Ravindran, R. Partial Differential Equations,

2.

Pinsky, M. A., Partial Differential Equations and Boundary Value Problems with
Applications.

3.

G. Stephemsion: An Introduction to Partial Differential Equations for Science
Students
.

4.

J. N. Sneddon: Elements of Partial Differential Equations.

5.

N. Minorsky: Non
-
linear Oscillations.

6.

D.W. Jordan and P. Smith : Non Linear OD Equations.

6

Course Code:

Math 6201

Course Title:

Fluid Dynamics

I

Credit Hours:

3.
00

Eulerian and Lagrangian method of description of fluid; Analytic approach of
deformations; Derivation of equations of conservation of mass. Momentum and
energy. Basic equations in different coordinate system, boundary conditions.
Irrotational and rota
tional flows. Bernoulli's equation and its applications. Two
dimensional irrotational incompressible flows with circulation; sources and sinks;
Vortex motion. Combination of basic flows, mapping of flows in complex
coordinates. Aerofoil theory, Schwartz
-
Ch
ristoffel theory, Nervier
-
Stokes equations.
Gravity waves, One dimensional compressible flows of sound waves, Shock waves;
Vibrations and Waves; Plane Waves, Acoustic waves in a layer, Elastic Plane Waves,
Dilatational waves. Two dimensional irrotational f
lows. Hypersonic flows; Viscous
compressible fluid flows. Incompressible fluid flow between two parallel plates; flow
through a circular pipe and annulus. Flow between a plane and a cone; Flow through
convergent and divergent channel flow in the vicinity o
flows.

Reference
s
:

1. G.K Batchelor : An Introduction to Fluid Dynamic

2. Kundu, Cohen: Fluid Mechanics

3. L. M. Milne Thomson: Theoretical Hydrodynamics

4. D. F. Parker, Fields, Flows and Waves. An introduction to con
tinuum Models.
Springer; New York.

7

Course Code:

Math 6202

Course Title:

Fluid Dynamics

II

Credit Hours:

3.00

Small Raynold's number flows; flows over a sphere; flow over a cylinder through
porous media; Lubrication theor
y. Boundary layer theory; properties of Navier
-
Stokes
equations; two dimensional boundary equations; displacement, momentum and
energy thickness for two dimensional flows. Von Mises transformation. Similarity
solutions of boundary layer equations. Boundary

layer flow over a flat plate, boundary
layer flow with pressure gradient; Approximate solutions of boundary layer equations,
including Von
-
karman's method. Stability theory; Basic concepts of stability theory;
Stability of Quett's flow; Stability of flow
between two parallel plates; Reyleigh
-

Taylor instability; Kelvin
-
Helmholtz instability. Temperature Boundary Layer in
Forced convection, Temperature Boundary layer in Natural Convection.

Turbulence : Reynolds stresses and basic equation for turbulent f
lows; Prandtl mixing
length theory; some simple turbulent flows; homogeneous turbulence; spectral theory
of homogeneous turbulence. Non Newtonian fluid flows; Riener
-
Rivlin fluids; power
law fluids; flows in ells fluids; flow in Binghan plastics; Visco
-
Ela
stic flows; general
visco elastic fluid flows. Supersonic Flow; Oblique Shock Wave, Reflection of
Oblique Shock waves, Prandtl
-

Meyer flow. Shock
Expansion

theory.

Reference
s
:

1)

Fremzini Finnemore : Fluid Mechanics

2)

White: Fluid Mechanics

3)

W. P. Boyle :
Fluid Mechanics

4)

Landau & Lipshits: Fluid Mechanics

5)

Joseph Spurk; Fluid Mechanics, Springer. London.

8

Course Code:

Math 6301

Course Title:

Similarity Analysis

Credit Hours:

3.00

Principle and illustrations of dimensional analysi
s, systematic calculation of
dimensionless products, algebraic theory of dimensional analysis, different
procedures, (Rayleigh; Buckingham pie
-
theorem, stepwise, echelon,
proportionalities

etc.) for the determination of dimensionless groups and its
behavio
r

for some
boundary value problems; Method of
seminude

and introduction to fractional analysis
of overall equations, a free parameter method for similarity solution applied to two
dimensional boundary layer flows, method of separation of variables, similar
ity
requirements for three dimensional. Axisymmetric velocity and thermal boundary
layer laminar flows (both steady and unsteady), group theory method, absorption of
parameters and natural co
-
ordinates in similarity variables, reduction of independent
vari
ables, similarity and
natural co
-
ordinates on liberalized

compressible flow,
supersonic and transonic similarity rules. Karman similarity criteria for turbulent
shear layers.

Reference:

1)

G.K. Batchelor: Fluid Dynamics I.

Course Code:

Math 6302

Course T
itle:

Perturbation and Approximation Theory

Credit Hours:

3.00

The nature of perturbation theory, some regular perturbation problems, the technique
of perturbation theory, some singular perturbation in sirofoil theory, the method of
matched asymptotic
expansion, the method of strained co
-
ordinates in viscous flow at
high Reynolds number, some inviscid single perturbation problems, aspect of
perturbation theory. New classes of information by approximation theory,
classification of problems and difficulti
es in approximation theory, analysis of the
condition for approximation theory.

9

Course Code:

Math 6401

Course Title:

Optimization Techniques

I

Credit Hours:

3.00

Introduction; Convex sets, convex and concave functions & their properties Pseudo
convex

and quasi convex functions, tangent and support hyperplanes, convex cones,
Farkas Lemma, Method of Lagrange multipliers. Classical methods with single and
multivariables. Linear programming, Graphical method with mathematical definitions
and theorems; Sol
ution of a system of linear simultaneous equations, Pivotal
reduction of a general system of equations simplex method with theoretical
development. Transportation problem. Non linear programming: One dimensional
problems by elimination and interpolation me
thods; Unconstrained techniques; direct
search and descent methods; constrained techniques and indirect methods.

Course Code:

Math 6402

Course Title:

Optimization Techniques

II

Credit Hours:

3.00

Geometrical programming, Dynamic programming; Stochas
tic programming; Game
theory; CPM and PERT; Calculus of variations.

Reference:

1)

G. R. Walsh: Optimization Methods

2)

V.K. Kapoor: Operations Research

3)

R.V. Mital: Optimization Method in Operations Research and System
Analysis.

10

Course Code
:

Math 6501

Course Title:

I

Credit Hours:

3.00

Basic development of quantum Mechanics: Experimental background, Old quantum
theory, Uncertainty and complementary principle of superposition, Dynamical
variables and observabl
es, Representations of the quantum conditions, development
of Schrodinger equation, Approximate Methods for Stationary and time dependent of
Schrodinger equation, Solution of S.E, diff. Kinematical conditions, Hydrogen atoms
for Zeeman effect. Perturbatio
n theory, the Born Approximation. The variation
Method, Inelastic collisions, Adiabatic and sudden approximation. Ionization
Problems in atomic scattering. Theory of Radiation; Connection between Bosons and
Oscillators, Omission and absorption of Bosons, A
pplication to photons, the
interaction energy between photon and an atom. Emission, Absorption and scattering

Reference:

1)

L.I. Schiff : Quantum Mechanics

2)

Gupta kumar Sharma: Quantum Mechanics

3)

Pavling and Wilson : Intr
oduction to Quantum Mechanics

4)

Bransden: Relativistic Quantum Mechanics.

5)

P.A.M. Dirac : The Principles of quantum Mechanics .

6)

Landau & Lipshitz: Quantum Mechanics

7)

Pauling & Wilson: Introduction to Quantum Mechanics.

11

Course Code
:

Math 6502

Course Title:

II

Credit Hours:

3.00

Relativisitic theory of the Electron: Relativisitic treatment of a particle; Klein

Gordon equation, the wave equation for the electron and its solution, Invariance under
Lorent
z transformation, the motion of a free electron, existence of the spin, the fine
structure of the Energy levels of Hydrogen, theory of positrons. Quantum
Electrodynamics; The Electromagnetic field in the absence of matter, Relativisitic
form of the quantum

conditions, the supplementary conditions, electron and position,
difficulties of the theory.

Reference:

1)

P.A.M. Dirac : Relativistic Quantum Mechanics

2)

James D. Bjorken and Sindney Drell: Relativistic Quantum Mechanics

3)

P.M Mathews, K Venka Tesan: A Text
Book of Quantum Mechanics

4)

GeoRge L. Trigg: Quantum Mechanics

5)

K. Gottfried: Quantum Mechanics.

6)

12

Course Code:

Math 6503

Course Title:

Quantum Field Theory
-
I

Credit Hours:

3.00

Canonical formalism and Quantization for fields.

Symmetries and Conservation laws,
The Klein
-
Gordon Field, Second Quantization of the scalar field, electromagnatic
field and spinor field, The Feynman Propagator, Interaction with an external field,
Symmetry properties of interactions, Symmetries of stran
ge particles, Vacuum
expectation values, The S
-
matrix and Asymptotic Theory, General properties of the S
-
matrix, Unitarily and partial wave decomposition, Causality and Analylicity,
Perturbation theory, Interaction representation and Feynman Rules, Electro
n

Electron and Electron
-
Positron and electron
-
Medium to Heavy atomi Scattering.
Derivation of covariant Perturbation rules and Computation of elementary process in
scalar electrodynamics, Dispersion Relations.

Reference:

1)

Claude Itzykson , Jean
-
Bernardu
ber : Quantum Field theory, Mc G Ran

Hill
-

International Editions Physics series.

2)

Michael E. Peskin. Daniel V. Schroeder: An Introduchir to Quantum Field
theory : The Advanced Book Program Levant Books Kolkata: India.

3)

Amitabha Lahiri
-

Dalash B. Pal : A

first Book of Quantum Field theory
Narosa Publishing House Kolkata., India.

4)

Ernest M. Henley, Walter Thirring : Elementary Quantum Field theory

Mc. GRAW
-

Hill Book Company , Inc, New York

5)

S. Schweber: Relativistic Quantum Field theory

6)

S. Weinberg: On t
he Quantum theory of Fields. Vol. I and Vol. II

7)

P. Ramond: Introduction to Quantum Field theory .

13

Course Code:

Math 6504

Course Title:

Quantum Field Theory
-
II

Credit Hours:

3.00

Path Intergrals. Trajectores in the Bargmann
-

F
ock space. Relativistic formulation. S
-
matrix and Green Functions in terms of Path Integrals. Constrained systems: The
Electromagnetic Field as an example. Large orders in perturbation theory,

Symmetries: Quantum Implementation of Symmetries, Mass spectru
m, Multiplets and
Goldstone Bosons. Current Algebra and Commutators, Axial Current and Chiral
Symmetry. Regularization and Power counting. Furry’s theorem, Renormalization.
Massless theories and Weinberg's theorem. Renormalization in case of Quantum
Electr
odynamics. The

-
model and renormalizations, Anomalies: Axial anomaly in
the

-
model, Classical theory of non
-
abelian Gauge fields, Quantization of Gauge
Fields. Feynman Rules. Massive Gauge fields. The Weinberg
-
Salam Model.

Reference:

1)

Itzykson , Zuber: Q
uantum Field Theory

2)

T.D. Lee Introduction to Particle Physics.

3)

D.V. S shirkov and N.N. Bogoliuov: Introduction to Quantized Field.

4)

Bjorken and Drell: Relativistic Quantum Fields.

5)

Quantum chromo dynamic
-

cheug & lie

14

Co
urse Code:

Math 6505

Course Title:

Mathematical Cosmology

Credit Hours:

3.00

The

Robertson
-

Walker model, the Einstein static model of the Universe, Einstein
equations
,

Friedmann Models, Hubbles Constant and the deceleration parameter
,
Models with a co
smological constant
,
Singularities in cosmology: The Schwarzschild
metric with its properties
,
The early Universe
,
The very early Universe and Inflation

Quantum Cosmology,
The distant future of the Universe.

References:

1.

J. N. Islam: An Introduction to M
athematical Cosmology.

2.

J. V. Narlikar: Introduction to cosmology.

3.

S. Weinberg: First three minutes.

4.

A. K. Raychaudhuri : Theoretical cosmology: The Expanding Universe.

5.

P.T. Peebles: Physical Cosmology.

6.

Lurdan & Lifshitn: Classical Theory of fields

7.

M. P
. Ryam Jr. & L.C. Sheply: Homogeneous Relativistic Cosmology.

15

Course Code:

Math 6506

Course Title:

Classical Theory of Fields

Credit Hours:

3.00

Special theory of relativity, relativistic quantum Mechanics, Maxwell’s equations,
relat
ivistic electrodynamics, eledromagnetic radiation, tensor Calculus (covariant
differentiation, Christoffel symbols, Riemann and Ricci tensor ), Principle of
Equivalence, Eiensten’s equations, Schwarz Schild solution and its properties,
Spherically Symmetr
ic gravitational collapse, Kerr solution and its properties,
-
elements, Friedmann and Lemaitre
cosmological models, non isotropic hermogenus cosmologies, Equation of geodesic
deviation and introduction to Penro
se
-

Hawking singularity theorem, Introduction to
Inflationary cosmologies.

Reference:

1.

L. D. Landau and E.M. Lifshitz: Chemical theory of Fields.

2.

J.D. Jackson: Classical Electrodynamics.

3.

Misner, Thorne and Wheeler: Gravitation.

4.

A. K. Ray chaudhuri: Theore
tical Cosmology.

5.

Gibbons, Hawking and Siklos (Eds): The very Early Universe

16

Course Code:

Math 6507

Course Title:

Geometrical Methods in Mathematical Physics

Credit Hours:

3.00

Differential Manifolds and tensors, Rie
mann Manifolds,
L
ie derivatives and
L
ie
groups, Differential forms: Riemannian Corrections on manifold. Applications to
thermodynamics, Hamiltonia
n

mechanics, electromagnetism, dynamics of a perfect
fluid and cosmology; Gauge theories, Tiber
bundles
, connections, Principa
l bundle,

Singer Inlax Theorem.

Reference:

1.

B. Schurtz: Geometrical Methods of Mathematical physics.

2.

Y. Ghognet
-

Bruhat C. Deintt
-

Analysic, of Mathods and Mhorett and M.
Dillard

Bleick Physics.

3.

Misner, thorne and Wheeler: Grav
itation.

4.

M. Sivak: A corrprehesive Introduction to Differential Geometry.

17

Course Code:

Math
6601

Course Title:

Credit Hours:

3.00

Matrix
Operations:

Direct sum of matrices, Kronecker product, Jordan product, Lie

product, Khatri
-
Rao product, Vec operation and their properties. Canonical Forms and
Matrix
Factorization:

Jordan canonical form, Smith’s canonical form, Full rank
factorization, Shur’s Triangularization, LU factorization, QR Factorization, Spectral
decom
position. Norms and Measures of Matrices.

Matrix calculus :

Matrix sequence, series and their convergence. Computation of
matrix function by different methods; limit, continuity, differentiation of matrices.
Solving ODE using matrix.

Generalized inverse

of matrices:

Classification and properties. Different methods
of computing generalized inverse of matrices: using property, Decell’s method,
Fedeev
-
Leverrier’s method, Penrose method, Graybill
-
Meyer
-
Painter method, Drazin
pseudoinverse, Moore
-
Penrose
-
Cli
ne inverse, Urquhart computation of various
inverses from {1} inverse. Stochastic Matrices: Limiting
Behavioer of

physical
system Expected values, Expected values of Squares.

References:

1.

Hazra, A. K., Matrix : Algebra, Calculus and Generalized Inverse
,

2.

Ben
-
Israel, A. and Grevilla, T. N. E, Generalized Inverses: Theory and
Applications.

3.

Richard Bellman: Introduction to Matrix Analysis. Tata Mc Graw
-
Hill publishing
corp. Bellonan Ltd.

18

Course Code:

Math
6602

Course Title:

Graph Theory

I

Credit Hou
rs:

3.00

Graphs and Subgraphs:

Graphs and Simple graphs, The Incidence and Adjacency
Matrices, Subgraphs, Vertex degrees, Paths and Connection, Cycles.

Trees and Forests:

Connectivity
:

Complementary graphs, Cut
-
vertices and Bridges, Blocks.
Construction
of Reliable Communication Networks.

Euler Tours and Hamilton Cycles : Euler Tours, Hamilton Cycles, The Chinese
Postman Problem, The Travelling Saleman problem.

Vertex Colourings:

Chromatic number, Chromatic Polynomials, Brooks Theorem, A
Storage Problem,

Edge Colorings:

Edge chromatic number, Vizing’s theorem, The time tabling
problem.

Reference:

1.

J. A. Bondy and U.S.R. Murty, Graph Theory with Applications.

2.

Mehdi Behzad, Gary Chartrand and Linda Lesniak Foster, Graphs and Digraphs.

3.

John Clark and Derek

Allan Holton, A First Look at Graph Theory.

19

Course Code:

Math
6603

Course Title:

Graph Theory

II

Credit Hours:

3.00

Matchings, Factorization and coverings:

The personal assignment problem. Planar
and nonplanar Graphs: Eule
r’s formula, Dual graphs, Characterization of planar
graphs, The Five colour theorem and the Four colour conjecture. NonHamiltonian
planar graphs. Independent sets and Cliques: Independent sets, Ramsey’s theorem,
Turan’s theorem, Schur’s theorem. Algorithm
s and Applications: Tuttes I
-
facter
theorem. Edmonds Blossom Algorithm (optional).

Directed Graphs:

Directed graphs, Directed paths and cycles, A job sequencing
problem.

Networks :

Flows, Cuts, The Max
-
flow Min
-
cut theorem, Manger’s theorem.

Tournament
s: Elementary properties of tournaments, Hamiltonian tournaments, Score
sequences.

Reference:

1.

J. A. Bondy and U.S.R. Murty, Graph Theory with Applications.

2.

Mehdi Behzad, Gary Chartrand and Linda Lesniak Foster, Graphs and Digraphs.

3.

John Clark and Derek Al
lan Holton, A First Look at Graph Theory.

4.

Douglas B. West: Introduction to Graph theory.

20

Course Code:

Math
6701

Course Title:

-
I

Credit Hours:

3.00

Richardson extrapolation of differentiation, Rom
berg integration,

Predictor
-
corrector
methods, Runge
-
Kutta Methods, Multistep methods (Adam Bashforth
-
Moulton
method, Adams method for initial value problem, Milne
-
Simpson method); Stability,
time stability, stiffness. Hybrid (Gragg and Stetter, Butcher,

Nordsieck) and
extrapolation (Bulirsch and Stoer) methods for two point boundary value problem,
Linear shooting, shooting for nonlinear problems, finite difference methods for linear
and nonlinear problems. Systems of ODE, stiffness, A
-
stability, Gear’s m
ethod. Finite
difference methods for Elliptic, Parabolic & Hyperbolic PDEs.

Reference:

1.

Balagurusamy, E.: Numerical Methods

2.

Smith: Numerical Solution of Partial Differential Equations.

3.

S C Chapra, R P. Canale : Numerical Methods for Engineers

21

Course C
ode:

Math
6702

Course Title:

-
II

Credit Hours:

3.00

Approximants. Algebraic and Differential Approximants. Approximate Solution
of Linear Differential Equations. Approximate Solution of Nonlinear Differential
Equations. A
symptotic Expansion of Integrals. Perturbation Series. Summation of
Series. WKB Theory. Multiple Scale Analysis. Keller Box methods. MAPLE and
MATLAB.

References:

1.

M K Jain , S.R.K Lyenger , R. K Jain Numerical Methods

2.

S C Chapra, R P. Canale , Numeric
al Methods for Engineers

3.

C F Gerald, P.O. Wheatly, Applied Numerical Analysis

22

Course Code:

Math
6703

Course Title:

Computational Numerical Methods

Credit Hours:

3.00

Error Analysis Numerical Solution of Transcenden
tal equations for two or more
independent variables with computer programming, Computational numerical
solution of system of linear equations by direct and iterative methods. Evaluation of
single and double integration by suitable computational methods.
Solutions of
ordinary differential equations with boundary conditions by finite deference method.
Orthogonalization and Orthonormalization process. Numerical solution of Different
types (Elliptic, Parabolic and Hyperbolic) of Partial Differential Equation
s
by finite

difference method. Eigen values and Eigen function of Boundary value problems by
finite difference method. Gradient vector and Hessian matrix, Non
-
linear
Programming problems. Fuzzy sets and logic.

References:

1)

M K Jain , S.R.K Lyenger ,
R. K Jain Numerical Methods

2)

S C Chapra, R P. Canale , Numerical Methods for Engineers

3)

C F Gerald, P.O. Wheatly, Applied Numerical Analysis

4)

S O S , Operations Research

5)

Xevier FORTRAN Language and Numerical Methods.

6)

Xevier C Language and Numerical Meth
ods.

23

Course Code:

Math
6801

Course Title:

Group Theory

Credit Hours:

3.00

Definitions and simple properties of groupoids quasigroups, semigroups, Groups and
Subgroups, Klein four
-
group dihydral groups, quaternion group, Quotien
t group
symmetric group alternating groups. Lagrange’s theorem, normal subgroups,
homomorphism theorem and isomorphism theorem, Cyclic groups of permutations,
Cayley’s theorem. Direct products of groups.

The Centraliser and the normaliser of a subset of a

groups, the center of group, the
coinmutator subgroup of a group, automorphisms of groups. Normal series,
subnormal series and composition, series of groups. Schreier’s subgroup theorem,
Jordan Holder’s theorem.

Finite groups, p
-
groups, the derived serie
s of a group, solvable groups. Cauch’s
theorem, Sylow subgroups, Sylow theorems, st
ructure theory of finite abelian groups.
Frattini subgrop /Φ subgroup, Cintral Chain (Lower and Upper
) Free

Abelian group.

Course Code:

Math
6802

Course Title:

Rings and Modulus

Credit Hours:

3.00

Rings, ideals, ring homomorphism, general ideal theo
ry of commutative rings,
Edclidean ring, factorizations in a rings, Unique factorization domain polynomial
domains in one and several variable, irreducibility criteria. Structure of semi
-
simple
and simple rings with minimum condition.

Modules: Direct sum,

projective modules, inje
ctive modules, Exact sequence of
modules, tensor and Homofunctions on modoues. Five lemmas, short five lemma,
strong lemma, 3x3 lemma. Sake diagram. Modules with chain conditions: Artinian
Modules, Noe therian Modules, Modules of Finite length, Artiniam Ri