M.Phil/Ph.D. FT/PT - APPLIED MATHEMATICS
PART-I – SYLLABUS
(For the candidates admitted from the academic year 2018-19 onwards)
Paper I : Research Methodology
Paper II: Computational Methods
Paper III: Special paper (Any One of the following)
1. Intuitionistic fuzzy sets
2. Fuzzy Sets, Logic, and Theory of Neural Networks
3. Convection Heat Transfer and Magnetohydrodynamics
4. Hamiltonian Dynamics and Chaos
5. Dynamic Neural Networks
PAPER - I –RESEARCH METHODOLOGY
Unit I
Research Methodology: Meaning of Research – Objectives of Research – Motivation in
Research – Types of Research – Research Approaches – Significance of Research – Research
Methods versus Methodology – Research and Scientific method – Importance of Knowing How
Research is done – Research Process – Criteria of Good Research – Problems Encountered by
Researchers in India.
Defining the Research Problem: Research problem – Selecting the Problem – Necessity of
Defining the Problem – Technique Involved in Defining the problem.
Report Writing: Significance of Report Writing – Different steps in writing Report – Layout of
the Research Report – Types of Reports – Oral Presentation - Mechanics of writing a Research
Report – Precautions for Writing Research Reports.
Unit II
Modules: Free Modules – Project Modules – Tensor product – Flat Modules.
Unit III
Localization:Ideals- Local Rings- Localization-Applications.
UNIT IV
Elementary Properties of Holomorphic Functions: Complex differentiation –
Integration over paths – The local Cauchy theorem – The power series representation – The open
mapping theorem – The global Cauchy theorem – The calculus of residues.
UNIT V
Fourier Transforms: Formal properties – The inversion theorem – The Plancheral
Theorem - The Banach algebra L1
.
Text Books:
Unit I: C.R. Kothari, Research Methodology, New age international publishers.
Units II & III: N. S. Gopalakrishnan, Commutative Algebra, Oxonian Press, New Delhi,
Second Printing 1988.
Units IV & V: W.Rudin, Real and Complex Analysis, Tata Mc-Graw Hill, Third Edition, 2006.
PAPER-II –COMPUTATIONAL METHODS
Unit I
Nonlinear Systems: Local Theory: Preliminary concepts and definitions-The
fundamental Existence-Uniqueness Theorem-Dependence on Initial conditions and parameters -
The Maximal interval of existence-The flow is defined by a differential equation-Linearization-The
stable manifold theorem-The Hartman-Grobman theorem.
Unit II
Nonlinear Systems: Local Theory: Stability and Liapunov functions-Saddles,Nodes, Foci
and Centers-Nonhyperbolic critical points in R2
–Center Manifold theory-Normal form theory-
Gradient and Hamiltonian systems.
Unit III
Second-order Elliptic Equations: Definitions- Existence of weak solutions –
Regularity- Maximum principles- Eigenvalues and Eigenfunctions – Problems.
UNIT IV
Finite Volume Method for Diffusion: Finite volume formulation for steady state One,
Two and Three -dimensional diffusionproblems. One dimensional unsteady heat conduction
through Explicit, Crank – Nicolson andfully implicit schemes.
UNIT V
Finite Volume Method for Convection Diffusion: Steady one-dimensional convection
and diffusion – Central, upwind differencing schemes-Properties of discretization schemes –
Conservativeness, Boundedness, Trasnportiveness,Hybrid, Power-law.
Text Books:
UnitsI& II: L.Perko, Differential equations and Dynamical systems, Third edition, Springer
Verlag, New York, 2001.
Unit III: L. C. Evans, Partial Differential Equations, American Mathematical Society,
Providence, 1998.
Units IV & V: H. K. Versteeg and W. Malalasekera, An Introduction to Computational
FluidDynamics: The finite volume Method, Longman, England, 1998.
Reference books:
1. R. C. McOwen, Partial Differential Equations: Methods and Applications, Second
Edition, Pearson Education, New Delhi 2005.
2. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations,
Springer,New York, 2004.
PAPER - III : 1. INTUITIONISTIC FUZZY SETS
UNIT I
Intuitionistic fuzzy sets: Definition – Operations and Relations - Properties –
Intuitionistic Fuzzy sets of a certain level - Cartesian product and Intuitionistic Fuzzy Relations -
Necessity and Possibility Operators - Topological Operators.
UNIT II
Interval valued intuitionistic fuzzy sets: Intuitionistic Fuzzy Sets and Interval Valued
Fuzzy Sets - Definition, Operations, and Relations on Interval Valued Intuitionistic Fuzzy Sets -
Norms and Metrics on Interval Valued Intuitionistic Fuzzy Sets.
UNIT III
Other extensions of intuitionistic fuzzy sets: Intuitionistic L-Fuzzy Sets - Intuitionistic
Fuzzy Sets over Different Universes - Temporal Intuitionistic Fuzzy Sets - Intuitionistic Fuzzy
Sets of Second Type - Some Future Extensions of Intuitionistic Fuzzy Sets.
UNIT IV
Distances: Norms and Metrics Over the Intuitionistic Fuzzy Sets– The Two Term
Approach- Distances between the Intuitionistic Fuzzy Sets – The Three Term Approach -
Ranking of the Intuitionistic Fuzzy Alternatives.
Similarity measures between intuitionistic fuzzy sets: Similarity Measures and Their
Axiomatic Relation to Distance Measures- Intuitive Results Given by the Traditional Similarity
Measures -Correlation of Intuitionistic Fuzzy Sets
UNIT V
Multi attribute decision-making methods: The Linear Weighted Averaging Method of
Multiattribute Decision-Making with Weights and Ratings Expressed by Intuitionistic Fuzzy Sets
- Multi attribute Decision-Making with Intuitionistic Fuzzy sets : TOPSIS- Optimum seeking
method- Linear Programming Method- LINMAP- Fraction Mathematical Programming Method.
Text Books:
Units I, II andIII :
Krassimir T Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications, Physica - Verlag,
Heidelberg, 1999.
Unit IV:
EulaliaSzmidt, Distances_and_Similarities in Intuitionistic Fuzzy Sets, Springer International
Publishing, Switzerland, 2014.
Unit V:
Deng Feng Li, Decision and Game Theory in Management with Intuitionistic Fuzzy Sets,
Springer- Verlag Berlin Heidelberg, 2014.
PAPER III - 2. FUZZY SETS, LOGIC, AND THEORY OF NEURAL NETWORKS
Unit I
Fuzzy sets and Fuzzy relations Fuzzy sets – Basic types and basic concepts – Properties of α -
cuts – Representations of fuzzy sets – Decomposition Theorems – Extension principle for fuzzy
sets . Crisp and fuzzy relations – Projections and cylindric extensions – Binary fuzzy relations –
Binary relations on a single set – Fuzzy equivalence relations – Fuzzy compatibility relations –
Fuzzy ordering relations – Fuzzy Morphisms – Sup-i compositions of fuzzy relations. Inf-wi
compositions of fuzzy relations.
Unit II
Fuzzy Relation Equations Introduction- Problem Partitioning-Solution Method-Fuzzy Relation
Equations Based on Sup-i Compositions-Fuzzy Relation Equations Based on Inf-
wiCompositionsApproximate Solutions- The Use of Neural Networks.
Unit III
Fuzzy Logic Introduction – Fuzzy Propositions – Fuzzy Quantifiers – Linguistic Hedges –
Inference from Conditional Fuzzy Propositions – Inference from Conditional and Qualified
Propositions – Inference from Quantified Propositions.
Unit IV
Fuzzy Control Origin and Objective-Automatic Control-The Fuzzy Controllers., Types of Fuzzy
Controllers-The Mamdani Controller- Defuzzification-The Sugeno Controller., Design
Parameters-Scaling Factors-Fuzzy Sets-Rules-Adaptive Fuzzy Control-Applications.
Unit V
Neural Network Theory Neuronal Dynamics : Activations and Signals –Neurons As Functions-
Signal Monotonicity-Biological Activations and Signals-Competitive Neuronal Signals-Neuron
Fields-Neuronal Dynamical Systems-Common Signal Functions-Pulse-Coded Signal Functions.
Activations Models- Neuronal Dynamical Systems-Additive Neuronal Dynamics-Additive
Neuronal Feedback-Additive Activation Models- Additive Bivalent Models.-Bivalent Additive
BAM-Bidirectional Stability-Lyapunov Functions- Bivalent BAM Theorem.
Text Books:
Units I, II & III:Klir G.J and Yaun Bo “Fuzzy sets and fuzzy logic: Theory and applications”,
Prentice Hall of India, New Delhi, (2002). (Relevant Sections only)
Unit IV:
Zimmermann H.J., “Fuzzy Set Theory and its Applications”, Fourth Edition, Kluwer
Academic Publishers, London,(2001). (Relevant Sections only)
Unit V:
Bart Kosko, “Neural Networks and Fuzzy Systems”, Prentice Hall of India, New Delhi,
(2001). (Relevant Sections only)
Reference Books:
1. Kaufmann “Introduction to the theory of fuzzy sets”, Volume 1 -, Academic Press, Inc.,
Orlando, Florida,(1973).
2. John N. Moderson and Premchand S. Nair., “Fuzzy Mathematics: An introduction for
Engineers and Scientists”, – PhysicaVerlag, Heidelberg, Germany, (1998).
3. S.Rajasekaran and G.A. VijayalakshmiPai., “Neural Networks, Fuzzy Logic and Genetic
Algorithms Synthesis and Applications”. Prentice-Hall of India, New Delhi,(2004).
PAPER III – 3. CONVECTION HEAT TRANSFER AND
MAGNETOHYDRODYNAMICS
Unit I
Laminar Boundary Layer Flow:Fundamental Problem in Convective Heat Transfer - Concept
of Boundary Layer- Velocity and Thermal Boundary Layers - Integral Solutions - Similarity
Solutions-Methods- Flow Solution - Heat Transfer Solution.
Unit II
Laminar Boundary Layer Flow: Other wall heating conditions - Unheated starting
length - Arbitrary wall Temperature - Uniform Heat flux - Film Temperature - Effect of
longitudinal Pressure Gradient: Flow past a wedge and stagnation flow - Effect of flow through
the wall: Blowing and suction - Effect of conduction across a solid coating deposited on a wall.
Laminar Duct Flow :Hydrodynamic Entrance length - Fully Developed Flow - Hydraulic
Diameter and Pressure Drop.
Unit III
Laminar Duct Flow: Heat Transfer to Fully Developed Duct Flow - Mean Temperature
- Fully Developed Temperature Profile- Uniform Wall Heat Flux - Uniform Wall Temperature -
Tube Surrounded by Isothermal Fluid - Heat Transfer to Developing Flow - Scale Analysis -
Thermally Developed Uniform (Slug) Flow - Thermally Developing Hagen - Poiseuille Flow.
Unit IV
Introduction and Fundamental equations of MHD and Steady Laminar Flow: The
electrodynamics of moving media - The electromagnetic effects and the magnetic Reynolds
number - Alfven’s theorem - The magnetic energy - The mechanical equations - The mechanical
effects - The electromagnetic stresses - Steady laminar motion.
Unit V
Magnetohydrodynamic waves and stability: Waves in an infinite fluid of infinite
electrical conductivity - Alfven waves -Magnetohydrodynamic waves in a compressible fluid –
Stability – Introduction - Simple illustrative examples - Instability of linear pinch - Flute
instability - A general stability criterion- The method of small oscillations - Boundary conditions
- Solution of the equations - Illustrative example.
Text books:
Units I, II, III :A.Bejan, “Convection Heat Transfer”, Third Edition, John Wiley & Sons,
Hoboken, (2004).
Unit I – Sections 2.1 to 2.5 from Chapter 2.
Unit II – Sections 2.6 to 2.9 from Chapter 2 and Sections 3.1 to 3.3 from Chapter 3.
Unit III – Sections 3.4 to 3.5.3 from Chapter 3.
Units IV & V:
V.C.A Ferraro & C. Plumpton, “An introduction to Magneto-Fluid Mechanics” Clanendon Press,
Oxford, (1966).
Unit IV – Sections 1.1 to 1.7 from Chapter I and Section 2.5 from Chapter II.
Unit V – Sections 3.1 to 3.3 from Chapter III and Sections 5.1 to 5.3 from Chapter V.
PAPER III - 4. HAMILTONIAN DYNAMICS AND CHAOS
Unit I
The Dynamics of Differential Equations: Integration of linear second-order equations -
Integration of nonlinear second-order equations - Dynamics in the phase plane - Linear Stability
analysis.
Unit II
Hamiltonian Dynamics: Lagrangianformulation of Mechanics - Hamiltonian formulation
of Mechanics Canonical transformations - Hamilton-Jacobi equation and action-angle variables
-integrable Hamiltonians.
Unit III
Classical Perturbation Theory: Elementary perturbation theory - Canonical perturbation
theory - Many degrees of freedom and the problem of small divisors - The Kolmogorov- Arnold-
Moser theorem.
Unit IV
Chaos in Hamiltonian systems and area-preserving mapping: Area
preserving mapping-Fixed points and the Poincare-Birkhoff fixed point theorem Homoclinic and
heteroclinic points-Criteria for local Chaos.
Unit V
Nonlinear Evolution Equations and Solitons: Basic properties of the Kdv equation -
The inverse Scattering transforms Basic principles, KdV equation - Other soliton systems -
Hamiltonian structure of integrable systems.
Textbook:
Chaos and Integrability in Nonlinear Dynamics by M.Tabor, John Wiley and Sons, New York,
1989.
Unit I : Chapter 1 Sections 1.1 - 1.4,
Unit II : Chapter 2 Sections 2.1 - 2.5
Unit III: Chapter 3 Sections 3.1 - 3.4
Unit IV: Chapter 4 Sections 4.2 -4.5
Unit V : Chapter 7 Sections 7.2 – 7.6
PAPER III - 5. DYNAMIC NEURAL NETWORKS
Unit I
Dynamic Neural Units (DNUs) Nonlinear Models and Dynamics: Models of Dynamic Neural
Units (DNUs)- Models and Circuits of Isolated DNUs- Neuron with Excitatory and Inhibitory
Dynamics- Neuron with Multiple Nonlinear feedback- Dynamic Temporal behavior of DNN-
Nonlinear analysis for DNUs.
Unit II
Continuous-Time Dynamic Neural Networks Dynamic Neural Networks Structures: An
Introduction- Hopfield Dynamic Neural Network (DNN) and its Implementation- Hopfield
Dynamic Neural Networks (DNNs) as Gradient-like systems- Modifications of Hopfield
Dynamic Neural Networks- Other DNN models- Conditions for Equilibrium points in DNN.
Unit III
Learning and Adaptation in Dynamic Neural Networks Some observation on Dynamic Neural
Filter Behaviors- Temporal Learning Process I (Dynamic Backpropagation)- Temporal Learning
Process II (Dynamic Forward Propagation)- Dynamic Backpropagation for Continuous-Time
Dynamic Neural Networks.
Unit IV
Stability of Continuous-Time Dynamic Neural Networks Local Asymptotic Stability- Global
Asymptotic Stability of Dynamic Neural Networks Local Exponential Stability of DNNs- Global
Exponential Stability of DNNs.
Unit V
Discrete-Time Dynamic Neural Networks and their Stability General Class of Discrete-Time
Dynamic Neural Networks- Lyapunov Stability of Discrete-Time Nonlinear Systems- Stability
conditions for Discrete-Time DNNs- More General Results on Global Asymptotic Stability
Textbook:
M. M. Gupta, L. Jin, N. Homma, Static and Dynamic Neural Networks: From Fundamentals to
Advanced Theory, John Wiley & Sons, Inc. Publications, New Jersey, 2003.
Reference books:
1. A. AntoSpiritusKingsly, Neural network, and fuzzy logic control, Anuradha
publications, Chennai, 2009.
2. Robert J.Schalkoff, Artificial Neural Networks, TATA Mcgraw Hill Education, 2011.
3. Satish Kumar, Neural Networks, Mcgrawhill Higher Education,2012.
