BHARATHIAR UNIVERSITY, COIMBATORE -641046
M.Phil./ Ph.D. Applied Mathematics
FT / PT with effective from 2009–10
Paper I : Research Methodology
Paper II : Computational Methods
Paper III : Special Paper (anyone of the following)
1. Heat Transfer and Magnetohydrodynamics.
2. Fuzzy Sets, Logic and Theory of Neural Networks.
Paper-I : Research Methodology
UNIT I: Dimensional analysis and scaling
Dimensional analysis – The program of Applied Mathematics –
Dimensional Methods – The Buckingham Pi theorem – Formulation – Application to a
Diffusion Problem – Proof of the Pi theorem – Scaling – Characteristic Scales – A
Chemical Reactor Problem – The Projectile Problem – Population Models.
UNIT II: Regular Perturbation Method
The Perturbation Method – Motion in a Nonlinear Resistive Medium – A Non
linear Oscillator – The Poincare-Lindsted Method – Asymptotics.
UNIT III: Singular Perturbation and boundary-layer analysis
Failure of Regular Perturbation – Inner and outer approximations – Algebraic
equations and Balancing – The inner approximation – Matching – Uniform
approximations – Worked example – Boundary Layer Phenomena
UNIT – IV: WKB Approximation & Asymptotic Expansion of Integrals
The WKB Approximation - The Nonoscillatory Case - The Oscillatory Case.
Asymptotic Expansion of Integrals - Laplace Integrals - Integration by parts -
Generalizations.
UNIT – V:Wave Phenomena in Continuous Systems
Wave propagation - Waves - Linear Waves - Nonlinear Waves – Burgers’
Equation - The Korteweg-deVries Equation.
Text book
J.David Logan “Applied Mathematics”, Second Edition, John Wiley & Sons, Inc.
(1997). (Relevant Sections Only)
Reference Books
1. A.H. Nayfeh, “Perturbation Methods”, John Wiley & Sons, New York,
(1973).
2. R. Bellman, “Perturbation Techniques in Mathematics, Physics &
Engineering”, Holt, Rinehart & Winston, Inc. New York. (1963).
Paper- II : Computational Methods
UNIT I: Finite Difference Method
Two-dimensional parabolic equations – Alternating Direction implicit methodThe parabolic equation in cylindrical and in spherical polar co-ordinates – Miscellaneous
methods for improving accuracy – Reduction of the local truncation error – Use of Three
time –level difference equation – Solution of Non-linear parabolic equation – A three
time-level method .
UNIT II: Finite Element Method for One Dimensional Stress Deformation
Local and global coordinate system for the One-Dimensional Problem-OneDimensional Problem-Stress-Strain Relation-Principle of Minimum Potential EnergyPotential Energy Approach (for assembly)-Direct Stiffness Method-Boundary
Conditions-Strains and Stresses-Formulation by Galerkin’s Method-Complementary
Energy Approach-Mixed Approach.
UNIT III: Finite Element Method for Two Dimensional Stress Deformation
Introduction-Plane Deformations-Plane Stress Idealization-Plane Strain
Idealization-Axisymmetric Idealization-Strain-Displacement Relations-Finite Element
Formulation-Requirements for Approximation Function-Plane Stress IdealizationTriangular element-Comment on convergence.
UNIT IV: The Finite Volume Method for Diffusion Problems
Summary of conservative form of the governing equations of fluid flowDifferential and integral forms of the general transport equations-Finite volume method
for Diffusion problems-Introduction-Finite volume method for one dimensional steady
state diffusion-worked examples-Finite volume method for two dimensional diffusion
problems-Finite volume method for three dimensional diffusion problems.
UNIT V: The Finite Volume Method for Convection –Diffusion Problems
Introduction-steady one dimensional convection and diffusion-The central
differencing scheme-Properties of discretization schemes-Assessment-The upwind
differencing scheme-The hybrid Differencing scheme-Assessment-Higher Differencing
scheme for multi dimensional convection diffusion-The power law scheme
Text book for Unit I
G.D.Smith, “Numerical Solution of Partial Differential Equations – Finite
Difference Methods”, Clarendon Press, Oxford, (1978). (Relevant Sections only)
Text book for Unit II & Unit III
C.S.Desai, “Elementary Finite Element Method” Prentice Hall, Inc. (1979).
(Relevant Sections only)
Text book for Unit IV & Unit V
H.K.Versteey & W. Malalasekara, “An Introduction to CFD-The Finite Volume
Method” Longman Scientific &Technical, England. (1995). (Relevant Sections only)
Reference Books:
1. T.J. Chung,“Computational Fluid Dynamics”, Cambridge University Press, (2003).
2. Joel H. Ferzigen & Milovan Peric “Computational Methods for Fluid Dynamics”, Springer, (2002).
3. J.N.Reddy, “An Introduction to the Finite Element Method”, McGraw-Hill, (2005).
Paper - III : Special Paper
1. Heat Transfer and Magnetohydrodynamics
UNIT I: Flow along surfaces and in channels
Boundary layer and turbulence – The momentum equation of the boundary
layer – The laminar-flow boundary-layer equation - The plane plate in longitudinal flow -
Pressure gradients along a surface - Exact solutions of the laminar boundary-layer
equations for a flat plate
UNIT II: Forced Convection in Laminar Flow
The heat-flow equation of the boundary layer – Laminar boundary-layer
energy equation – The plane plate in longitudinal flow – The plane plate with arbitrarily
varying wall temperature– Exact solutions of the laminar- boundary- layer energy
equation – Flow through a tube.
UNIT III: Free Convection
Laminar heat transfer on a vertical plate and horizontal tube – Turbulent
heat transfer on a vertical plate – Derivation of the boundary-layer equations – Free
convection in a fluid enclosed between two plane walls – Mixed free and forced
convection.
UNIT IV:Introduction and fundamental Equations of Magnetohydrodynamics
and Steady Laminar motion
Introduction and fundamental equations: The electrodynamics of moving mediaThe electromagnetic effects and the magnetic Reynolds number-Alfven’s theoremThe magnetic energy-The mechanical Equations - The mechanical effects-The
Electromagnetic stresses-Steady Laminar motion.
UNIT V: Magnetohydrodynamic waves and stability
Magnetohydrodynamic waves-Waves in an infinite fluid of infinite electrical
conductivity-Alfven waves- Magnetohydrodynamic waves in a compressible fluidStability-Introduction—Simple illustrative examples-The Method of small Oscillations
Text book for Units I, II, III
E.R.G.Eckert & Robert M. Drake, “Heat and Mass Transfer” McGraw-Hill,
Tokyo, (1979). (Relevant Sections only)
Textbook for Units IV & V
V.C.A Ferraro & C. Plumpton, “An Introduction to Magneto-Fluid Mechanics”
Clanendon Press, Oxford, (1966). (Relevant Sections only)
Books for Reference:
1. B. Gebhart, “Heat Transfer”, McGraw-Hill, NewYork, (1971).
2. H .Schlichiting, “Boundary Layer Theory”, Mc Graw Hill, (1979).
3. Alan Jeffrey, “Magnetohydrodynamoics”, Oliver & Boyd, London, (1966).
Paper - III : Special Paper
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-wi CompositionsApproximate 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 Book for 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)
Text Book for Unit IV
Zimmermann H.J., “Fuzzy Set Theory and its Applications”, Fourth Edition,
Kluwer Academic Publishers, London,(2001). (Relevant Sections only)
Text Book for 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”, – Physica Verlag, Heidelberg, Germany, (1998).
3. S.Rajasekaran and G.A. Vijayalakshmi Pai., “Neural Networks, Fuzzy Logic and
Genetic Algorithms Synthesis and Applications”. Prentice-Hall of India, New Delhi,(2004).
