User Rating: 5 / 5

Course Code  :           AMAT 42793

Title                :           Fluid Dynamics

Pre-requisites :           PMAT 41763

Learning Outcomes   :

At the end of this course the student will

• recognize the difference between discrete mass-points and continuous matter in mechanics
• define the fundamental properties of two-dimensional and Axi-symmetric motion, three dimensional motion of a perfect fluid, and motion of a viscous fluid.

 Course Content         :

Further Vector Analysis: Orthogonal curvilinear coordinates, Gradient, divergence and curl.

Basic Principles of  Dynamics: Fluid pressure, Velocity, Acceleration, Stream lines, Equation of continuity, conservation of momentum, conservation of energy, Euler’s equations of motion,  Bernoulli’s theorem, Vorticity, Irrotational motion under conservative forces, Kinetic energy in irrotational motion, Uniqueness theorems, Velocity circulation round a closed curve, Kelvin’s theorem, Vortex lines,  Helmholtz vorticity equation, Naviers stokes theorem, Cyclic and acyclic motions, Kinetic energy in irrotational motion, Uniqueness theorems.

Two Dimensional Motion: Stream function and plotting stream lines, Complex potential, Sources and sinks, Vortices, Doublets and image systems, Milne-Thompson theorem, Flow past a cylinder, Applications of conformal transformations including Schwarz-Christoeffel transformation, Blassius theorem.

Axi-symmetric Motion: Stokes’ stream function (3D).

Three Dimensional Motion:  Irrotational motion, Laplace’s Equation, Spherical Harmonics, Flow of a stream past a fixed sphere, Motion of a sphere in a fluid, Impulsive motion.

 Method of Teaching and Learning : A combination of lectures, tutorial discussions and presentations.

 Assessment     : Based on tutorials, tests, presentations and end of course Examination.

 Recommended Reading       :

1. Feistauer ,M. Mathematical Methods in Fluid Dynamics, chapman and Hall/CRC,1993.
2. A. J. Chorin, J. E. Marsden. A Mathematical Introduction to Fluid Mechanics, Springer Science & Business Media, 2012.
3. Dan, H. , Martin, B. Fluid Dynamics Theory and Computation, Stokholm 2005.
4. Chorlton, F. Textbook of Fluid Dynamics, CBS Publishers & Distributors, 2005

User Rating: 5 / 5

Course Unit Code     : AMAT 41773

Course Title               : Advanced Computational Mathematics

Pre-requisites             : AMAT 22582

 Learning outcomes:

Upon successful completion of the course unit the student will be able to:

• Investigate the criteria such as convergence, rate of convergence, accuracy and appropriate consistency and stability
• Apply numerical algorithms to solve initial boundary value problems in the form of partial differential equations.

 Course Content:

Finite Difference Methods: Introduction to finite difference schemes, Solve parabolic, hyperbolic and elliptic partial differential equations, Dirichlet boundary conditions and Neumann boundary conditions, Convergence: Consistency and Stability using Von Neumann Analysis, Lax Equivalent Theorem.

Finite Element Methods: Variational formulation of problem-classification of partial differential operators, Weighted residual methods: Collocation method, least square method, Galerkin method.

Practicals: Implement Finite difference Schemes using an appropriate software, Implement Finite element schemes using Free FEM++

 Method of Teaching and Learning : A combination of lectures and tutorial discussions

 Assessment     :           Based on tutorials, presentation, tests and end of course examination

 Recommended Textbook:

1. Burden, R.L., Faires, J.D, Burden, M.L. Numerical Analysis (10e), Cengage Learning. (2015).
2. J. Davies, Finite Element Method: An Introduction to Partial Differential Equations (2e), OUP Oxford (2011)
3. M. Desai, Finite Element Method with Applications in Engineering. Pearson Education India (2011)
4. Pavel Ŝolín, Partial Differential Equations and the Finite Element Method. (2013)

User Rating: 5 / 5

Course Unit Code     : AMAT 42783 

Course Title               : Advanced Mathematical Modeling

Pre-requisites             : AMAT 41763

Learning outcomes:

Upon successful completion of the course unit the student will be able to:

• explain how the general principals arise in the context of Mathematical Modeling
• analyze some existingmathematical models and  construct simple models for real world situations.
• explain and apply the basic concepts of Mathematics and their uses in analyzing and solving real-world problems

 Course Content:

Introduction to Modeling: Philosophy of modeling, Modeling Methodology, Problem formulation, Mathematical Description, Analysis, Interpretation

Mathematical Modeling Using Ordinary Differential Equations: Classification of ODE, Equilibrium points, Qualitative analysis of equilibrium points.

First order Differential Equations: Mixing, chemical reactions, Population models: Logistic growth model, Harvesting models, Traffic Dynamic models: Microscopic and macroscopic models

System of Differential equations: Interacting population models (Predator –Prey models, Competition models), Compartment models (Dynamic of infectious disease, Age structured models, Reaction kinetics)

Mathematical Modeling Using Difference Equations: First order difference equations, Equilibrium points, asymptotic stability of equilibrium points, System of linear difference equations: Autonomous systems, Discrete analogue of Putzer algorithm, Jordan form, linear periodic systems

Applications: Markov chains, Population dynamics, Trade models, Age classes, Business cycle models.

Group Project: Mathematical model formulation for a real world problem

 Method of teaching and learning    : A combination of lectures and tutorial discussions

 Assessment                 : Continuous assessment and/or end of course unit examination

 Recommended Textbook:

1. Kapur, J.N., Mathematical Modeling, New Age International. (2015).
2. A., Bender, An introduction to Mathematical Modeling, Courier Corporation, 2012
3. Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow. SIAM ,(1998)

User Rating: 5 / 5

Course Code     : AMAT 41763

Title                  : Qualitative and Quantitative Behavior of the Solutions of Ordinary Differential        

                              Equations

Pre-requisites   : PMAT 22572

Learning Outcomes:

At the end of this course, the student should be able to obtain the numerical solutions of differential equations and their implementations using appropriate software.

 Course Contents:     

Introduction to Software: Basic procedures in using an appropriate software, Handling numbers and matrices, Control structures, Program design, Script and function files, Plotting.           

Basic Properties of the Solutions of Ordinary Differential Equations: Stability of the solution and State-Space Analysis, Qualitative behavior of the solution of Ordinary Differential Equations.

Numerical Solutions of Ordinary Differential Equations: Concept of consistency, Stability and convergence properties of numerical schemes, Single and multistep methods, Solving Stiff systems, Shooting method, Gear's implementation of automatic ordinary differential equation solver, Finite difference discretizations for second order boundary value problems.                                                                                                                                   

Lab work using appropriate software: Algorithms studied in AMAT 21553, AMAT 22562 and in this unit will be implemented  using an appropriate software.

 Method of Teaching and Learning: A combination of lectures, computer laboratory sessions, tutorial discussions and presentations.

 Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading      :

1. Hanselman, D. & Littlefield, B.R.. Mastering MATLAB, Pearson Education Limited, (2014).
2. Ferdinand, V. Nonlinear differential equations and dynamical systems, Springer Science & Business Media (2012).
3. Vanloan, C.F., Introduction to scientific computing: a matrix-vector approach using MATLAB, Prentice Hall, New York. (2000).