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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

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Course Unit Code     : AMAT 41373

Course Title               : Advanced Computational Mathematics

Pre-requisites             : AMAT 22292

 Learning outcomes:

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

• calculate finite difference operators to approximate the derivatives and corresponding truncation errors
• identify initial and boundary conditions of PDE
• apply finite difference methods to obtain the approximate solution of PDEs together with prescribed boundary and/or initial conditions
• analyze the stability, consistency and convergence of numerical scheme
• compare the accuracy of the approximate solution obtained by finite difference scheme with exact solution using simulation results
• solve boundary value problems using basic finite element methods
• solve one dimensional PDEs using finite element methods
• solve boundary value problems using free FEM++.

 Course Content:

Finite Difference Methods: Introduction, Classification of Partial Differential Equations (PDE): parabolic, hyperbolic and elliptic, Taylor series expansion: analysis of truncation error. Initial and boundary conditions: Dirichlet and Neumann boundary conditions. Finite difference methods: Forward, Backward, Centered and Crank-Nicholson schemes, Implicit and Explicit methods. Stability and Convergence analysis of numerical schemes: Von Neumann Analysis, Consistency and Stability, Lax Equivalent Theorem, Comparison of Numerical Schemes.

Finite Element Methods: Introduction, Weak Formulation. Solving one and two dimensional PDEs using finite element method: Weighted residual methods: Collocation method, least square method, Galerkin method.

Practical: Implement Finite Difference Schemes using appropriate software, Implement Finite element method 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. (10th Ed., 2016). Numerical Analysis, Cengage Learning.
2. Smith, G.D. (3rd Ed., 1986). Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon press.
3. Evans, J., Blackledge, J. & Yardley, P. (2000). Numerical Methods for Partial Differential Equation, Springer.
4. Davies, A.J. (2nd Ed., 2011). Finite Element Method: An Introduction to Partial Differential Equations, OUP Oxford.
5. Desai, Y.M. (2011). Finite Element Method with Applications in Engineering, Pearson Education India.
6. Ŝolín, P. (2013). Partial Differential Equations and the Finite Element Method, Wiley.

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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)

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Course Code     : AMAT 41363

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

                              Equations

Pre-requisites   : AMAT 22292

Learning Outcomes:

At the end of this course, the student should be able to 

1. obtain the numerical solutions of differential equations

2. examine the existence, uniqueness and stability of the solutions

3. analyze the asymptotic behavior of the solutions

4. implement the numerical methods of ordinary differential equations using appropriate software

5. use appropriate techniques for solving real-world problems.

 Course Contents:     

Linear Differential Equations: First-Order Linear Differential Equations, Higher-Order Linear Differential Equations, Routh-Hurwitz Criteria, Converting Higher-Order Equations to First-Order Systems, First-Order Linear Systems: Constant Coefficients, Diagonalizations, Methods for Computing the Matrix Exponential, The Fundamental Theorem for Linear Systems, Phase-Plane Analysis, Gershgorin's Theorem, An Example: Pharmacokinetics Model.

Qualitative Theory of Nonlinear Ordinary Differential Equations: Introduction to Nonlinear Ordinary Differential Equations, The Fundamental Existence-Uniqueness Theorem, The Maximal Interval of Existence, Linearization, Stability and Liapunov Functions, Phase Plane Analysis, Stable and Unstable Manifolds, Bifurcations, Periodic Solutions, Poincaré-Bendixson Theorem, Dulac’s Criteria

Numerical Solutions of Ordinary Differential Equations: Review of Numerical Methods, Stability and Convergence Properties of Numerical Schemes, Absolute Stability and Stiff Equations, Implementation of Analytical and Numerical Solutions Using 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. Chapra, S.C. (4th Ed., 2017). Applied Numerical Methods with MATLAB for Engineers and Scientists, McGraw Hill.
2. Verhulst, F. (2012). Nonlinear Differential Equations and Dynamical Systems, Springer.
3.  Allen, L.J.S. (2007). An Introduction to Mathematical Biology, Pearson.
4. Perko, L. (2001). Differential Equations and Dynamical Systems, Springer.
5. Van Loan, C.F. (2000). Introduction to Scientific Computing: A Matrix-Vector Approach Using MATLAB, Prentice Hall.