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Course Code    : AMAT 22292

Title                   : Scientific Computing using appropriate software II

Pre-requisites   : AMAT 21262

Co-requisites   : AMAT 22282

 Learning Outcomes:

On completion of this unit, the student should be able to

• develop computer programs for curve fitting applications
• use software environment for numerical integration and optimization
• define function files for solving linear systems of equations
• solve initial value problems numerically.

 Course Contents:

Curve Fitting: Linear Least-Squares Regression, Linearization of nonlinear relationships: exponential, power and Saturation Growth Rate models, interpolation, extrapolation, interpolation hazards: multiple curve fitting.

Numerical Integration: Trapezoidal, Simpson’s methods, Gauss quadrature.

Optimization: One-Dimensional optimization and multidimensional optimization.

Solving Linear Systems: Solving linear algebraic equations using the software, Direct methods: naive Gauss elimination, pivoting, Gauss Elimination as LU Factorization, Iterative methods: Jacobi method, Gauss-Seidel method, Richardson method, and SOR methods.

Initial-Value Problems: Euler’s method, Heun’s method, Midpoint method, Runge Kutta Methods, adaptive methods for solving initial value problems, comparison of described methods.

Appropriate built-in functions in the software environment for selected numerical methods.

 Method of Teaching and Learning: A combination of lectures and computer laboratory sessions.

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

 Recommended Reading                

1. C. Chapra, Applied Numerical Methods with MATLAB for Engineers and Scientists, McGraw-Hill, (2017).
2. Jaan Kiusalaas, Numerical Methods in Engineering with MATLAB (3e), Cambridge University Press, 2015
3. R. Otto and J.P. Denier, An Introduction to Programming and Numerical Methods in MATLAB, Springer-Verlag London Limited 2005

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Course Code    : AMAT 21272

Title                   :  Mechanics II

Pre-requisites   : AMAT 11232

 Learning Outcomes : 

Upon successful completion of this course, the student should be able to

1. describe and derive the moment of inertia of rigid bodies
2. collect and organize the knowledge for solving problems in motion of lamina
3. describe, derive and apply Euler’s equation of motion
4. collect and organize a sound knowledge of Lagrangian approach to mechanics
5. determine the Lagrangian functions for a physical systems
6. describe and derive the Lagrange equation of motion for impulsive motion.

 Course Content  :

Rigid Body Motion: Rigid bodies, Moments and products of inertia, Principal axes, Equimomental systems,
Motion of a lamina, Instantaneous centre, Body and space centrodes, Uniplanar motion of a rigid body, Impulsive
motion, Euler’s equations of Motion.

Lagrangian Mechanics: Generalized coordinates, Lagrange’s equations of motion for elementary systems,
Constraint forces, Lagrange’s equation of motion for holonomic systems, Determination of holonomic constraint
forces, generalized force functions, Lagrange equations, Constants of motion in the Lagrangian formalism,
Lagrange equation of motion for impulsive motion.

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

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

 Recommended Reading:

1. Chorlton, F. (2nd Ed., 2019). Textbook of Dynamics, D. Van Nostrand.
2. Desloge, E.A. (1982). Classical Mechanics, John Wiley, New York.
3. Gignoux C & Silvestre-Brac, B. (2014). Solved Problems in Lagrangian and Hamiltonian Mechanics,
Springer Netherlands.
4. Goldstein, H. (2011). Classical Mechanics, Addison Wesley.
5. Kelley, J.D. & Leventhal, J.J. (2016). Problems in Classical and Quantum Mechanics: Extracting the
Underlying Concepts. Springer.
6. Ramsey, A.S. (1975). Dynamics, Parts I & II, Cambridge University Press

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Course Code    : AMAT 22282

Title                 : Numerical Methods II

Pre-requisites   : AMAT 12253

Learning Outcomes:

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

1. define vector norm, matrix norm, and their general properties
2. use numerical methods for differentiation and integration
3. find numerical solutions to a system of equations using iterative methods
4. calculate approximate solutions to ordinary differential equations using numerical methods
5. discuss the convergence and stability of the solution.

 Course Contents:

Numerical Linear Algebra: Vector Norms, Matrix norms, General properties of vector and matrix norms, norm
convergence.   

 Numerical Differentiation and Integration: Numerical differentiation, open and closed Newton-Cotes formulae,
Trapezoidal, Simpson’s 1/3 and 3/8 rules, Romberg integration method, Gaussian quadrature.

Solving Linear Systems of Equations (Iterative): Relative error bound, condition number, iterative and relaxation
methods: Jacobi, Gauss-Seidel methods and their convergence, Richardson, SOR Iterative methods. Conjugate
Gradient Method.

 Numerical Solutions of Ordinary Differential Equations: Explicit and Implicit numerical schemes, Taylor-
Series method, Picard’s method of successive approximations, Euler’s method, Heun’s method, Midpoint method,
Runge-Kutta methods, computation of error bound, stability of methods, predictor-corrector methods: Adams -
Moulton, Adams-Bashforth, Milne’s methods, global and local truncations errors.

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

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

 Recommended Reading : 

1. Burden, R.L., Faires, J.D, Burden, M.L. (10th Ed., 2016). Numerical Analysis, Cengage Learning.
2. Trefethen, L.N. & Bau, D. (1997). Numerical Linear Algebra, Philadelphia, USA.
3. Golub, H., Vanloan, C.F. (2013). Matrix computations, JHU Press.
4. Kreyszig, E. (10th Ed., 2010). Advanced Engineering Mathematics, John Wiley.
5. Sauer, T. (2012). Numerical Analysis, Pearson.
6. Epperson, J.F. (2013). An Introduction to Numerical Methods and Analysis, Wiley.
7. Faul, A.C. (2016). A Concise Introduction to Numerical Analysis, Chapman and Hall/CRC.

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Course Code    : AMAT 21262

Title                   : Scientific Computing using appropriate software I

Pre-requisites   : AMAT 12253

 Learning Outcomes:

On completion of this unit, the student should be able to

1. use the software for interactive mathematical computations
2. build 2D and 3D plots for a given application
3. create script and function files using the software development environment
4. use basic flow structures
5. select appropriate numerical methods in the software environment for root finding applications.

 Course Contents:

Introduction to software: software as a calculator, basic mathematical functions, variables, vectors, accessing
elements of arrays and array manipulation, built-in functions, scripts, plotting: 2D plot, sub-plot, 3D plot: mesh
generation, surface plot, and contour plot.

Programming: data types, Boolean evaluation, decision structures: if, if-else, if-elseif and switch structures, good
programming practices, function files, Loop: for and while, specific commands: break, continue and pause, error
checking and displaying, calling function files: function handles, anonymous functions, passing parameters.

Data import and export: input methods: handle user response and data import from files, output methods:
displaying output, writing, and amending output to a file.

Implementation of Numerical Methods: Roots of nonlinear equations using Graphical method, Bisection method,
False Position method, Newton Raphson method, Secant method and modified Secant method, built-in functions
for root finding problems, finding maxima and minima of functions using root-finding methods

 Method of Teaching and Learning: A combination of lectures and computer laboratory sessions.

 Assessment: Based on tutorials, tests 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. Kiusalaas, J. (3rd Ed., 2015). Numerical Methods in Engineering with MATLAB, Cambridge University
Press.
3. Otto, S.R. & Denier, J.P. (2005). An Introduction to Programming and Numerical Methods in MATLAB,
Springer-Verlag London Limited