Course Outline

Course Syllabus (Faculty of Engineering Sciences Handbook)

Matrices and Vector operations, linear homogenous systems, Eigen-vectors and values. Numerical errors, absolute and relative errors, stability and convergence of numerical algorithms. Interpolation Methods: Lagrange polynomials, finite differences, least square approximation. Numerical solutions to Nonlinear Equations: Newton Raphson method, secant, false position, bisection, fixed point algorithm. Numerical Integration: Simpson’s rule, trapezoidal rule, Newton-Cotes method. Numerical solutions to Ordinary Differential Equations: Taylor series method, Euler method, Runge-Kutta method. Numerical solutions to Partial Differential Equations: Second order quasi-linear equations, numerical solutions

Course Description

This course is a study of mathematical techniques used to model engineering systems. It involves the development of mathematical models and the application of the computer to solve engineering problems using the following computational techniques: Taylor Series approximation, numerical differentiation, root-finding using bracketing and open methods, linear and polynomial curve fitting, solution methods for matrix equations, numerical integration, and the solution of differential equations. Laboratory sessions involve the application of numerical analysis to physical systems involving statics, dynamics, fluid dynamics, heat transfer, electrical circuits, and vibratory systems.

Reference Books and Recommended Materials

    1. Applied Numerical Methods with MATLAB for Engineers and Scientists, 2nd Edition. Stephen C. Chapra, McGraw Hill, 2010
    2. Cleve Moler, Numerical Computing with MATLAB, Electronic edition: The MathWorks, Inc., Natick, MA, 2004, http://www.mathworks.com/moler. Print edition: SIAM, Philadelphia, 2004.http://ec-securehost.com/SIAM/ot87.html
    3. Possession of a personal laptop and student version of MATLAB is strongly recommended for this course.
    4. Installed proprietary MATLAB software in the lab will be utilized during laboratory sessions

Supplementary Textbooks:

      1. L. V. Fausett, Applied Numerical Analysis Using MATLAB® 2/E, Prentice Hall, ISBN: 0132397285
      2. S. Nakamura, Numerical Analysis and Graphic Visualization with MATLAB® , 2/e, Prentice Hall, 2002, ISBN:01306548921
      3. A. Gilat and V. Subramaniam, Numerical Methods for Engineers and Scientists, John Wiley & Sons, Inc., 2008, ISBN: 9780471734406
      4. J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB®, 3rd ed, Upper Saddle River, NJ: Prentice Hall, 2004, ISBN: 0130652482
      5. J. Kiusalaas, Numerical Methods in Engineering with MATLAB® , Cambridge University Press, 2005, ISBN: 0521852889

Course Goals:

    • Introduce MATLAB as a technical computing environment and mathematical software for engineers and scientists
    • Teach important aspects of mathematical modeling using differential equations and associated numerical methods for solutions.
    • Enhance students' programming skills using the MATLAB environment to implement numerical method algorithms
    • Teach the use of MATLAB as a tool (using built-in functions) for solving mathematical problems that require numerical solutions and introduce the simulation tools of MATLAB

Course Learning Outcomes

Upon successful completion of this course, the student will:

    • Be able to model engineering systems using first and second order differential equations, and solve the equations both analytically and numerically.
    • Be able to employ the Taylor Series for approximation and error analysis.
    • Be able to formulate and apply numerical techniques for root finding, curve fitting, differentiation, and integration.
    • Be able to write computer programs to solve engineering problems with MATLAB and C++ object oriented capabilities depending upon the nature of the problem.
    • Be able to perform both hand computation and programming applied in MATLAB

Course Topics

    1. Background for matrix and vector operations;
    2. Introduction to numerical methods; Systems of linear equations: Unsolvable and ill-conditioned systems, condition number
    3. Solving systems of Linear Equations: Background, Gauss elimination method, Pivoting, Gauss-Jordan method
    4. Solving systems of Linear Equations: LU decomposition method, inverse of a matrix, brief MATLAB review
    5. Solving systems of Linear Equations: Iterative methods, use of MATLAB built-in functions
    6. Matrix eigenvalues and eigenvectors; Power method
    7. Curve Fitting and interpolation; interpolation using a single polynomial, Lagrange and Newton’s polynomials, Piecewise interpolation, linear, quadratic, and cubic splines, use of MATLAB built-in functions for curve fitting and interpolation
    8. Nonlinear equations; background, estimation of error; Solving nonlinear equations; Fixed-point iteration method, Bisection method, Regula Falsi method, Secant method
    9. Multivariate systems of nonlinear equations; Newton’s method, use of MATLAB built-in functions; equations with multiple solutions
    10. Numerical differentiation; Differentiation using Lagrange polynomials, use of MATLAB built-in functions for numerical differentiation
    11. Numerical differentiation; Richardson’s extrapolation, error in numerical differentiation, numerical partial differentiation
    12. Numerical Integration; background, rectangle and midpoint methods, trapezoidal method, Simpson’s methods; use of MATLAB built-in functions for integration, Richardson extrapolation, Romberg integration
    13. ODE initial value problems; Runge-Kutta methods, multistep methods, predictor-corrector methods, system of first-order ODEs, higher-order IVP; local truncation error in 2nd-order Runge-Kutta method, step size for desired accuracy, stability, stiff ODEs
    • Introduction to modeling (2 classes)
    • Error analysis/Taylor Series (2 classes)
    • Root finding (3 classes)
    • Curve fitting (3 classes)
    • Optimization and Matrix applications (3 classes)
    • Numerical differentiation (3 classes)
    • Numerical integration (3 classes)
    • Differential equations (7 classes)
    • Partial differential equations & boundary value problems (2 classes)
    • Testing and review (2 classes)

Prerequisites by topic

    • Programming
    • Differential equations
    • Differential and integral calculus

Laboratory topics

    • Programming/computing techniques
    • Matrix solution methods
    • Solution of simultaneous equations using MATLAB
    • Modeling of first and second order mechanical/electrical/thermal systems
    • Applications of root-finding to vehicle dynamics & thermal insulation
    • Applications of curve-fitting to experimental data
    • Applications of numerical integration to evaluate moments of inertia, friction work and volumetric fluid flow

Course Format

    1. The course consists of classroom instruction including lectures using classical lecture style, power point slides, and simultaneous MATLAB and C++ applications via projection.
    2. Homeworks are given take-home style to increase students' numerical analysis skills using MATLAB and C++.

Exams and Grading

    1. There will be weekly assignments and quizzes during the semester. For full credit homeworks must be submitted on the agreed date for submission. Solutions to home work and the quizzes will be discussed at the following lecture. The overall grade for this course will be determined as follows:
    • Assignments – 10%
    • Laboratory/Course Project – 10%
    • Quizzes – 10%
    • Final Exam including Laboratory Component – 70%

Homework Assignments and Course Projects

    • Homework assignments will be announced in class and posted on the web. All homework is due in class on the assigned date, which will be announced in class and posted to the course website. Homework may be submitted as pdf files by email and printed hardcopies before class. Please do not send obscure formats, zipped files, or extremely long files.
    • Late homework will NOT be accepted.
    • Work submitted should be the student’s own.
    • All necessary steps towards obtaining the solution, as well as any MATLAB/Simulink code, must be included in the write-up for full credit.
    • There will be approximately ten homework assignments during the course of the semester.
    • Students are allowed, even encouraged, to work on the homework in small groups, but each student must hand in an individual set of answers, which must be their own work.
    • Students are referred to the University’s code of student conduct at http://www.ug.edu.gh/ or in the students’ handbook.

Projects:

The projects to be assigned during the course of the semester will cover a wide range of computational problems arising from science and engineering to be solved using MATLAB and its associated toolboxes.

This course provides twelve (12) computational projects aimed at numerically solving problems from a broad range of applications areas including:

      1. Electrical Circuits
      2. Fluid Mechanics
      3. Chemistry
      4. Elasticity
      5. Thermal Science
      6. Computer Aided Design
      7. Signal and Image Processing.