Course Details

Course Information Package

Course Unit TitleNUMERICAL METHODS
Course Unit CodeAMAT314
Course Unit DetailsBSc Mechanical Engineering (Required Courses) - BSc Civil Engineering (Required Courses) -
Number of ECTS credits allocated5
Learning Outcomes of the course unitBy the end of the course, the students should be able to:
  1. Explain the various methods for finding approximation of roots of nonlinear equations, employ these methods to solve applied engineering problems, and identify the advantages and disadvantages of each method through the solutions.
  2. Define the concept of interpolation and least squares for curve fitting, employ the two methods to obtain the interpolation polynomials for given data sets and various functions, and generate a set of criteria that allow the use of each method.
  3. Describe the concept of numerical integration, apply different techniques for the calculation of integral approximations, and identify when the relative errors become minimal.
  4. Explain the need for approximation of derivatives of any order, define the important approximation formulas and employ various methods to calculate approximate solutions of first and second order differential equations.
  5. Analyse approximate solutions and based on the analysis classify the different methods based on their order of approximation.
  6. Explain the concept of finite difference methods in two dimensions and relate to simple problems that arise in Engineering.
  7. Employ a computer programming language (Matlab) to solve applied engineering problems discussed throughout the course, and compare the approximate solutions with the ones obtained by hand.
Mode of DeliveryFace-to-face
PrerequisitesAMAT223,AMAT204Co-requisitesNONE
Recommended optional program componentsNONE
Course Contents

Introduction: Use of mathematical modelling in engineering problem solving; Overview of modern engineering tools used in engineering practice (such as MATLAB); Approximations of errors.

 

Roots of Equations: Bracketing Methods(Graphical, Bisection and False Position Methods), Open Methods(Fixed-Point Iteration, Newton-Rapson and Secant Methods, Multiple Roots and Systems of Nonlinear Equations), Roots of Polynomials(Conventional, Muller’s, and Bairstow’ Methods).

 

Curve Fitting: Interpolation Methods, Least-Squares Regression. 

 

Numerical Integration: Newton-Cotes Integration Formulas (Trapezoidal Rule, Simpson’s Rules, Integration with unequally spaced data, Open Integration Formulas), Integration of Equations (Newton-Cotes Algorithms for Equations, Romberg Integration, Gauss Quadrature).

 

Numerical Differentiation: High-Accuracy Differentiation Formulas, Richardson Extrapolation, Derivatives of Unequally Spaced Data. 

 

Numerical Solution of Ordinary Differential Equations: Initial value problems, single and multiple step problems, convergence and stability. Boundary value problems, finite difference methods using simple routines. The Euler Method, the Runge-Kutta Methods, and Multi-step Methods.

 

Numerical solution of field problems: Finite difference methods, applications using simple routines.

 

Applied Engineering Problems using MATLAB  

Recommended and/or required reading:
Textbooks
  • Steven C. Chapra, Raymond Canale, Numerical Methods for Engineers: With Software and Programming Applications, McGraw-Hill, 4th Edition, 2001.
References
  • Cleve Moler, Numerical Computing with MATLAB, Society for Industrial and Applied Mathematics, 2008.
  • Singiresu S. Rao, Applied Numerical Methods for Engineers and Scientists, Prentice Hall, 2002.
  • Laurene V. Fausett, Applied Numerical Analysis Using MATLAB, Prentice Hall 1999.
  • George Lindfield and John Penny, Numerical Methods Using MATLAB, Prentice Hall, 1999.
Planned learning activities and teaching methodsThe method of interactive teaching is used for the delivery of the teaching material. A variety of strategies are employed to ensure that all students have equal opportunities to learn. The lesson plans are carried out in several ways such as by questioning, explaining, collaborating and demonstrating. Students can take in class lecture notes and are given additional practice problems either in class or through the website.
Assessment methods and criteria
Tests40%
Final Exam60%
Language of instructionEnglish
Work placement(s)NO

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