Syllabus

This course investigates mathematical solution methods applied to problems encountered in various engineering applications. These applications include electrical circuits, fluid mechanics, heat and mass transfer, chemical kinetics, production systems optimization and stress analysis. The mathematical techniques studied include solving linear systems of simple-algebraic or differential equations using vector-matrix operations, and the application of eigenvectors, eigenvalues and singular values to engineering problems. Familiarity with computer programming is recommended. Students will be familiarized with the use of the Matlab? software and programming language.

Upon completion of this module, the student should be able to:

1. Translate engineering problems into mathematical statements and systems.
2. When necessary, determine appropriate numerical methods for analyzing specific mathematical systems.
3. Write, debug and run suitable computer codes (in MATLAB or another suitable language) to arrive at solutions to mathematical systems.
4. Interpret and present solutions to mathematical systems in graphical, tabular or other format suitable to the engineering field of study.

Required: Advanced Engineering Mathematics - 9th Edition, Erwin Kreyszig, Wiley, 2006

Recommended: Advanced Engineering Mathematics - 4th Edition, Stroud K. A., Booth D. J., Macmillan, 2003

Your final grade in the course is based upon the total number of points you have earned.??Final Grades will be assigned using the following scale:

This course revolves around a series of UNITS.

We will cover a total of 11 units -- each of which is followed by in-class exercises to enhance your level of understanding.

Upon successful completion of the module, the student should be able to:

1. State the typical notations and definitions related to matrices and arrays.
2. State and apply the various properties and types of matrices and their determinants.
3. Apply the various rules and operations of matrix algebra systems.

1. Derive a linear algebraic system from a common engineering scenario.
2. State the criteria for which direct elimination methods are used as opposed to iterative methods.
3. Solve linear systems using various direct methods including Cramer's Rule, The Matrix Inverse Method and Gauss and Gauss-Jordon Elimination.
4. Choose the most appropriate direct method based on the benefits and limitations of each.
5. State the meaning of ill-conditioning, assess the degree of ill-conditioning and apply suitable strategies to address it.
6. Solve linear systems using iterative methods.

1. Derive a homogeneous linear algebraic system (eigenproblem) from a common engineering scenario.
2. State the various mathematical characteristics of eigenproblems.
3. Solve eigenproblems using the iterative or direct methods.
4. Explain the physical meaning of eigensolutions, i.e. eigenvalues and eigenvectors.

1. Give examples of engineering problems that require root-finding techniques to solve.
2. State the general characteristics and rules of root-finding.
3. Characterize various root-finding techniques as either closed-domain or open-domain methods.
4. Solve non-linear problems by applying any of the following methods: Bisection, Regula-Falsi, Newton-Raphson, Secant, Muller.
5. State and explain the most common problems associated with root-finding methods.
6. Apply Newton's method to non-linear systems (multiple equations).

1. State the conditions under which the interpolation technique is used.
2. State what is a polynomial and the characteristics of polynomials that makes them suitable for interpolation and curve-fitting.
3. Demonstrate how Lagrange Polynomials accomplish interpolation for estimating values for multi-point systems.
4. Use Neville's algorithm to accomplish polynomial interpolation.
5. Define a difference and construct a difference table.
6. List the advantages and disadvantages of difference polynomials and apply Newton's forward-difference and backward-difference polynomials.
7. List the advantages and disadvantages of cubic splines and apply cubic splines to a set of tabular data.

1. Describe the general features of numerical differentiation.
2. Explain and apply direct fit polynomials to evaluate a derivative in a set of tabular data.
3. Apply Lagrange polynomials to evaluate a derivative in a set of tabular data.
4. Apply divided difference polynomials to evaluate a derivative in a set of tabular data.
5. Describe the procedure for numerical differentiation using Newton forward- and backward-difference.
6. Apply the procedure for developing difference formulas from Newton difference polynomials or Taylor series.

1. Describe the general features of numerical integration
2. Explain and apply the procedure for numerical integration of a set of tabular data using direct fit polynomials.
3. Describe the procedure for numerical integration using Newton forward-difference polynomials.
4. Derive and apply the trapezoid rule, Simpson?s 1/3 rule and Simpson?s 3/8 rules.
5. Describe the relative advantage and disadvantages of the Simpson?s rules.

1. Describe the general features and classifications of Ordinary Differential Equations.
2. Classify physical problems and identify them as initial-value and boundary-value ODEs.
3. Identify and categorize common engineering scenarios as ODEs problems.
4. Explain the following concepts: 1st vs. higher order ODEs, linear vs. non-linear ODEs, stable vs. unstable ODEs, complimentary vs. particular solutions.
5. Solve initial-value ODEs using the Taylor-Series methods.
6. Explain the difference between explicit and implicit Finite Difference Equations (FDEs).
7. Solve initial-value ODEs using the Euler and Runga-Kutta methods.
8. Utilize the Taylor-Series for solving a nonlinear implicit FDEs.
9. Solve higher-order ODEs by transforming into a system of 1st order ODEs.
10. Explain what is stiffness.
11. Describe the general feature of different types of boundary-value (BV) ODEs including linear, nonlinear, second-order and higher-order BV ODEs.
12. Discuss the number and the types of boundary conditions required to solve a BV ODE: Dirichlet, Neumann, and mixed
13. Describe how higher-order ODEs and systems of second-order ODEs can be solved using the procedures for solving a single 2nd-order ODE
14. Explain and apply the shooting (initial-value) method in solving BV ODEs
15. Apply the equilibrium method in solving linear 2nd-order BV ODEs including homogeneous ODEs (eigenproblems).

1. Describe the general features and classifications of PDEs.
2. Classify the types of physical problems occuring with with PDEs and the relevance of the type of physical problem in the solution approach for the PDE.
3. Identify the following types of PDEs: Elliptic PDEs, Parabolic PDEs, Hyperbolic PDEs, the Convection-Diffusion Equation.
4. Apply the finite difference method to elliptic PDEs, namely the Laplace and Poisson Equations.
5. Apply iterative methods to elliptic PDEs.
6. Apply the control volume method to typical engineering PDE problems.
7. Apply PDEs solution methods to non-rectangular domains.
8. Apply the finite difference method to hyperbolic PDEs.