Module 2: System of Equations

Overview

Many complex engineering systems cannot be described by only one equation, and are rather expressed as a series of inter-related equations. These equations represent a system of equations and can generally be seen through the matrix notation:

$$[A][X]=[B]$$

or

$$\left[ \begin{matrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\\ \end{matrix}  \right]  \left[ \begin{matrix} x_1 \\x_2\\x_3 \end{matrix} \right]=  \left[ \begin{matrix} b_1 \\b_2\\b_3 \end{matrix} \right]$$

Where aij are coefficients of the variables xi with constants bi. In high school, system of equations can be solved using substitution or elimination methods. Substitution can be easily done for a small number of equations, but can become quite difficult for systems that have hundreds of equations, which can happen with simulations in computational fluid dynamics (CFD). This module will focus on elimination methods as well as an iterative method for solving SOE. Most methods discussed in this module will focus on systems of linear equations. A small section at the end of the module will discuss how to approach a system of non-linear equations.

Learning Objectives

At the end of this module, you will be able to:

  • SOE|LO|01: Identify a system of equation problem.
  • SOE|LO|02: Solve an SOE using Gauss Elimination.
  • SOE|LO|03: Identify when decomposition methods should be used. 
  • SOE|LO|04: Solve a problem with Crout Decomposition.
  • SOE|LO|05: Solve a problem with Thomas Algorithm.
  • SOE|LO|06: Solve an SOE with an iterative Method.
  • SOE|LO|07: Identify a system of non-linear equations. 

Knowledge Test

Before you jump into the module, complete this test to learn what concepts you will learn in this module.