Chapter 2: Linear System

A “system” of equations is a set or collection of equations that you deal with all together at once. Linear equations (ones that graph as straight lines) are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables.


Chapter 1. Nonlinear Equation

In physical sciences, a nonlinear system is a system in which the output is not directly proportional to the input. Nonlinear problems are of interest to engineers, physicists and mathematicians and many other scientists because most systems are inherently nonlinear in nature. Nonlinear systems may appear chaotic, unpredictable or counterintuitive, contrasting with the much simpler linear systems.

Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. It does not matter if nonlinear known functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.


Numerical Methods: Introduction

Analytical methods are typically only solvable in cases of “simple” models. Numerical methods, in contrast, provide ways to manipulate complex math problems so that they may be solved by simple processes. These methods allow for imperfect and complex models to be approximated, usually with great accuracy.  Numerical methods can account for more variables and dimensions than would be solvable when using analytical methods.

Note that the numerical solution is an approximation.  In most cases it will be close enough, which is fine for most engineering problems.  Typically mathematicians have more time and funding to find an analytical solution, but the demands of business usually necessitates an approximate solution for engineers working in industry.