Saturday, 16 August 2008

Definition of Linear | Non-Linear Terms

In mathematics, we may sometime encounter the terms "Linear" and "Non-linear" used in equation or expression.

What are they referring to?

And how do they look like?

Linear equations are equations that produce predicatable output given a certain input.

Mathematical examples are:

  • y = 2x

  • y = 3x - 4

  • y = 4(x - 3)

Real life examples are:

  • Audio amplifier ==> The louder the input, the louder the output.

  • Force==> The faster the acceleration of an object, the stronger the force it has

Graphically, the linear equation is a straight line with predictable amount of output change for a given change at the input (See diagram 1).

Diagram 1

What is Non-linear expression?

It is an expression whereby the output is unpredictable given certain known input. The rate of change for the output to the expression varies with the input.

Mathematical examples are:

  • y = x2

  • y = ex ( an exponential equation)

  • y = 2x3 - 1

Real life examples:

  • Characteristics of electronic component Diode ==> exponential change of current flow versus voltage drop across it.

  • Battery discharging rate ==> exponential in nature

  • Capacitor charging ==> exponential nature

  • Water ripple ==> Trigonometric change (sine wave)

  • Weather change ==> affected by many factors (unpredictable)

A point to note is that our world consists of mostly non-linear events.

There are many graphs that can demonstrate the nature of non-linearity. See below.

The response of the output (y) is such that the rate of change of output varies with change in input (x). It is not a straight line type of graph!

More examples:

From the above many examples, we can see that non-linear equations basically produce curve in their response and are not predictable since their rate of change varies.

Mathematics come in useful in this aspect since it can model the performance for ease of analysis. Though the result of the modelling may not be accurate to some extent, it does provide users a better picture of the system or component.

It is also with this mathematical equations that computer simulation can be carried out with software written based on these expressions.


1 comment:

What are Linear Equations said...

Very knowledgeable and descriptive blog.The linear equation can be solve by three methods.My favorite is graphical method.But there is some complex situations come when the solutions are imaginary.Can you please help me about it.