What is Nonlinear Math?

Date: 10/02/97 at 17:25:41
From: Andy Jones
Subject: Nonlinear math

I have been wondering for a while what exactly nonlinear math is and 
what it is used for.  I have heard of things such as the fractal curve 
and I know it is mostly theories. I was just wondering if you could 
explain some of the facts about it to me.  If you can help, I'll 
appreciate it. Thank you.

Date: 10/02/97 at 18:58:51
From: Doctor Anthony
Subject: Re: Nonlinear math

The word 'LINEAR' appears in many guises in mathematics.  A linear 
equation like ax + by + c = 0 represents a straight line, but you also 
get linear differential equations like 

  a(d^2y/(dx)^2 + b(dy/dx) + cy = f(x) 

and linear combinations of vectors. A non-linear differential equation 
would have the constants a, b, c replaced by functions of x or y.  

We also talk of linear algebra, which deals with the mathematics of 
linear equations, matrices, determinants, and vector spaces. I suppose 
the single defining property of linearity is that if you superimpose 
two solutions you get another solution.

Non-linear equations can be extremely difficult, and are often 
insoluble. For example the long-term behaviour of three bodies moving 
under gravity cannot be solved mathematically, though we can work a 
long way into the future. Unfortunately, non-linear, partial 
differential equations are exactly the equations that describe most 
real-life situations, like weather systems, frictional or turbulent 
motion, and so on. Before computers came along these equations were 
handled by making sweeping simplifications - the trick was to know 
which simplifications were justified. Although we still can't SOLVE 
the equations we can run computer models which give a very good idea 
of what will actually happen.

The fact that we now tackle such problems is one reason why chaos 
theory and complexity have become such popular fields of study.  In 
the past the non-linear world was largely ignored - it was just too 
difficult.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/