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/ |
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