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Linearity is a term used to describe certain easily resolved matehmatical systems or equations, where the output meets certain criteria, namely additivity, and homogeneity. In such a system, features such as superimposition exist and the equation can often be split into "sum of its parts" which means certain kinds of assumption, approximation and mathematical approaches are possible.
By contrast, a non-linear equation or system is not subject to such limits and usually cannot be simplified this way. It may exhibit behaviour and results which are extremely hard (or impossible) to calculate or predict under current knowledge or technology, chaos effects, strange attractors, and freak effects. Whilst some non-linear systems and equations of general interest have been extensively studied, the vast majority are poorly understood if at all.
Non-Linear systems are probably easiest understood as "everything except the relatively few systems which prove to be linear".
Background
Linear systems
In mathematics, a linear function f(x) is one which satisfies the following two properties:
- Additivity: f(x + y) = f(x) + f(y)
- Homogeneity: f(αx) = αf(x) for all α
Systems that satisfy both additivity and homogeneity are considered to be linear systems. These two rules, taken together, are often referred to as the principle of superposition. Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian. When anequation can be expressed in linear form, it becomes particularly easy to solve because it can be broken down into smaller pieces, then solving each of those pieces separately.
Non-linear systems
Nonlinear equations and functions are of interest to physicists and mathematicians because they are hard to solve and give rise to interesting phenomena such as chaos. To use a modern language, a linear equation can be described by a using a linear operator, L. A linear equation in some unknown u have the form
Lu=0.
Example of linear operators
are matrices or linear combinations of powers of partial derivatives e.g.
L=d_x^2 + d_y, where x and y are real variables.
A map F(u) is a generalization of a linear operator. Equations involving maps includes linear equations, and nonlinear equations as well as nonlinear systems (the last is a misdenomer steaming from matrix equations 'systems', a nonlinear equation can be a scalar valued or matrix valued equation). Examples of a maps are
F(x)=x^2, x a real number
F(u)=-d_x^2 u + g(u), where u is a function u(x) and x is a real number and g is a function.
F(u,v)=(u+v,u^2), where u, v are functions or numbers.
A nonlinear equation is an equation of the form
F(u)=0. For some unknown u.
In order to solve any equation, one needs to decide in what mathematical space we try to find a solution u in. It might be that u is a real number, a vector or perhaps a function with some properties.
The solutions of linear equations can in general be described as a superposition of other solutions of the same equation. This makes linear equations particularly easy to solve and reason about.
Nonlinear equations are more complex, and much harder to understand because of their lack of simple superposed solutions. For nonlinear equations the solutions to the equations do not in general form a vector space and cannot (in general) be superposed (added together) to produce new solutions. This makes solving the equations much harder than in linear systems.
Specific non-linear equations
Some nonlinear equations, like x^2-1=0 and other polynomial equations
are well understood. System of nonlinear polynomial equations are more complex . For Differential equations the picture is similar. First order nonlinear ordinary differential equation like
d_x u = u^2
are easily solved by e.g. separation of variables and are well understood. Higher order differential equations like
d_x^2 u + g(u)=0, where g is any nonlinear function.
are much less understood. For partial differential equations the picture is even poorer. A number of results involving existence of solutions,
stability of a solution and dynamic of solutions have been proved for
partial differential equations.
Tools for solving certain non-linear systems
Today there are several tools for analyzing nonlinear equations,
to mention a few: Implicit function theorem, Contraction mapping principle and the theory of bifurcations.
Examples of nonlinear equations
To do:
See also:
External links
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