A universal differential equation (UDE) is a non-trivial

differential algebraic equation In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. The set of the solutions of such a system is a ...

with the property that its solutions can

approximate An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...

any

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

on any interval of the real line to any desired level of accuracy. Precisely, a (possibly implicit) differential equation

P(y', y'', y, ..., y^) = 0

is a UDE if for any continuous real-valued function

f

and for any positive continuous function

\varepsilon

there exist a smooth solution

y

P(y', y'', y, ..., y^) = 0

with

, y(x) - f(x),  < \varepsilon (x)

for all

x \in \R

. The existence of an UDE has been initially regarded as an analogue of the

universal Turing machine In computer science, a universal Turing machine (UTM) is a Turing machine capable of computing any computable sequence, as described by Alan Turing in his seminal paper "On Computable Numbers, with an Application to the Entscheidungsproblem". Co ...

for analog computers, because of a result of Shannon that identifies the outputs of the

general purpose analog computer The general purpose analog computer (GPAC) is a mathematical model of analog computers first introduced in 1941 by Claude Shannon. This model consists of circuits where several basic units are interconnected in order to compute some function. The G ...

with the solutions of algebraic differential equations. However, in contrast to universal Turing machines, UDEs do not dictate the evolution of a system, but rather sets out certain conditions that any evolution must fulfill.

Examples

* Rubel found the first known UDE in 1981. It is given by the following implicit differential equation of fourth-order:

3 y^ y^ y^-4 y^ y^ y^+6 y^ y^ y^ y^+24 y^ y^ y^-12 y^ y^ y^-29 y^ y^ y^+12 y^=0

* Duffin obtained a family of UDEs given by: :

n^2 y^ y^+3 n(1-n) y^ y^ y^+\left(2 n^2-3 n+1\right) y^=0

and

n y^ y^+(2-3 n) y^ y^ y^+2(n-1) y^=0

, whose solutions are of class ''

C^n

'' for ''n'' > 3. * Briggs proposed another family of UDEs whose construction is based on

Jacobi elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are define ...

: :

y^ y^-3 y^ y^ y^+2\left(1-n^\right) y^=0

, where ''n'' > 3. * Bournez and Pouly proved the existence of a fixed polynomial vector field ''p'' such that for any ''f'' and ''ε'' there exists some initial condition of the differential equation y' = p(y) that yields a unique and analytic solution satisfying , ''y''(''x'') − ''f''(''x''), < ''ε''(''x'') for all ''x'' in R.

References

External links

Wolfram Mathworld page on UDEs
Differential equations Approximation theory {{mathanalysis-stub

Examples

See also

References

External links