A universal differential equation (UDE) is a non-trivial
differential algebraic equation
In electrical engineering, a differential-algebraic system of equations (DAEs) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system.
In mathematics these are examples of ``d ...
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 continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
on any interval of the real line to any desired level of accuracy.
Precisely, a (possibly implicit) differential equation ''P''(''y''
', ''y''
'', ''y''
, ... , ''y''
(''n'')) = 0 is a UDE if for any continuous real-valued function ''f'' and for any positive continuous function ''ε'' there exist a
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
solution ''y'' of ''P''(''y''
', ''y''
'', ''y''
, ... , ''y''
(''n'')) = 0 with , ''y''(''x'') − ''f''(''x''), < ''ε''(''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 that can simulate an arbitrary Turing machine on arbitrary input. The universal machine essentially achieves this by reading both the description of the machine to be simu ...
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 Computation, compute some Fu ...
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:
* Duffin obtained a family of UDEs given by:
:
and
, whose solutions are of class ''
'' 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 (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tri ...
:
:
, 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.
See also
*
Zeta function universality
In mathematics, the universality of List of zeta functions, zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate arbitrary non-vanishing holomorphic fun ...
*
Hölder's theorem In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Hölder in 1887; several alternative proofs have su ...
References
External links
Wolfram Mathworld page on UDEs
Differential equations
Approximation theory
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