Universal differential equation
A universal differential equation (UDE) is a non-trivial differential algebraic equation with the property that its solutions can approximate any continuous function on any interval of the real line to any desired level of accuracy. Precisely, a (possibly implicit) differential equation [math]\displaystyle{ P(y', y'', y''', ..., y^{(n)}) = 0 }[/math] is a UDE if for any continuous real-valued function [math]\displaystyle{ f }[/math] and for any positive continuous function [math]\displaystyle{ \varepsilon }[/math] there exist a smooth solution [math]\displaystyle{ y }[/math] of [math]\displaystyle{ P(y', y'', y''', ..., y^{(n)}) = 0 }[/math] with [math]\displaystyle{ |y(x) - f(x)| \lt \varepsilon (x) }[/math] for all [math]\displaystyle{ x \in \R }[/math].[1]
The existence of an UDE has been initially regarded as an analogue of the universal Turing machine for analog computers, because of a result of Shannon that identifies the outputs of the general purpose analog computer with the solutions of algebraic differential equations.[1] 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.[2]
Examples
- Rubel found the first known UDE in 1981. It is given by the following implicit differential equation of fourth-order:[1][2] [math]\displaystyle{ 3 y^{\prime 4} y^{\prime \prime} y^{\prime \prime \prime \prime 2}-4 y^{\prime 4} y^{\prime \prime \prime 2} y^{\prime \prime \prime \prime}+6 y^{\prime 3} y^{\prime \prime 2} y^{\prime \prime \prime} y^{\prime \prime \prime \prime}+24 y^{\prime 2} y^{\prime \prime 4} y^{\prime \prime \prime \prime}-12 y^{\prime 3} y^{\prime \prime} y^{\prime \prime \prime 3}-29 y^{\prime 2} y^{\prime \prime 3} y^{\prime \prime \prime 2}+12 y^{\prime \prime 7}=0 }[/math]
- Duffin obtained a family of UDEs given by:[3]
- [math]\displaystyle{ n^2 y^{\prime \prime \prime \prime} y^{\prime 2}+3 n(1-n) y^{\prime \prime \prime} y^{\prime \prime} y^{\prime}+\left(2 n^2-3 n+1\right) y^{\prime \prime 3}=0 }[/math] and [math]\displaystyle{ n y^{\prime \prime \prime \prime} y^{\prime 2}+(2-3 n) y^{\prime \prime \prime} y^{\prime \prime} y^{\prime}+2(n-1) y^{\prime \prime 3}=0 }[/math], whose solutions are of class [math]\displaystyle{ C^n }[/math] for n > 3.
- Briggs proposed another family of UDEs whose construction is based on Jacobi elliptic functions:[4]
- [math]\displaystyle{ y^{\prime \prime \prime \prime} y^{\prime 2}-3 y^{\prime \prime \prime \prime} y^{\prime \prime} y^{\prime}+2\left(1-n^{-2}\right) y^{\prime \prime 3}=0 }[/math], 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.[2]
See also
References
- ↑ 1.0 1.1 1.2 Rubel, Lee A. (1981). "A universal differential equation" (in en). Bulletin of the American Mathematical Society 4 (3): 345–349. doi:10.1090/S0273-0979-1981-14910-7. ISSN 0273-0979. https://www.ams.org/bull/1981-04-03/S0273-0979-1981-14910-7/.
- ↑ 2.0 2.1 2.2 Pouly, Amaury; Bournez, Olivier (2020-02-28). "A Universal Ordinary Differential Equation". Logical Methods in Computer Science 16 (1). doi:10.23638/LMCS-16(1:28)2020. https://lmcs.episciences.org/6168/pdf.
- ↑ Duffin, R. J. (1981). "Rubel's universal differential equation". Proceedings of the National Academy of Sciences 78 (8): 4661–4662. doi:10.1073/pnas.78.8.4661. ISSN 0027-8424. PMID 16593068. Bibcode: 1981PNAS...78.4661D.
- ↑ Briggs, Keith (2002-11-08). "Another universal differential equation". arXiv:math/0211142.
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