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 $P(y',y'',y''',...,y^{(n)})=0$ is a UDE if for any continuous real-valued function $f$ and for any positive continuous function $\varepsilon$ there exist a smooth solution $y$ of $P(y',y'',y''',...,y^{(n)})=0$ with $|y(x)-f(x)|<\varepsilon (x)$ for all $x\in \mathbb {R}$ .[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] $3y^{\prime 4}y^{\prime \prime }y^{\prime \prime \prime \prime 2}-4y^{\prime 4}y^{\prime \prime \prime 2}y^{\prime \prime \prime \prime }+6y^{\prime 3}y^{\prime \prime 2}y^{\prime \prime \prime }y^{\prime \prime \prime \prime }+24y^{\prime 2}y^{\prime \prime 4}y^{\prime \prime \prime \prime }-12y^{\prime 3}y^{\prime \prime }y^{\prime \prime \prime 3}-29y^{\prime 2}y^{\prime \prime 3}y^{\prime \prime \prime 2}+12y^{\prime \prime 7}=0$
Duffin obtained a family of UDEs given by:[3]

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

and

ny^{\prime \prime \prime \prime }y^{\prime 2}+(2-3n)y^{\prime \prime \prime }y^{\prime \prime }y^{\prime }+2(n-1)y^{\prime \prime 3}=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:[4]

y^{\prime \prime \prime \prime }y^{\prime 2}-3y^{\prime \prime \prime \prime }y^{\prime \prime }y^{\prime }+2\left(1-n^{-2}\right)y^{\prime \prime 3}=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.[2]

References

Rubel, Lee A. (1981). "A universal differential equation". Bulletin of the American Mathematical Society. 4 (3): 345–349. doi:10.1090/S0273-0979-1981-14910-7. ISSN 0273-0979.
Pouly, Amaury; Bournez, Olivier (2020-02-28). "A Universal Ordinary Differential Equation". Logical Methods in Computer Science. 16 (1). arXiv:1702.08328. doi:10.23638/LMCS-16(1:28)2020.
Duffin, R. J. (1981). "Rubel's universal differential equation". Proceedings of the National Academy of Sciences. 78 (8): 4661–4662. Bibcode:1981PNAS...78.4661D. doi:10.1073/pnas.78.8.4661. ISSN 0027-8424. PMC 320216. PMID 16593068.
Briggs, Keith (2002-11-08). "Another universal differential equation". arXiv:math/0211142.

External links

Wolfram Mathworld page on UDEs

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[:0-1] Rubel, Lee A. (1981). "A universal differential equation". Bulletin of the American Mathematical Society. 4 (3): 345–349. doi:10.1090/S0273-0979-1981-14910-7. ISSN 0273-0979.

[:1-2] Pouly, Amaury; Bournez, Olivier (2020-02-28). "A Universal Ordinary Differential Equation". Logical Methods in Computer Science. 16 (1). arXiv:1702.08328. doi:10.23638/LMCS-16(1:28)2020.

[3] Duffin, R. J. (1981). "Rubel's universal differential equation". Proceedings of the National Academy of Sciences. 78 (8): 4661–4662. Bibcode:1981PNAS...78.4661D. doi:10.1073/pnas.78.8.4661. ISSN 0027-8424. PMC 320216. PMID 16593068.

[4] Briggs, Keith (2002-11-08). "Another universal differential equation". arXiv:math/0211142.

Universal differential equation

Examples

See also

References

External links