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 {\displaystyle P(y',y'',y''',...,y^{(n)})=0} is a UDE if for any continuous real-valued function f {\displaystyle f} and for any positive continuous function ε {\displaystyle \varepsilon } there exist a smooth solution y {\displaystyle y} of P ( y , y , y , . . . , y ( n ) ) = 0 {\displaystyle P(y',y'',y''',...,y^{(n)})=0} with | y ( x ) f ( x ) | < ε ( x ) {\displaystyle |y(x)-f(x)|<\varepsilon (x)} for all x R {\displaystyle 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] 3 y 4 y y 2 4 y 4 y 2 y + 6 y 3 y 2 y y + 24 y 2 y 4 y 12 y 3 y y 3 29 y 2 y 3 y 2 + 12 y 7 = 0 {\displaystyle 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 y 2 + 3 n ( 1 n ) y y y + ( 2 n 2 3 n + 1 ) y 3 = 0 {\displaystyle 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 n y y 2 + ( 2 3 n ) y y y + 2 ( n 1 ) y 3 = 0 {\displaystyle 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 {\displaystyle C^{n}} for n > 3.
  • Briggs proposed another family of UDEs whose construction is based on Jacobi elliptic functions:[4]
y y 2 3 y y y + 2 ( 1 n 2 ) y 3 = 0 {\displaystyle 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]

See also

References

  1. ^ a b c 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.
  2. ^ a b c 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. S2CID 4736209.
  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.

External links

  • Wolfram Mathworld page on UDEs


  • v
  • t
  • e