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Ordinary Differentiation Equations

  • Ker-I Ko
Part of the Progress in Theoretical Computer Science book series (PTCS)

Abstract

In this chapter, we investigate the computational complexity of the solutions y of an ordinary differential equation with initial condition
$$\begin{array}{*{20}{c}} {y\prime (x) = f(x,y(x)),} & {y(0) = 0} \\ \end{array}$$
(7.1)
defined by a polynomial-time computable function f on the rectangle [0,1] × [-1,1]. We consider only ordinary differential equations of the first order, and only equations with initial conditions. The complexity of the solutions y of equation (7.1) depends on certain properties of the function f. First, if equation (7.1) does not have a unique solution, then it is possible that all of its solutions y are noncomputable. If, on the other hand, equation (7.1) has a unique solution, then its solution y must be computable but the complexity could be arbitrarily high. This suggests that we consider those equations (7.1) where the function f satisfies the Lipschitz condition. The Lipschitz condition provides immediately an upper bound of polynomial space on the solution y (for example, by the use of the Euler method). The main result of this chapter proves that polynomial space is also a lower bound for the solution y of equation (7.1) if the function f is polynomial-time computable and satisfies a weak form of local Lipschitz condition in the neighborhood of the solution y.

Keywords

Lipschitz Condition Polynomial Space Quantify Boolean Formula Local Lipschitz Condition Global Lipschitz Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • Ker-I Ko
    • 1
  1. 1.Department of Computer ScienceState University of New York at Stony BrookStony BrookUSA

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