# 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
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