Complexity Theory of Real Functions pp 215-246 | Cite as

# Ordinary Differentiation Equations

Chapter

## Abstract

In this chapter, we investigate the computational complexity of the solutions defined by a polynomial-time computable function

*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)

*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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© Birkhäuser Boston 1991