Abstract
The computational complexity of the solution y of the differential equation y ′( x ) = f ( x, y ( x )), with the initial value y (0) = 0, relative to the computational complexity of the function f is investigated. The Lipschitz condition on the function f is shown to play an important role in this problem. On the one hand, examples are given in which f is polynomial time computable but none of the solutions y is computable. On the other hand, if f is polynomial time computable and if f satisfies a weak form of the Lipschitz condition then the (unique) solution y is polynomial space computable. Furthermore, there exists a polynomial time computable function f which satisfies this weak Lipschitz condition such that the (unique) solution y is not polynomial time computable unless P = PSPACE.
Published Version
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