On the computational complexity of ordinary differential equations
Information and Control
Differential equations and dynamical systems
Differential equations and dynamical systems
Complexity theory of real functions
Complexity theory of real functions
Computability on computable metric spaces
Theoretical Computer Science
Relatively recursive reals and real functions
Theoretical Computer Science - Special issue on real numbers and computers
Computable analysis: an introduction
Computable analysis: an introduction
Computability theory of generalized functions
Journal of the ACM (JACM)
Computability on subsets of metric spaces
Theoretical Computer Science - Topology in computer science
An Algorithm for Computing Fundamental Solutions
SIAM Journal on Computing
Computing the solution of the Korteweg-de Vries equation with arbitrary precision on turing machines
Theoretical Computer Science
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Consider the initial value problem of the first-order ordinary differential equationddtx(t)=f(t,x(t)),x(t"0)=x"0 where the locally Lipschitz continuous function f:R^l^+^1-R^l with open domain and the initial datum (t"0,x"0)@?R^l^+^1 are given. It is shown that the solution operator producing the maximal ''time'' interval of existence and the solution on it is computable. Furthermore, the complexity of the blowup problem is studied for functions f defined on the whole space. For each such function f the set Z of initial conditions (t"0,x"0) for which the positive solution does not blow up in finite time is a G"@d-set. There is even a computable operator determining Z from f. For l=2 this upper G"@d-complexity bound is sharp. For l=1 the blowup problem is simpler.