Computing the solution of the Korteweg-de Vries equation with arbitrary precision on Turing machines
type: Item
instance of: scholarly article
title: Computing the solution of the Korteweg-de Vries equation with arbitrary precision on Turing machines
zbMATH Open document ID: 1079.03034
author: Ning Zhong
author: Klaus Weihrauch
published in: Theoretical Computer Science
publication date: 2005-04-06T00:00:00Z
review text: The Korteweg-de Vries (KdV) equation \(u_t + u u_t + u_{ttt} =0\) (\(t, x \in \mathbb{R}\)) with the initial condition \(u(x, 0)= \varphi(x)\) defines an operator which maps a function \(\varphi\) to the solution of the KdV equation. In Math. Log. Q. 47, No. 1, 93--110 (2001; Zbl 0970.03047), \textit{W. Gay, B.-Y. Zhang} and the second author show that this operator preserves the computability, i.e., it maps a computable initial function to the computable solution. This paper extends this result to a more uniform version, namely, this operator is actually a computable function from the Sobolev space \(H^s(\mathbb{R})\) with \(s\geq 3\) to the space \(C(\mathbb{R}, H^s(\mathbb{R}))\) of continuous function. This answers an open question of \textit{M. B. Pour-El} and \textit{J. I. Richards} [Computability in analysis and physics. Berlin etc.: Springer (1989; Zbl 0678.03027)].
Mathematics Subject Classification ID: 03D80
Mathematics Subject Classification ID: 03F60
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 2153191
zbMATH Keywords: KdV equation
zbMATH Keywords: computable operator
zbMATH Keywords: Turing machine
zbMATH Keywords: type 2 theory of effectivity
zbMATH Keywords: initial value problem
zbMATH Keywords: Sobolev spaces
reviewed by: Xizhong Zheng
MaRDI profile type: Publication
full work available at URL: j.tcs.2004.11.005
OpenAlex ID: W2059334683
cites work: Q5182135
cites work: Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces
cites work: The initial-value problem for the Korteweg-de Vries equation
cites work: Order-free Recursion on the Real Numbers
cites work: Computability of Solutions of the Korteweg-de Vries Equation
cites work: Weak Nonlinear Dispersive Waves: A Discussion Centered Around the Korteweg–De Vries Equation
cites work: On the Korteweg-de Vries equation
cites work: Well-Posedness of the Initial Value Problem for the Korteweg-de Vries Equation
cites work: The Korteweg–deVries Equation: A Survey of Results
cites work: Q3994907
cites work: Q4191015
cites work: Remarks on the Korteweg-de Vries equation
cites work: On the Korteweg - de Vries equation: Existence and uniqueness
cites work: A simple and powerful approach for studying constructivity, computability, and complexity
cites work: Q4485693
cites work: IS WAVE PROPAGATION COMPUTABLE OR CAN WAVE COMPUTERS BEAT THE TURING MACHINE?
cites work: Computability theory of generalized functions
DOI: 10.1016/J.TCS.2004.11.005
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description: scientific article; zbMATH DE number 2153191