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An exponentially convergent algorithm for nonlinear differential equations in Banach spaces

Authors: Ivan P. Gavrilyuk and Volodymyr L. Makarov
Journal: Math. Comp. 76 (2007), 1895-1923
MSC (2000): Primary 65J15, 65M15; Secondary 34G20, 35K90
Published electronically: April 19, 2007
MathSciNet review: 2336273
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Abstract: An exponentially convergent approximation to the solution of a nonlinear first order differential equation with an operator coefficient in Banach space is proposed. The algorithm is based on an equivalent Volterra integral equation including the operator exponential generated by the operator coefficient. The operator exponential is represented by a Dunford-Cauchy integral along a hyperbola enveloping the spectrum of the operator coefficient, and then the integrals involved are approximated using the Chebyshev interpolation and an appropriate Sinc quadrature. Numerical examples are given which confirm theoretical results.

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Additional Information

Ivan P. Gavrilyuk
Affiliation: Staatliche Studienakademie Thueringen-Berufsakademie Eisenach, University of Cooperative Edukation, Am Wartenberg 2, D-99817 Eisenach, Germany

Volodymyr L. Makarov
Affiliation: National Academy of Sciences of Ukraine, Institute of Mathematics, Tereschenkivska 3, 01601 Kiev, Ukraine

Keywords: Nonlinear evolution equation, exponentially convergent algorithms, Sinc-methods
Received by editor(s): March 15, 2005
Received by editor(s) in revised form: June 30, 2006
Published electronically: April 19, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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