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

DOI:
https://doi.org/10.1090/S0025-5718-07-01987-4

Published electronically:
April 19, 2007

MathSciNet review:
2336273

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Email:
ipg@ba-eisenach.de

**Volodymyr L. Makarov**

Affiliation:
National Academy of Sciences of Ukraine, Institute of Mathematics, Tereschenkivska 3, 01601 Kiev, Ukraine

Email:
makarov@imath.kiev.ua

DOI:
https://doi.org/10.1090/S0025-5718-07-01987-4

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.