## An exponentially convergent algorithm for nonlinear differential equations in Banach spaces

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- by Ivan P. Gavrilyuk and Volodymyr L. Makarov;
- Math. Comp.
**76**(2007), 1895-1923 - DOI: https://doi.org/10.1090/S0025-5718-07-01987-4
- Published electronically: April 19, 2007
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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.## References

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## Bibliographic 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
- Received by editor(s): March 15, 2005
- Received by editor(s) in revised form: June 30, 2006
- Published electronically: April 19, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2336273