**Preprint series:** 05-02, Reports on Numerical Mathematics

I. P. and Makarov Gavrilyuk

**MSC:**- 65J15 Equations with nonlinear operators (do not use 65Hxx)
- 65M15 Error bounds
- 34G20 Nonlinear equations [See also 47Hxx, 47Jxx]
- 35K90 Abstract parabolic evolution equations

**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.

**Keywords:** *nonlinear evolution equation, exponentially convergent algorithms, Sinc-methods*

**Upload:** 2005-03-30

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