An exponentially convergent algorithm for nonlinear differential equations in Banach spaces

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 coeffic...

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Bibliographic Details
Published in:Mathematics of computation 2007-10, Vol.76 (260), p.1895-1923
Main Authors: Gavrilyuk, Ivan P., Makarov, Volodymyr L.
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
Banach space
Coefficients
Differential equations
Exact sciences and technology
Functional analysis
Hyperbolas
Interpolation
Mathematical analysis
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Ordinary differential equations
Perceptron convergence procedure
Polynomials
Sciences and techniques of general use
Sine function
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Summary: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.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-07-01987-4