Abstract
In this paper we give a simple, but high order and rapid convergence method for computing the Cauchy principal value integrals of the form \(\int_{-1}^{1}e^{i\omega x}\frac{f(x)}{x-\tau}dx\) and its error bounds, where f(x) is a given smooth function, ω∈R + may be large and −1<τ<1. The proposed method is constructed by approximating \((\frac{f(x)-f(\tau)}{x-\tau})^{(s)}\) by using the special Hermite interpolation polynomial, which is a Taylor series. The validity of the method has been demonstrated by the results of several numerical experiments and the comparisons with other methods.
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The author is deeply grateful to the associated editor and referees for their valuable comments and suggestions for great improvement of this paper. Remarks are cited from the referee’s comments.
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This work was supported partially Doctor Funds of Guangdong Ocean University (No. E09230).
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Chen, R. Fast Integration for Cauchy Principal Value Integrals of Oscillatory Kind. Acta Appl Math 123, 21–30 (2013). https://doi.org/10.1007/s10440-012-9709-z
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DOI: https://doi.org/10.1007/s10440-012-9709-z