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Error bounds for low-rank approximations of the first exponential integral kernel. (English) Zbl 1270.33013

The authors studies two methods to approximate the integral operator defined by the formula \[ (Tx)(\tau)= \int^{t^{*}}_{0} {\frac{\omega}{2}E_{1}(t-\sigma)x(\sigma)}\,d{\sigma}, \qquad \tau\in[0,\tau^{*}], \] where \[ E_{\nu}(\tau)= \int^{\infty}_{1}{\frac{\exp(-\tau \mu)}{\mu^{\nu}}\,d\mu} \] and \( \omega\in {[0,1]}, \tau >{0}, t^{*}< \infty, \nu \geq {0}. \) The first method uses the Taylor approximation for the kernel and the second method uses the polynomial interpolation of the kernel under the system of Chebyshev nodes. For both cases, the error bounds of each approximation are obtained. The approximate operators can be used to compute the approximate eigenvalues of the integral operator.

MSC:

33F05 Numerical approximation and evaluation of special functions
45C05 Eigenvalue problems for integral equations
45E99 Singular integral equations
68P05 Data structures
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65R20 Numerical methods for integral equations
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