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Superconvergence of the composite Simpson’s rule for a certain finite-part integral and its applications. (English) Zbl 1158.65022

Let \[ I_p(a,b;s,f):=\;\;{=\!\!\!\!\!\!\!\int}_a^b \frac{f(t)}{(t-s)^{p+1}}, \] \(s\in (a,b)\), \(p=1,2\), where \(=\!\!\!\!\!\!\!\int\) denotes a Hadamard finite-part integral with \(p+1\) order singularity and \(s\) is the singular point.
The paper is devoted to estimate \(I_2(a,b;s,f)\), by applying the composite Simpson’s rule. The accuracy of this rule, which in general is \(0(h)\), can reach the order \(0(h^2)\) when \(s\) coincides with some special points. This phenomenon is called superconvergence.
The main result consists in obtaining an estimate for the Simpson’s quadrature error, whose order \(0(h^{1+\alpha})\), \(0<\alpha\leq 1\), is found according to the smoothness of \(f(t)\). This result is proved for \(s=t_m+h/2\pm h/3\), where \(t_m\) is a point belonging to a uniform mesh.
Two algorithms with a second order accuracy are described for the general case. The superconvergence result is also used to solve a finite-part integral equation. Numerical examples are shown to illustrate the applications.

MSC:

65D32 Numerical quadrature and cubature formulas
65R20 Numerical methods for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
41A55 Approximate quadratures
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