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The superconvergence of the composite trapezoidal rule for Hadamard finite part integrals. (English) Zbl 1092.65023

Several papers have dealt with quadrature formulas for evaluating Hadamard finite part integrals (or hypersingular integrals) with \((p+1)\)-order singularity. On the other hand the composite trapezoidal method has been applied for numerical integration and numerical solution of integral equations with smooth or weakly singular kernels.
The paper deals with the superconvergence of the composite trapezoidal rule for a Hadamard finite part integral for \(p=1\) and \(p=2\). The superconvergence estimates at some special points and uniqueness of the superconvergence points are proved. Numerical results are reported for some examples to confirm the theoretical analysis. The method can be extended to integral and integro-differential equations.

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45J05 Integro-ordinary differential equations
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