Compact numerical quadrature formulas for hypersingular integrals and integral equations. (English) Zbl 1264.65033
Compact numerical quadrature formulas are developed for integrals having singularities at the end-points based on a novel technique recently created by the author using Euler-Maclaurin expansions. It is shown that the accuracy of the proposed quadrature formulas can be increased using the Richardson extrapolation process. The investigation includes the periodic hypersingular case providing a very high accuracy, the spectral accuracy. It is proved that the proposed quadrature formulas are exact for some periodic hypersingular integrals involving trigonometric polynomials. Moreover, it is proved that the accuracy becomes exponential when the integrands are analytic in a strip of the complex plane that includes the real axis. These quadrature formulas are applied to hypersingular integral equations both in periodic and nonperiodic case.
The first part of this long and nice paper ends with a suitable numerical example that illustrates the accuracy of the quadrature formulas and confirm the presented theory. The second part of the paper is devoted to singular integrals having algebraic endpoints singularities extending the approach of the first part based on Euler-Maclaurin expansions. It is proved that the obtained quadratures are exact for some periodic Cauchy principal value integrals involving a family of trigonometric polynomials and the accuracy is proved to be spectral and it is illustrated on a numerical example involving periodic integrands.
The first part of this long and nice paper ends with a suitable numerical example that illustrates the accuracy of the quadrature formulas and confirm the presented theory. The second part of the paper is devoted to singular integrals having algebraic endpoints singularities extending the approach of the first part based on Euler-Maclaurin expansions. It is proved that the obtained quadratures are exact for some periodic Cauchy principal value integrals involving a family of trigonometric polynomials and the accuracy is proved to be spectral and it is illustrated on a numerical example involving periodic integrands.
Reviewer: Alexandru Mihai Bica (Oradea)
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 |
Keywords:
Hadamard finite part integral; hypersingular integrals; hypersingular integral equations; numerical quadrature; trapezoidal rule; midpoint rule; Euler-Maclaurin expansions; asymptotic expansions; Richardson extrapolation; numerical example; Cauchy principal value integralReferences:
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