# American Institute of Mathematical Sciences

July  2015, 20(5): 1355-1375. doi: 10.3934/dcdsb.2015.20.1355

## Euler-Maclaurin expansions and approximations of hypersingular integrals

 1 College of Mathematics, Sichuan University, Chengdu,Sichuan, 610064, China, China 2 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65401, United States

Received  March 2013 Revised  January 2015 Published  May 2015

This article presents the Euler-Maclaurin expansions of the hypersingular integrals $\int_{a}^{b}\frac{g(x)}{|x-t|^{m+1}}dx$ and $\int_{a}^{b}% \frac{g(x)}{(x-t)^{m+1}}dx$ with arbitrary singular point $t$ and arbitrary non-negative integer $m$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.
Citation: Chaolang Hu, Xiaoming He, Tao Lü. Euler-Maclaurin expansions and approximations of hypersingular integrals. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1355-1375. doi: 10.3934/dcdsb.2015.20.1355
##### References:

show all references

##### References:
 [1] Jingzhi Li, Hongyu Liu, Hongpeng Sun, Jun Zou. Imaging acoustic obstacles by singular and hypersingular point sources. Inverse Problems and Imaging, 2013, 7 (2) : 545-563. doi: 10.3934/ipi.2013.7.545 [2] Zainidin Eshkuvatov. Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations. Numerical Algebra, Control and Optimization, 2018, 8 (3) : 337-350. doi: 10.3934/naco.2018022 [3] Gabriella Pinzari. Euler integral and perihelion librations. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6919-6943. doi: 10.3934/dcds.2020165 [4] Jacek Banasiak, Aleksandra Puchalska. Generalized network transport and Euler-Hille formula. Discrete and Continuous Dynamical Systems - B, 2018, 23 (5) : 1873-1893. doi: 10.3934/dcdsb.2018185 [5] José Natário. An elementary derivation of the Montgomery phase formula for the Euler top. Journal of Geometric Mechanics, 2010, 2 (1) : 113-118. doi: 10.3934/jgm.2010.2.113 [6] Ju Ge, Wancheng Sheng. The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2733-2748. doi: 10.3934/cpaa.2014.13.2733 [7] Patricia J.Y. Wong. Existence of solutions to singular integral equations. Conference Publications, 2009, 2009 (Special) : 818-827. doi: 10.3934/proc.2009.2009.818 [8] Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017 [9] Xing Huang, Yulin Song, Feng-Yu Wang. Bismut formula for intrinsic/Lions derivatives of distribution dependent SDEs with singular coefficients. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4597-4614. doi: 10.3934/dcds.2022065 [10] Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure and Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241 [11] Thomas Hudson. Gamma-expansion for a 1D confined Lennard-Jones model with point defect. Networks and Heterogeneous Media, 2013, 8 (2) : 501-527. doi: 10.3934/nhm.2013.8.501 [12] Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248 [13] Wenji Zhang. Global generalized solvability in the Keller-Segel system with singular sensitivity and arbitrary superlinear degradation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022121 [14] William Ott, Qiudong Wang. Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1035-1054. doi: 10.3934/dcds.2010.26.1035 [15] Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094 [16] P. De Maesschalck, Freddy Dumortier. Detectable canard cycles with singular slow dynamics of any order at the turning point. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 109-140. doi: 10.3934/dcds.2011.29.109 [17] Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1947-1955. doi: 10.3934/dcdss.2020152 [18] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [19] Simona Fornaro, Maria Sosio, Vincenzo Vespri. $L^r_{ loc}-L^\infty_{ loc}$ estimates and expansion of positivity for a class of doubly non linear singular parabolic equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 737-760. doi: 10.3934/dcdss.2014.7.737 [20] A. Pedas, G. Vainikko. Smoothing transformation and piecewise polynomial projection methods for weakly singular Fredholm integral equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 395-413. doi: 10.3934/cpaa.2006.5.395

2021 Impact Factor: 1.497