On uniform approximations to hypersingular finite-part integrals

Abstract In this paper, new uniform approximation schemes for computation of hypersingular finite-part integrals are studied. The methods are verified to be supremely qualified for oscillatory integrands. In addition, based on the results on Chebyshev approximations, the uniform convergence and computational error bounds are considered. Preliminary numerical results show the stability and efficiency of the schemes.

[2]  Qi‐Kui Du,et al.  Evaluations of certain hypersingular integrals on interval , 2001 .

[3]  Shuhuang Xiang,et al.  On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals , 2015, Adv. Comput. Math..

[4]  Nikolaos I. Ioakimidis,et al.  On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives , 1985 .

[5]  J. Oliver,et al.  The numerical solution of linear recurrence relations , 1968 .

[6]  Daan Huybrechs,et al.  Asymptotic expansions and fast computation of oscillatory Hilbert transforms , 2011, Numerische Mathematik.

[7]  U. Jin Choi,et al.  Improvement of the asymptotic behaviour of the Euler–Maclaurin formula for Cauchy principal value and Hadamard finite‐part integrals , 2004 .

[8]  Elisabetta Santi,et al.  On the evaluation of Hilbert transforms by means of a particular class of Turán quadrature rules , 2005, Numerical Algorithms.

[9]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[10]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[11]  Shuhuang Xiang,et al.  Efficient evaluation of oscillatory Bessel Hilbert transforms , 2014, J. Comput. Appl. Math..

[12]  Shuhuang Xiang,et al.  Uniform approximations to Cauchy principal value integrals of oscillatory functions , 2009, Appl. Math. Comput..

[13]  Giuliana Criscuolo,et al.  A new algorithm for Cauchy principal value and Hadamard finite-part integrals , 1997 .

[14]  Shuhuang Xiang,et al.  Efficient quadrature of highly oscillatory integrals with algebraic singularities , 2013, J. Comput. Appl. Math..

[15]  Pawel Keller,et al.  A practical algorithm for computing Cauchy principal value integrals of oscillatory functions , 2012, Appl. Math. Comput..

[16]  Giovanni Monegato,et al.  Numerical evaluation of hypersingular integrals , 1994 .

[17]  Tatsuo Torii,et al.  Hilbert and Hadamard transforms by generalized Chebyshev expansion , 1994 .

[18]  Chung-Yuen Hui,et al.  EVALUATIONS OF HYPERSINGULAR INTEGRALS USING GAUSSIAN QUADRATURE , 1999 .

[19]  Shuhuang Xiang,et al.  On the calculation of highly oscillatory integrals with an algebraic singularity , 2010, Appl. Math. Comput..

[20]  Peter Linz,et al.  On the approximate computation of certain strongly singular integrals , 1985, Computing.

[21]  L. Ahlfors.  Complex analysis : an introduction to the theory of analytic functions of one complex variable / Lars V. Ahlfors , 1984 .

[22]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[23]  Congpei An,et al.  On evaluation of Bessel transforms with oscillatory and algebraic singular integrands , 2014, J. Comput. Appl. Math..

[24]  T. Kim,et al.  Filon-Clenshaw-Curtis Rules for Highly Oscillatory Integrals with Algebraic Singularities and Stationary Points , 2012, SIAM J. Numer. Anal..

[25]  G. E. Okecha.  HERMITE INTERPOLATION AND A METHOD FOR EVALUATING CAUCHY PRINCIPAL VALUE INTEGRALS OF OSCILLATORY KIND , 2006 .

[26]  Mark Ainsworth,et al.  An additive Schwarz preconditioner for p-version boundary element approximation of the hypersingular operator in three dimensions , 2000, Numerische Mathematik.

[27]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[28]  K. Diethelm.  A method for the practical evaluation of the Hilbert transform on the real line , 1999 .

[29]  Robert Piessens,et al.  On the computation of Fourier transforms of singular functions , 1992 .

[30]  L. Gori,et al.  Quadrature Rules Based on s-Orthogonal Polynomials for Evaluating Integrals with Strong Singularities , 1999 .

[31]  M. R. Capobianco,et al.  On quadrature for Cauchy principal value integrals of oscillatory functions , 2003 .

[32]  M. M. Chawla,et al.  Quadrature formulas for cauchy principal value integrals , 2005, Computing.

[33]  N. G. Zamani,et al.  Adaptive mesh redistribution for the boundary element in elastostatics , 1990 .

[34]  Víctor Domínguez,et al.  Filon-Clenshaw-Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities , 2013, J. Comput. Appl. Math..

[35]  G. E. Okecha.  Quadrature formulae for Cauchy principal value integrals of oscillatory kind , 1987 .

[36]  David Elliott,et al.  Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals , 1997 .

[37]  A. I. Hasçelik.  Suitable Gauss and Filon-type methods for oscillatory integrals with an algebraic singularity , 2009 .

[39]  Xiaojun Chen,et al.  Error bounds for approximation in Chebyshev points , 2010, Numerische Mathematik.

[40]  M. M. Chawla,et al.  Convergence of quadratures for Cauchy principal value integrals , 2005, Computing.

[41]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

[42]  Weiwei Sun,et al.  A Fast Algorithm for the Electromagnetic Scattering from a Large Cavity , 2005, SIAM J. Sci. Comput..

[43]  Ivan G. Graham,et al.  Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals , 2011 .

[44]  Germund Dahlquist,et al.  Numerical methods in scientific computing , 2008 .

[45]  Shuhuang Xiang,et al.  An improved algorithm for the evaluation of Cauchy principal value integrals of oscillatory functions and its application , 2015, J. Comput. Appl. Math..

[46]  R. Piessens,et al.  An extension of Clenshaw-Curtis quadrature , 1975 .

[47]  Tatsuo Torii,et al.  An automatic quadrature for Cauchy principal value integrals , 1991 .

[48]  Weiwei Sun,et al.  Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence , 2007 .

[49]  Weiwei Sun,et al.  The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval , 2008, Numerische Mathematik.

[50]  Takemitsu Hasegawa,et al.  Uniform approximations to finite Hilbert transform and its derivative , 2004 .

[51]  Shuhuang Xiang,et al.  On the evaluation of Cauchy principal value integrals of oscillatory functions , 2010, J. Comput. Appl. Math..

[52]  N. Ioakimidis.  Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity , 1982 .

[53]  C. W. Clenshaw,et al.  A method for numerical integration on an automatic computer , 1960 .

[54]  Giovanni Monegato,et al.  Definitions, properties and applications of finite-part integrals , 2009 .

[55]  David Elliott,et al.  Truncation Errors in Two Chebyshev Series Approximations , 1965 .