The trapezoidal rule for computing supersingular integral on interval

Abstract The modified trapezoidal rule for the computation of supersingular integrals in boundary element methods is discussed. For the case of the mesh-point coinciding with singular point, a new quadrature rule is presented and the asymptotic expansion of error function is obtained. Some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms.

[1]  Dan Givoli,et al.  Natural Boundary Integral Method and Its Applications , 2002 .

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

[3]  Weiwei Sun,et al.  The Superconvergence of the Composite Trapezoidal Rule for Hadamard Finite Part Integrals , 2005, Numerische Mathematik.

[4]  Jiming Wu,et al.  Generalized Extrapolation for Computation of Hypersingular Integrals in Boundary Element Methods , 2009 .

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

[6]  Dehao Yu,et al.  Numerical solution of hypersingular equation using recursive wavelet on invariant set , 2010, Appl. Math. Comput..

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

[8]  Dehao Yu,et al.  The Superconvergence of Certain Two-Dimensional Cauchy Principal Value Integrals , 2011 .

[9]  A. Frangi,et al.  A direct approach for boundary integral equations with high-order singularities , 2000 .

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

[11]  Xiaoping Zhang,et al.  Superconvergence and ultraconvergence of Newton-Cotes rules for supersingular integrals , 2010, J. Comput. Appl. Math..

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

[13]  Larry C. Andrews,et al.  Special Functions Of Mathematics For Engineers , 2022 .

[14]  Johan H. de Klerk,et al.  Solving strongly singular integral equations by Lp approximation methods , 2002, Appl. Math. Comput..

[15]  Mohamed S. Akel,et al.  Numerical treatment of solving singular integral equations by using Sinc approximations , 2011, Appl. Math. Comput..

[16]  Dehao Yu,et al.  The Superconvergence of Certain Two-Dimensional Hilbert Singular Integrals , 2011 .

[17]  Philsu Kim,et al.  Two trigonometric quadrature formulae for evaluating hypersingular integrals , 2003 .

[18]  Weiwei Sun,et al.  Toeplitz-type approximations to the Hadamard integral operator and their applications to electromagnetic cavity problems , 2008 .

[19]  Dehao Yu,et al.  Superconvergence of the composite Simpson's rule for a certain finite-part integral and its applications , 2009 .

[20]  L. Gray,et al.  Evaluation of supersingular integrals: second‐order boundary derivatives , 2007 .

[21]  Prof. Dr. Mohamed Abdalla Darwish.  Fredholm-Volterra integral equation with singular kernel , 1999 .

[22]  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 .

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