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Journal articles on the topic 'Quadrature by expansion'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Ahrens, Cory, and Gregory Beylkin. "Rotationally invariant quadratures for the sphere." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2110 (2009): 3103–25. http://dx.doi.org/10.1098/rspa.2009.0104.

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We construct near-optimal quadratures for the sphere that are invariant under the icosahedral rotation group. These quadratures integrate all ( N +1) 2 linearly independent functions in a rotationally invariant subspace of maximal order and degree N . The nodes of these quadratures are nearly uniformly distributed, and the number of nodes is only marginally more than the optimal ( N +1) 2 /3 nodes. Using these quadratures, we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of Lagrange interpolation on the sphere. This representation uses function

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2

Rachh, Manas, Andreas Klöckner, and Michael O'Neil. "Fast algorithms for Quadrature by Expansion I: Globally valid expansions." Journal of Computational Physics 345 (September 2017): 706–31. http://dx.doi.org/10.1016/j.jcp.2017.04.062.

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3

Kyei, Yaw, and Kossi Edoh. "Higher-order accurate finite volume discretization of Helmholtz equations with Pollution Effects Reductions." International Journal for Innovation Education and Research 6, no. 4 (2018): 130–48. http://dx.doi.org/10.31686/ijier.vol6.iss4.1017.

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Higher-order accurate finite volume schemes are developed for Helmholtz equations in two dimensions. Through minimizations of local equation error expansions for the flux integral formulation of the equation, we determine quadrature weights for the discretization of the equation. Collocations of local expansions of the solution and the source terms are utilized to formulate weighted quadratures of all local compact fluxes to describe the equation error expansion within the computational domain. In using the source term distribution to account for fluxes along all compact directions about each

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4

Favati, P., G. Lotti, and F. Romani. "Asymptotic expansion of error in interpolatory quadrature." Computers & Mathematics with Applications 24, no. 10 (1992): 99–104. http://dx.doi.org/10.1016/0898-1221(92)90022-a.

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5

Kyei, Yaw. "Higher-Order Accurate Finite Volume Discretization of the Three-Dimensional Poisson Equation Based on An Equation Error Method." International Journal for Innovation Education and Research 6, no. 6 (2018): 107–23. http://dx.doi.org/10.31686/ijier.vol6.iss6.1076.

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Efficient higher-order accurate finite volume schemes are developed for the threedimensional Poisson’s equation based on optimizations of an equation error expansion on local control volumes. A weighted quadrature of local compact fluxes and the flux integral form of the equation are utilized to formulate the local equation error expansions. Efficient quadrature weights for the schemes are then determined through a minimization of the error expansion for higher-order accurate discretizations of the equation. Consequently, the leading numerical viscosity coefficients are more accurately and com

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6

SOLIMAN, AHMED M. "GENERATION OF THIRD-ORDER QUADRATURE OSCILLATOR CIRCUITS USING NAM EXPANSION." Journal of Circuits, Systems and Computers 22, no. 07 (2013): 1350060. http://dx.doi.org/10.1142/s0218126613500606.

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A systematic synthesis procedure for generating third-order grounded passive element quadrature oscillators is given. The synthesis procedure is based on using nodal admittance matrix (NAM) expansion applied to the Y matrix of a recently reported three Op Amp third-order oscillator circuit. Four new circuits using current conveyors (CCII) are reported. In addition four more new circuits using inverting current conveyors (ICCII) are also given. Many more quadrature third-order oscillator circuits using combinations of CCII and ICCII can be obtained. Simulation results demonstrating the practica

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7

Wala, Matt, and Andreas Klöckner. "Optimization of fast algorithms for global Quadrature by Expansion using target-specific expansions." Journal of Computational Physics 403 (February 2020): 108976. http://dx.doi.org/10.1016/j.jcp.2019.108976.

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8

Iserles, Arieh, and David Levin. "Asymptotic expansion and quadrature of composite highly oscillatory integrals." Mathematics of Computation 80, no. 273 (2010): 279–96. http://dx.doi.org/10.1090/s0025-5718-2010-02386-5.

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9

Ito, Takashi. "High-Order Analytic Expansion of Disturbing Function for Doubly Averaged Circular Restricted Three-Body Problem." Advances in Astronomy 2016 (2016): 1–23. http://dx.doi.org/10.1155/2016/8945090.

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Terms in the analytic expansion of the doubly averaged disturbing function for the circular restricted three-body problem using the Legendre polynomial are explicitly calculated up to the fourteenth order of semimajor axis ratio (α) between perturbed and perturbing bodies in the inner case (α<1), and up to the fifteenth order in the outer case (α>1). The expansion outcome is compared with results from numerical quadrature on an equipotential surface. Comparison with direct numerical integration of equations of motion is also presented. Overall, the high-order analytic expansion of the do

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10

Petras, K. "An asymptotic expansion for the weights of Gaussian quadrature formulae." Acta Mathematica Hungarica 70, no. 1-2 (1996): 89–100. http://dx.doi.org/10.1007/bf00113915.

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11

Wala, Matt, and Andreas Klöckner. "A fast algorithm for Quadrature by Expansion in three dimensions." Journal of Computational Physics 388 (July 2019): 655–89. http://dx.doi.org/10.1016/j.jcp.2019.03.024.

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12

Klinteberg, Ludvig af, and Anna-Karin Tornberg. "Error estimation for quadrature by expansion in layer potential evaluation." Advances in Computational Mathematics 43, no. 1 (2016): 195–234. http://dx.doi.org/10.1007/s10444-016-9484-x.

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13

Wala, Matt, and Andreas Klöckner. "A fast algorithm with error bounds for Quadrature by Expansion." Journal of Computational Physics 374 (December 2018): 135–62. http://dx.doi.org/10.1016/j.jcp.2018.05.006.

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14

Li, Yong-An. "Synthesis Approach for Compact VDTA Quadrature Sine-Wave Oscillators with Orthogonal Control." Journal of Circuits, Systems and Computers 28, no. 12 (2019): 1950205. http://dx.doi.org/10.1142/s0218126619502050.

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From the nodal admittance matrix (NAM) expansion approach and 16 pathological patterns of the voltage differencing transconductance amplifier (VDTA), two different categories, namely sort A and sort B, of the quadrature sine-wave oscillators-employed VDTAs and grounded capacitors in the current mode are synthesized. In all, four quadrature sine-wave oscillators with two sets of quadrature current outputs are attained. Because canonic number is used for component quantity, the oscillators are appropriate to integrated circuit realization and also provide electronic, linear, and orthogonal adjus

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15

af Klinteberg, Ludvig, and Anna-Karin Tornberg. "Adaptive Quadrature by Expansion for Layer Potential Evaluation in Two Dimensions." SIAM Journal on Scientific Computing 40, no. 3 (2018): A1225—A1249. http://dx.doi.org/10.1137/17m1121615.

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16

Rahimian, Abtin, Alex Barnett, and Denis Zorin. "Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion." BIT Numerical Mathematics 58, no. 2 (2017): 423–56. http://dx.doi.org/10.1007/s10543-017-0689-2.

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17

Barani, Saeed, Davood Poorveis, and Shapoor Moradi. "Buckling Analysis of Ring-Stiffened Laminated Composite Cylindrical Shells by Fourier-Expansion Based Differential Quadrature Method." Applied Mechanics and Materials 225 (November 2012): 207–12. http://dx.doi.org/10.4028/www.scientific.net/amm.225.207.

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This article focuses on the application of the Fourier-expansion based differential quadrature method (FDQM) for the buckling analysis of ring-stiffened composite laminated cylindrical shells. Displacements and rotations are expressed in terms of Fourier series expansions in longitudinal direction and their first order derivatives are approximated with FDQM in circumferential direction. The 'smeared stiffener' approach is adopted for the stiffeners modeling. Two FORTRAN programs prepared for linear and nonlinear analysis and results were compared by ABAQUS finite element software. Buckling loa

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18

Meng, Jianping, Xiao-Jun Gu, David R. Emerson, Yong Peng, and Jianmin Zhang. "Discrete Boltzmann model of shallow water equations with polynomial equilibria." International Journal of Modern Physics C 29, no. 09 (2018): 1850080. http://dx.doi.org/10.1142/s0129183118500808.

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A type of discrete Boltzmann model for simulating shallow water flows is derived by using the Hermite expansion approach. Through analytical analysis, we study the impact of truncating distribution function and discretizing particle velocity space. It is found that the convergence behavior of expansion is nontrivial while the conservation laws are naturally satisfied. Moreover, the balance of source terms and flux terms for steady solutions is not sacrificed. Further numerical validations show that the capability of simulating supercritical flows is enhanced by employing higher-order expansion

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19

Pan, Lu, Xiaoming He, and Tao Lü. "High Accuracy Combination Method for Solving the Systems of Nonlinear Volterra Integral and Integro-Differential Equations with Weakly Singular Kernels of the Second Kind." Mathematical Problems in Engineering 2010 (2010): 1–21. http://dx.doi.org/10.1155/2010/901587.

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This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples

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20

Zhang, Li, Jin Huang, Yubin Pan, and Xiaoxia Wen. "A Mechanical Quadrature Method for Solving Delay Volterra Integral Equation with Weakly Singular Kernels." Complexity 2019 (June 16, 2019): 1–12. http://dx.doi.org/10.1155/2019/4813802.

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In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses a high accuracy but also has the posterior error estimate which can be used to design self-adaptive alg

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21

Mehditabar, Aref, Gholam H. Rahimi, and Kerameat Malekzadeh Fard. "Thermoelastic Analysis of Rotating Functionally Graded Truncated Conical Shell by the Methods of Polynomial Based Differential Quadrature and Fourier Expansion-Based Differential Quadrature." Mathematical Problems in Engineering 2018 (2018): 1–19. http://dx.doi.org/10.1155/2018/2804123.

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This paper focuses on the three-dimensional (3D) asymmetric problem of functionally graded (FG) truncated conical shell subjected to thermal field and inertia force due to the rotating part. The FG properties are assumed to be varied along the thickness according to power law distribution, whereas Poisson’s ratio is assumed to be constant. On the basis of 3D Green-Lagrange theory in general curvilinear coordinate, the fundamental equations are formulated and then two versions of differential quadrature method (DQM) including polynomial based differential quadrature (PDQ) and Fourier expansion-

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22

Cheng, Pan, and Ling Zhang. "Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity." Mathematical Problems in Engineering 2018 (May 31, 2018): 1–8. http://dx.doi.org/10.1155/2018/6932164.

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This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equations will be converted into nonlinear boundary integral equations by the potential theory, in which logarithmic singularity and Cauchy singularity are calculated simultaneously. Mechanical quadrature methods (MQMs) are presented to solve the nonlinear equations where the accuracy of the solutions is of three orders. According to the asymptotical compact convergence theory, the errors with odd powers asymptotic expansion are obtained. Following the asymptotic expa

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23

Wei-Qin, Zhao, and Ju Chang-Sheng. "Different Versions of Perturbation Expansion Based on the Single-Trajectory Quadrature Method." Communications in Theoretical Physics 38, no. 3 (2002): 271–80. http://dx.doi.org/10.1088/0253-6102/38/3/271.

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24

SHU, C., and Y. T. CHEW. "Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems." Communications in Numerical Methods in Engineering 13, no. 8 (1997): 643–53. http://dx.doi.org/10.1002/(sici)1099-0887(199708)13:8<643::aid-cnm92>3.0.co;2-f.

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25

Klöckner, Andreas, Alexander Barnett, Leslie Greengard, and Michael OʼNeil. "Quadrature by expansion: A new method for the evaluation of layer potentials." Journal of Computational Physics 252 (November 2013): 332–49. http://dx.doi.org/10.1016/j.jcp.2013.06.027.

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26

Thiagarajan, Ponkrshnan, and K. V. Nagendra Gopal. "Asymptotic expansion differential quadrature method for the analysis of laminated hemispherical shells." Composite Structures 228 (November 2019): 111275. http://dx.doi.org/10.1016/j.compstruct.2019.111275.

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27

Li, Hu, and Yanying Ma. "Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/812505.

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We study the numerical solution of Helmholtz equation with Dirichlet boundary condition. Based on the potential theory, the problem can be converted into a boundary integral equation. We propose the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote byhmthe mesh width of a curved edgeΓm (m=1,…,d)of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied withO(hm3)for all mesh widthshmis obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical ap

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28

OLVER, SHEEHAN. "Moment-free numerical approximation of highly oscillatory integrals with stationary points." European Journal of Applied Mathematics 18, no. 4 (2007): 435–47. http://dx.doi.org/10.1017/s0956792507007012.

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This article presents a method for the numerical quadrature of highly oscillatory integrals with stationary points. We begin with the derivation of a new asymptotic expansion, which has the property that the accuracy improves as the frequency of oscillations increases. This asymptotic expansion is closely related to the method of stationary phase, but presented in a way that allows the derivation of an alternate approximation method that has similar asymptotic behaviour, but with significantly greater accuracy. This approximation method does not require moments.

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29

Kyei, Yaw. "Effective Source Term Discretizations for Higher Accuracy Finite Volume Discretization of Parabolic Equations." International Journal for Innovation Education and Research 9, no. 8 (2021): 366–92. http://dx.doi.org/10.31686/ijier.vol9.iss8.3305.

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A finite volume method is applied to develop space-time discretizations for parabolic equations based on an equation error method.A space-time expansion of the local equation error based on flux integral formulation of the equation is first designed using a desiredframework of neighboring quadrature points for the solution and local source terms. The quadrature weights are then determined through aminimization process for the error which constitutes all local compact fluxes about each centroid within the computational domain.In utilizing a local source term distribution to account for diffusiv

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30

Ahmet, Rifat GORGUN, OZEN Sukru, and HELHEL Selcuk. "Implementation of Fourier Expansion Based Differential Quadrature Method (FDQM) and Polynomial Based Differential Quadrature Method (PDQM) for the 2D Helmholtz Problem." Scientific Research and Essays 8, no. 34 (2013): 1670–75. http://dx.doi.org/10.5897/sre12.560.

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31

Li, Jin, and Xiuzhen Li. "The Modified Trapezoidal Rule for Computing Hypersingular Integral on Interval." Journal of Applied Mathematics 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/736834.

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The modified trapezoidal rule for the computation of hypersingular integrals in boundary element methods is discussed. When the special function of the error functional equals zero, the convergence rate is one order higher than the general case. A new quadrature rule is presented and the asymptotic expansion of error function is obtained. Based on the error expansion, not only do we obtain a high order of accuracy, but also a posteriori error estimate is conveniently derived. Some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms.

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32

Walters, Gage, Andrew Wixom, and Sheri Martinelli. "Comparison of quadrature and regression based generalized polynomial chaos expansions for structural acoustics." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 263, no. 6 (2021): 863–74. http://dx.doi.org/10.3397/in-2021-1670.

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This work performs a direct comparison between generalized polynomial chaos (GPC) expansion techniques applied to structural acoustic problems. Broadly, the GPC techniques are grouped in two categories: , where the stochastic sampling is predetermined according to a quadrature rule; and , where an arbitrary selection of points is used as long as they are a representative sample of the random input. As a baseline comparison, Monte Carlo type simulations are also performed although they take many more sampling points. The test problems considered include both canonical and more applied cases tha

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33

Maleknejad, K., and T. Lotfi. "Numerical expansion methods for solving integral equations by interpolation and Gauss quadrature rules." Applied Mathematics and Computation 168, no. 1 (2005): 111–24. http://dx.doi.org/10.1016/j.amc.2004.08.048.

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34

Saka, Bülent. "Cosine expansion-based differential quadrature method for numerical solution of the KdV equation." Chaos, Solitons & Fractals 40, no. 5 (2009): 2181–90. http://dx.doi.org/10.1016/j.chaos.2007.10.004.

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35

Kropf, Pascal, and Amir Shmuel. "1D Current Source Density (CSD) Estimation in Inverse Theory: A Unified Framework for Higher-Order Spectral Regularization of Quadrature and Expansion-Type CSD Methods." Neural Computation 28, no. 7 (2016): 1305–55. http://dx.doi.org/10.1162/neco_a_00846.

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Estimation of current source density (CSD) from the low-frequency part of extracellular electric potential recordings is an unstable linear inverse problem. To make the estimation possible in an experimental setting where recordings are contaminated with noise, it is necessary to stabilize the inversion. Here we present a unified framework for zero- and higher-order singular-value-decomposition (SVD)—based spectral regularization of 1D (linear) CSD estimation from local field potentials. The framework is based on two general approaches commonly employed for solving inverse problems: quadrature

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36

Mehditabar, Aref, Gholam H. Rahimi, and Keramat Malekzadeh Fard. "Vibrational responses of antisymmetric angle-ply laminated conical shell by the methods of polynomial based differential quadrature and Fourier expansion based differential quadrature." Applied Mathematics and Computation 320 (March 2018): 580–95. http://dx.doi.org/10.1016/j.amc.2017.10.017.

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37

Syam, Muhammed I. "A new proof of the Euler-Maclaurin expansion for quadrature over implicitly defined curves." Journal of Computational and Applied Mathematics 104, no. 1 (1999): 19–25. http://dx.doi.org/10.1016/s0377-0427(98)00198-8.

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38

Li, Wanyou, Gang Wang, and Jingtao Du. "Vibration Analysis of Conical Shells by the Improved Fourier Expansion-Based Differential Quadrature Method." Shock and Vibration 2016 (2016): 1–10. http://dx.doi.org/10.1155/2016/9617957.

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An improved Fourier expansion-based differential quadrature (DQ) algorithm is proposed to study the free vibration behavior of truncated conical shells with different boundary conditions. The original function is expressed as the Fourier cosine series combined with close-form auxiliary functions. Those auxiliary functions are introduced to ensure and accelerate the convergence of series expansion. The grid points are uniformly distributed along the space. The weighting coefficients in the DQ method are easily obtained by the inverse of the coefficient matrix. The derivatives in both the govern

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39

Chen, Ruyun, Di Yu, and Juan Chen. "Asymptotic expansion and quadrature rule for a class of singular-oscillatory-Bessel-type transforms." Journal of Computational and Applied Mathematics 383 (February 2021): 113141. http://dx.doi.org/10.1016/j.cam.2020.113141.

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40

Khubezhty, Shalva S. "Approximate Solution of a Hypersingular Integral Equation of the First Kind Bounded at Both Ends of the Integration Segment Using Chebyshev Series." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 1 (209) (March 31, 2021): 33–38. http://dx.doi.org/10.18522/1026-2237-2021-1-33-38.

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A hypersingular integral equation on the interval of integration is considered. The hypersingular integral is understood in the sense of Hadamard, that is, in the finite part. The class of such equations is widely used in problems of mathematical physics, in technology, and most importantly: in recent years, they are one of the main devices for modeling problems in electrodynamics. With the use of Chebyshev polynomials of the second kind, the unknown function, the right-hand side and the kernel are replaced in the equation. The expansion coefficients of these functions are calculated using qua

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41

Fangyang, Yuan, and Chen Zhongli. "Direct expansion method of moments with n/3th moments for nanoparticle Brownian coagulation in the free molecule regime." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 1 (2015): 86–97. http://dx.doi.org/10.1108/hff-04-2013-0120.

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Purpose – The purpose of this paper is to develop new types of direct expansion method of moments (DEMM) by using the n/3th moments for simulating nanoparticle Brownian coagulation in the free molecule regime. The feasibilities of new proposed DEMMs with n/3th moments are investigated to describe the evolution of aerosol size distribution, and some of the models will be applied to further simulation of physical processes. Design/methodology/approach – The accuracy and efficiency of some kinds of methods of moments are mainly compared including the quadrature method of moments (QMOM), Taylor-ex

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42

Zhang, Yang-Hong, and Jiao-Kai Chen. "Error Estimates for the Nearly Singular Momentum-Space Bound-State Equations." Advances in Mathematical Physics 2020 (February 1, 2020): 1–5. http://dx.doi.org/10.1155/2020/2560239.

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We present errors of quadrature rules for the nearly singular integrals in the momentum-space bound-state equations and give the critical value of the nearly singular parameter. We give error estimates for the expansion method, the Nyström method, and the spectral method which arise from the near singularities in the momentum-space bound-state equations. We show the relations amongst the near singularities, the odd phenomena in the eigenfunctions, and the unreliability of the numerical solutions.

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43

Yusufoğlu (Agadjanov), Elçin. "Numerical expansion methods for solving systems of linear integral equations using interpolation and quadrature rules." International Journal of Computer Mathematics 84, no. 1 (2007): 133–49. http://dx.doi.org/10.1080/00207160601176905.

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44

af Klinteberg, Ludvig, and Anna-Karin Tornberg. "A fast integral equation method for solid particles in viscous flow using quadrature by expansion." Journal of Computational Physics 326 (December 2016): 420–45. http://dx.doi.org/10.1016/j.jcp.2016.09.006.

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45

Siegel, Michael, and Anna-Karin Tornberg. "A local target specific quadrature by expansion method for evaluation of layer potentials in 3D." Journal of Computational Physics 364 (July 2018): 365–92. http://dx.doi.org/10.1016/j.jcp.2018.03.006.

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46

Dubeau, François. "Revisited Optimal Error Bounds for Interpolatory Integration Rules." Advances in Numerical Analysis 2016 (November 16, 2016): 1–8. http://dx.doi.org/10.1155/2016/3170595.

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We present a unified way to obtain optimal error bounds for general interpolatory integration rules. The method is based on the Peano form of the error term when we use Taylor’s expansion. These bounds depend on the regularity of the integrand. The method of integration by parts “backwards” to obtain bounds is also discussed. The analysis includes quadrature rules with nodes outside the interval of integration. Best error bounds for composite integration rules are also obtained. Some consequences of symmetry are discussed.

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47

Son, Jeongeun, Dongping Du, and Yuncheng Du. "Modified Polynomial Chaos Expansion for Efficient Uncertainty Quantification in Biological Systems." Applied Mechanics 1, no. 3 (2020): 153–73. http://dx.doi.org/10.3390/applmech1030011.

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Uncertainty quantification (UQ) is an important part of mathematical modeling and simulations, which quantifies the impact of parametric uncertainty on model predictions. This paper presents an efficient approach for polynomial chaos expansion (PCE) based UQ method in biological systems. For PCE, the key step is the stochastic Galerkin (SG) projection, which yields a family of deterministic models of PCE coefficients to describe the original stochastic system. When dealing with systems that involve nonpolynomial terms and many uncertainties, the SG-based PCE is computationally prohibitive beca

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48

Li, Hu, and Guang Zeng. "Convergence of the High-Accuracy Algorithm for Solving the Dirichlet Problem of the Modified Helmholtz Equation." Complexity 2020 (January 20, 2020): 1–8. http://dx.doi.org/10.1155/2020/6484890.

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In this paper, we derive the convergence for the high-accuracy algorithm in solving the Dirichlet problem of the modified Helmholtz equation. By the boundary element method, we transform the system to be a boundary integral equation. The high-accuracy algorithm using the specific quadrature rule is developed to deal with weakly singular integrals. The convergence of the algorithm is proved based on Anselone’s collective compact theory. Moreover, an asymptotic error expansion shows that the algorithm is of order Oh03. The numerical examples support the theoretical analysis.

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49

Carley, Michael. "Closed-Form Evaluation of Potential Integrals in the Boundary Element Method." Journal of Theoretical and Computational Acoustics 28, no. 02 (2020): 1950014. http://dx.doi.org/10.1142/s2591728519500142.

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A method is presented for the closed-form evaluation of the singular and near-singular integrals arising in the Boundary Element Method solution of the Helmholtz equation. An error analysis is presented for the numerical evaluation of such integrals on a plane element, and used to develop a criterion for the selection of quadrature rules. The analytical approach is based on an optimized expansion of the Green’s function for the problem, selected to limit the error to some required tolerance. Results are presented showing accuracy to tolerances comparable to machine precision.

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50

Bhrawy, A. H., and M. A. Alghamdi. "A Shifted Jacobi-Gauss Collocation Scheme for Solving Fractional Neutral Functional-Differential Equations." Advances in Mathematical Physics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/595848.

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Abstract:

The shifted Jacobi-Gauss collocation (SJGC) scheme is proposed and implemented to solve the fractional neutral functional-differential equations with proportional delays. The technique we have proposed is based upon shifted Jacobi polynomials with the Gauss quadrature integration technique. The main advantage of the shifted Jacobi-Gauss scheme is to reduce solving the generalized fractional neutral functional-differential equations to a system of algebraic equations in the unknown expansion. Reasonable numerical results are achieved by choosing few shifted Jacobi-Gauss collocation nodes. Numer

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