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Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations

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Abstract

This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h 3) and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h 3-Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h 5) is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.

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Authors and Affiliations

  1. School of Science, Chongqing Jiaotong University, Chongqing, 400074, P. R. China

    Pan Cheng  (程 攀)

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, P. R. China

    Jin Huang  (黄 晋) & Zhu Wang  (王 柱)

  3. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    Zhu Wang  (王 柱)

Authors
  1. Pan Cheng  (程 攀)
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  2. Jin Huang  (黄 晋)
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  3. Zhu Wang  (王 柱)
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Corresponding author

Correspondence to Pan Cheng  (程 攀).

Additional information

Project supported by the National Natural Science Foundation of China (No. 10871034), the Natural Science Foundation Project of Chongqing (No.CSTC20-10BB8270), the Air Force Office of Scientific Research (No. FA9550-08-1-0136), and the National Science Foundation (No.OCE-0620464)

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Cheng, P., Huang, J. & Wang, Z. Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations. Appl. Math. Mech.-Engl. Ed. 32, 1505–1514 (2011). https://doi.org/10.1007/s10483-011-1519-7

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  • DOI: https://doi.org/10.1007/s10483-011-1519-7

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Chinese Library Classification

2010 Mathematics Subject Classification

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