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AIMS Mathematics

2020, Volume 5, Issue 6: 6211-6220. doi: 10.3934/math.2020399
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Research article

The modified quadrature method for Laplace equation with nonlinear boundary conditions

  • School of Mathematics, Chengdu Normal University, Chengdu, Sichuan, 611130, China
  • Received: 06 May 2020 Accepted: 29 July 2020 Published: 31 July 2020

  • MSC : 65N38, 65R20

  • Here, the numerical solutions for Laplace equation with nonlinear boundary conditions is studied. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equation. The modified quadrature method is presented for solving the nonlinear equation, which possesses high accuracy order $O(h^3)$ and low computing complexities. A nonlinear system is obtained by discretizing the nonlinear equation and the convergence of numerical solutions is proved by the theory of compact operators. Moreover, in order to solve the nonlinear system, the Newton iteration is provided by using the Ostrowski fixed point theorem. Finally, numerical examples support the theoretical results.

    Citation: Hu Li. The modified quadrature method for Laplace equation with nonlinear boundary conditions[J]. AIMS Mathematics, 2020, 5(6): 6211-6220. doi: 10.3934/math.2020399

    Related Papers:

  • Here, the numerical solutions for Laplace equation with nonlinear boundary conditions is studied. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equation. The modified quadrature method is presented for solving the nonlinear equation, which possesses high accuracy order $O(h^3)$ and low computing complexities. A nonlinear system is obtained by discretizing the nonlinear equation and the convergence of numerical solutions is proved by the theory of compact operators. Moreover, in order to solve the nonlinear system, the Newton iteration is provided by using the Ostrowski fixed point theorem. Finally, numerical examples support the theoretical results.


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    [1] S. Dohr, C. Kahle, S. Rogovs, et al. A FEM for an optimal control problem subject to the fractional Laplace equation, Calcolo, 56 (2019), 1-21. doi: 10.1007/s10092-018-0296-x
    [2] C. Kuo, W. Yeih, C. Ku, et al. The method of two-point angular basis function for solving Laplace equation, Eng. Anal. Bound. Elem., 106 (2019), 264-274. doi: 10.1016/j.enganabound.2019.05.018
    [3] P. Dehghan, M. Dehghan, On the Numerical Solution of Logarithmic Boundary Integral Equations Arising in Laplace's Equations Based on the Meshless Local Discrete Collocation Method, Adv. Appl. Math. Mech., 11 (2019), 807-837. doi: 10.4208/aamm.OA-2018-0050
    [4] M. A. Jankowska, Interval Nine-Point Finite Difference Method for Solving the Laplace Equation with the Dirichlet Boundary Conditions, In: Parallel Processing and Applied Mathematics, Springer, Cham, 2016, 474-484.
    [5] X. Li, J. Oh, Y. Wang, et al. The method of transformed angular basis function for solving the Laplace equation, Eng. Anal. Bound. Elem., 93 (2018), 72-82. doi: 10.1016/j.enganabound.2018.04.001
    [6] K. Ruotsalainen, W. Wendland, On the boundary element method for some nonlinear boundary value problems, Numer. Math., 53 (1988), 299-314. doi: 10.1007/BF01404466
    [7] K. Maleknejad, H. Mesgaran, Numerical method for a nonlinear boundary integral equation, Appl. Math. Comput., 182 (2006), 1006-1009.
    [8] P. Assari, M. Dehghan, Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions, Int. J. comput. Math., 96 (2019), 170-198. doi: 10.1080/00207160.2017.1420786
    [9] X. Chen, R. Wang, Y. Xu, Fast Fourier-Galerkin Methods for Nonlinear Boundary Integral Equations, J. Sci. Comput., 56 (2013), 494-514. doi: 10.1007/s10915-013-9687-y
    [10] I. H. Sloan, A. Spence, The Galerkin method for integral equations of the first kind with logarithmic kernel: theory, IMA J. Numer. Anal., 8 (1988), 105-122. doi: 10.1093/imanum/8.1.105
    [11] Y. Yan, The collocation method for first-kind boundary integral equations on polygonal regions, Math. Comput., 54 (1990), 139-154. doi: 10.1090/S0025-5718-1990-0995213-6
    [12] H. Li, J. Huang, High-accuracy quadrature methods for solving nonlinear boundary integral equations of axisymmetric Laplace's equation, Comput. Appl. Math., 37 (2018), 6838-6847. doi: 10.1007/s40314-018-0714-3
    [13] S. Christiansen, Numerical solution of an integral equation with a logarithmic kernel, BIT., 11 (1971), 276-287. doi: 10.1007/BF01931809
    [14] M. S. Abou El-Seoud, Numerische behandlung von schwach singulären Integralgleichungen 1. Art. Dissertation, Technische Hochschule Darmstadt, Federal Republic of Germany, Darmstadt, 1979.
    [15] J. Saranen, The modified quadrature method for logarithmic-kernel integral equations on closed curves, J. Integral Equ. Appl., 3 (1991), 575-600. doi: 10.1216/jiea/1181075650
    [16] P. M. Anselone, J. Davis, Collectively Compact Operator Approximation Theory and applications to integral equations, Prentice Hall, 1971.
    [17] H. Li, J. Huang, A novel approach to solve nonlinear Fredholm integral equations of the second kind, SpringerPlus, 5 (2016), 1-9. doi: 10.1186/s40064-015-1659-2
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  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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