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A meshless method for the numerical solution of nonlinear weakly singular integral equations using radial basis functions

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Abstract.

In this paper a simple and effective method is given for the numerical solution of nonlinear one- and two-dimensional Fredholm integral equations of the second kind with weakly singular kernels. The general framework of the new scheme is based on the collocation method together with radial basis functions (RBFs) constructed on scattered points in which all integrals are computed via quadrature formulae. In order to approximate the singular integrals appeared in the scheme, we introduce a special quadrature formula, since these integrals cannot be estimated by classical integration rules. The method does not require any cell structures, so it is meshless and consequently is independent of the geometry of the domain. We also present the error analysis of the proposed method and demonstrate that the convergence rate of the approach is arbitrarily high for infinitely smooth RBFs. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

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References

  1. H. Wendland, Scattered Data Approximation (Cambridge University Press, New York, 2005)

  2. G.E. Fasshauer, Meshfree methods, in Handbook of Theoretical and Computational Nanotechnology (American Scientific Publishers, 2005)

  3. R.L. Hardy, J. Geophys. Res. 176, 1905 (2006)

    Google Scholar 

  4. J. Meinguet, Z. Angew. Math. Phys. 30, 292 (1979)

    Article  MathSciNet  Google Scholar 

  5. G. Wahba, Convergence rates of “thin plate” smoothing splines wihen the data are noisy, in Smoothing Techniques for Curve Estimation, edited by T. Gasser, M. Rosenblatt, Lect. Notes Math., Vol. 757 Springer, Berlin, Heidelberg, 1979) pp. 233--245

  6. R. Franke, Math. Comput 38, 181 (1982)

    Google Scholar 

  7. E.J. Kansa, Comput. Math. Appl. 19, 127 (1990)

    Article  MathSciNet  Google Scholar 

  8. E.J. Kansa, Comput. Math. Appl. 19, 147 (1990)

    Article  MathSciNet  Google Scholar 

  9. Y.C. Hon, X.Z. Mao, Appl. Math. Comput. 95, 37 (1998)

    MathSciNet  Google Scholar 

  10. Y.C. Hon, K.F. Cheung, X. Mao, E.J. Kansa, ASCE J. Hydraul. Eng. 125, 524 (1999)

    Article  Google Scholar 

  11. M. Tatari, M. Dehghan, Eng. Anal. Bound. Elem. 34, 206 (2010)

    Article  MathSciNet  Google Scholar 

  12. M. Zerroukat, H. Power, C.S. Chen, Int. J. Numer. Methods Eng. 42, 1263 (1998)

    Article  Google Scholar 

  13. M. Tatari, M. Dehghan, Appl. Math. Model. 33, 1729 (2009)

    Article  MathSciNet  Google Scholar 

  14. Y.C. Hon, X.Z. Mao, Financ. Eng. 8, 31 (1999)

    Google Scholar 

  15. M.D. Marcozzi, S. Choi, C.S. Chen, Appl. Math. Comput. 124, 197 (2001)

    MathSciNet  Google Scholar 

  16. M. Dehghan, A. Shokri, Nonlinear Dyn. 50, 111 (2007)

    Article  Google Scholar 

  17. M. Dehghan, A. Shokri, J. Comput. Appl. Math. 230, 400 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  18. A. Shokri, M. Dehghan, Comput. Phys. Commun. 181, 1990 (2010)

    Article  ADS  Google Scholar 

  19. M. Dehghan, A. Shokri, Math. Comput. Simulat. 79, 700 (2008)

    Article  Google Scholar 

  20. G.E. Fasshauer, in Surface Fitting and Multiresolution Methods, edited by A. Le Méhauté, C. Rabut, L.L. Schumaker (Vanderbilt University Press, 1997) pp. 131--138

  21. P. Assari, H. Adibi, M. Dehghan, J. Comput. Appl. Math. 239, 72 (2013)

    Article  MathSciNet  Google Scholar 

  22. P. Assari, H. Adibi, M. Dehghan, Appl. Math. Model. 37, 9269 (2013)

    Article  MathSciNet  Google Scholar 

  23. A. Golbabai, S. Seifollahi, Appl. Math. Comput. 174, 877 (2006)

    MathSciNet  Google Scholar 

  24. K. Parand, J.A. Rad, Appl. Math. Comput. 218, 5292 (2012)

    MathSciNet  Google Scholar 

  25. P. Assari, H. Adibi, M. Dehghan, Appl. Numer. Math. 81, 76 (2014)

    Article  MathSciNet  Google Scholar 

  26. P. Assari, H. Adibi, M. Dehghan, Numer. Algor. 67, 423 (2014)

    Article  Google Scholar 

  27. D. Mirzaei, M. Dehghan, Appl. Numer. Math. 60, 245 (2010)

    Article  MathSciNet  Google Scholar 

  28. M. Dehghan, R. Salehi, J. Comput. Appl. Math. 236, 2367 (2012)

    Article  MathSciNet  Google Scholar 

  29. P. Assari, H. Adibi, M. Dehghan, J. Comput. Appl. Math. 267, 160 (2014)

    Article  MathSciNet  Google Scholar 

  30. X. Li, J. Zhu, J. Comput. Appl. Math. 230, 314 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  31. X. Li, Appl. Numer. Math. 61, 1237 (2011)

    Article  MathSciNet  Google Scholar 

  32. X. Li, J. Zhu, Eng. Anal. Bound. Elem. 33, 858 (2009)

    Article  MathSciNet  Google Scholar 

  33. B. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989)

  34. K. Mei, J. Van Bladel, IEEE Trans. Antennas Propagation 11, 52 (1963)

    Article  ADS  Google Scholar 

  35. G.T. Symm, Numer. Math. 9, 250 (1966)

    Article  MathSciNet  Google Scholar 

  36. M.G. Andreasen, IEEE Trans. AP 12, 235 (1964)

    Article  ADS  Google Scholar 

  37. K.K. Mei, J.G. van Bladel, IEEE Trans. Antennas Propagation 11, 185 (1963)

    Article  ADS  Google Scholar 

  38. R.P. Banaugh, W. Goldsmith, J. Appl. Mech. 30, 589 (1963)

    Article  ADS  Google Scholar 

  39. A.M. Wazwaz, Linear and Nonlinear Integral equations: Methods and Applications (Higher Education Press and Springer Verlag, Heidelberg, 2011)

  40. W. Chen, W. Lin, Appl. Math. Comput. 121, 75 (2001)

    MathSciNet  Google Scholar 

  41. A. Pedas, G. Vainikko, J. Integral Equations Appl. 9, 379 (1997)

    Article  MathSciNet  Google Scholar 

  42. A. Pedas, G. Vainikko, Commun. Pure Appl. Anal. 5, 395 (2006)

    Article  MathSciNet  Google Scholar 

  43. T. Okayama, T. Matsuo, M. Sugihara, J. Comput. Appl. Math. 234, 1211 (2010)

    Article  MathSciNet  Google Scholar 

  44. Y. Cao, M. Huang, L. Liu, Y. Xu, Appl. Numer. Math. 57, 549 (2007)

    Article  MathSciNet  Google Scholar 

  45. V. Dominguez, J. Comput. Appl. Math. 161, 145 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  46. G. Long, G. Nelakanti, X. Zhang, Appl. Numer. Math. 62, 201 (2012)

    Article  MathSciNet  Google Scholar 

  47. A.A. Gusenkova, N.B. Pleshchinskii, J. Appl. Maths. Mechs. 64, 435 (2000)

    Article  Google Scholar 

  48. S.A. Khuri, A.M. Wazwaz, Int. J. Comput. Math. 61, 103 (1996)

    Article  Google Scholar 

  49. A.M. Wazwaz, R. Rach, J. Duan, Cent. Eur. J. Eng. 3, 669 (2013)

    Google Scholar 

  50. U. Lepik, Appl. Math. Comput. 198, 326 (2008)

    MathSciNet  Google Scholar 

  51. H. Adibi, P. Assari, J. Vib. Control. 17, 689 (2011)

    Article  MathSciNet  Google Scholar 

  52. X. Tang, Z. Pang, T. Zhu, J. Liu, Wuhan Univ. J. Nat. Sci. 12, 437 (2007)

    Article  MathSciNet  Google Scholar 

  53. W. Fang, Y. Wang, Y. Xu, J. Sci. Comput. 20, 277 (2004)

    Article  MathSciNet  Google Scholar 

  54. J. Gao, Y. Jiang, J. Comput. Appl. Math. 215, 242 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  55. I.G. Graham, J. Austral. Math. Soc. Ser. B 22, 460 (1981)

    Article  Google Scholar 

  56. R. Farengo, Y.C. Lee, P.N. Guzdar, Phys. Fluids 26, 3515 (1983)

    Article  ADS  Google Scholar 

  57. K.F. Warnick, Numerical Analysis for Electromagnetic Integral Equations (Artech House, Boston, 2008)

  58. J. Radlow, Bull. Am. Math. Soc. 70, 596 (1964)

    Article  MathSciNet  Google Scholar 

  59. M.V. Mirkin, A.J. Bard, J. Electroad. Chem. 323, 1 (1992)

    Article  Google Scholar 

  60. B.M. Johnston, P.R. Johnston, Int. J. Numer. Math. Eng. 56, 589 (2003)

    Article  Google Scholar 

  61. I.G. Graham, C. Schneider, Product Integration for Weakly Singular Integral Equations In $R^{m}$ in Constructive Methods for the Practical Treatment of Integral Equations, edited by G. Hammerlin, K.H. Hoffman, International Series of Numerical Mathematics, Vol 73 (Springer, 1985) pp. 156--165

  62. A. Pedas, G. Vainikko, in Integral Methods in Science and Engineering, edited by B. Bertram, C. Constanda, A. Struthers (Chapman and Hall/CRC, 2000) pp. 280--285

  63. H. Kaneko, Y. Xu, Math. Comp. 62, 739 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  64. X. Li, Int. J. Numer. Methods Eng. 88, 442 (2011)

    Article  Google Scholar 

  65. X. Li, J. Zhu, Comput. Methods Appl. Mech. Eng. 198, 2874 (2009)

    Article  ADS  Google Scholar 

  66. K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind (Cambridge University Press, Cambridge, 1997)

  67. S.A. Khuri, A. Sayfy, J. Comput. Appl. Math. 233, 2081 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  68. M.D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge University Press, Cambridge, 2003)

  69. I.J. Schoenberg, Ann. Math. 39, 811 (1938)

    Article  MathSciNet  Google Scholar 

  70. A.H. Cheng, M.A. Golberg, E.J. Kansa, Q. Zammito, Numer. Meth. Partial. Differ. Eqs. 19, 571 (2003)

    Article  Google Scholar 

  71. G.E. Fasshauer, J.G. Zhang, Numer. Algor. 45, 345 (2007)

    Article  ADS  Google Scholar 

  72. K.E. Atkinson, SIAM J. Numer. Anal. 10, 799 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  73. R. Weiss, SIAM J. Numer. Anal. 11, 550 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  74. K.E. Atkinson, J. Flores, IMA J. Numer. Anal. 13, 195 (1993)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Bu-Ali Sina University, 65178, Hamedan, Iran

    Pouria Assari

  2. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Mehdi Dehghan

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  1. Pouria Assari
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  2. Mehdi Dehghan
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Assari, P., Dehghan, M. A meshless method for the numerical solution of nonlinear weakly singular integral equations using radial basis functions. Eur. Phys. J. Plus 132, 199 (2017). https://doi.org/10.1140/epjp/i2017-11467-y

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