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Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation

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Abstract

In this article, a general type of two-dimensional time-fractional telegraph equation explained by the Caputo derivative sense for (1 < α ≤ 2) is considered and analyzed by a method based on the Galerkin weak form and local radial point interpolant (LRPI) approximation subject to given appropriate initial and Dirichlet boundary conditions. In the proposed method, so-called meshless local radial point interpolation (MLRPI) method, a meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. The point interpolation method is proposed to construct shape functions using the radial basis functions. In the MLRPI method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares. Two numerical examples are presented and satisfactory agreements are achieved.

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References

  1. K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley & Sons Inc, New York, 1993).

  2. S. Samko, A. Kilbas, O. Maxitchev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, in Nauka i Tekhnika (Minsk, 1987) in Russian.

  3. Z. Jiao, Y. Chen, I. Podlubny, Distributed-order Dunamic Systems: Stability, Simulation, Applications and Perspectives (Springer, 2012).

  4. I. Podlubny, Fract. Calc. Appl. Anal. 3, 359 (2000).

    MATH  MathSciNet  Google Scholar 

  5. I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, B.M.V. Jara, J. Comput. Phys. 228, 3137 (2009).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. C. Tadjeran, M.M. Meerschaert, H.-P. Scheffler, J. Comput. Phys. 213, 205 (2006).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers (Birkhäuser, Boston, 1997).

  8. A. Metaxas, R. Meredith, Industrial Microwave, Heating (Peter Peregrinus, London, 1983).

  9. O. Agrawal, Nonlinear Dyn. 29, 145 (2002).

    Article  MATH  Google Scholar 

  10. Z. Zhao, C. Li, Appl. Math. Comput. 219, 2975 (2012).

    Article  MathSciNet  Google Scholar 

  11. G. Liu, Y. Gu, An Introduction to Meshfree Methods and Their Programing (Springer, 2005).

  12. T. Belytschko, Y.Y. Lu, L. Gu, Int. J. Numer. Methods Eng. 37, 229 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Belytschko, Y.Y. Lu, L. Gu, Int. J. Solids Struct. 32, 2547 (1995).

    Article  MATH  Google Scholar 

  14. E. Shivanian, Eng. Anal. Bound. Elem. 37, 1693 (2013).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. K. Mramor, R. Vertnik, B. Sarler, Comput. Mater. Contin. 36, 1 (2013).

    Google Scholar 

  18. Y. Hon, L. Ling, K. Liew, Comput. Mater. Contin. 2, 39 (2005).

    MATH  Google Scholar 

  19. X. Xiong, M. Li, M.Q. Wang, J. Eng. Math. 80, 189 (2013).

    Article  MathSciNet  Google Scholar 

  20. B. Nayroles, G. Touzot, P. Villon, Comput. Mech. 10, 307 (1992).

    Article  MATH  Google Scholar 

  21. A. Bratsos, Int. J. Numer. Methods Eng. 75, 787 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  22. P. Clear, Appl. Math. Model. 22, 981 (1998).

    Article  Google Scholar 

  23. W. Liu, S. Jun, Y. Zhang, Int. J. Numer. Methods Eng. 20, 1081 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  24. Y. Mukherjee, S. Mukherjee, Int. J. Numer. Methods Eng. 40, 797 (1997).

    Article  MATH  Google Scholar 

  25. J. Melenk, I. Babuska, Comput. Methods Appl. Mech. Eng. 139, 289 (1996).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. S. De, K. Bathe, Comput. Mech. 25, 329 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  27. Y. Gu, G. Liu, Comput. Mech. 28, 47 (2002).

    Article  MATH  Google Scholar 

  28. Y. Gu, G. Liu, Struct. Eng. Mech. 15, 535 (2003).

    Article  Google Scholar 

  29. S. Atluri, T. Zhu, Comput. Mech. 22, 117 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  30. S. Atluri, T. Zhu, Comput. Model. Simulat. Eng. 3, 187 (1998).

    Google Scholar 

  31. S. Atluri, T. Zhu, Int. J. Numer. Methods Eng. 13, 537 (2000).

    Article  MathSciNet  Google Scholar 

  32. S. Atluri, T. Zhu, Comput. Mech. 25, 169 (2000).

    Article  MATH  Google Scholar 

  33. M. Dehghan, D. Mirzaei, Eng. Anal. Bound. Elem. 32, 747 (2008).

    Article  MATH  Google Scholar 

  34. M. Dehghan, D. Mirzaei, Appl. Numer. Math. 59, 1043 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  35. Y. Gu, G. Liu, Comput. Mech. 27, 188 (2001).

    Article  MATH  Google Scholar 

  36. S. Abbasbandy, A. Shirzadi, Eng. Anal. Bound. Elem. 34, 1031 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  37. S. Abbasbandy, A. Shirzadi, Appl. Numer. Math. 61, 170 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  38. A. Shirzadi, L. Ling, S. Abbasbandy, Eng. Anal. Bound. Elem. 36, 1522 (2012).

    Article  MathSciNet  Google Scholar 

  39. A. Shirzadi, V. Sladek, J. Sladek, Eng. Anal. Bound. Elem. 37, 8 (2013).

    Article  MathSciNet  Google Scholar 

  40. G. Liu, L. Yan, J. Wang, Y. Gu, Struct. Eng. Mech. 14, 713 (2002).

    Article  Google Scholar 

  41. G. Liu, Y. Gu, J. Sound Vib. 246, 29 (2001).

    Article  ADS  Google Scholar 

  42. M. Dehghan, A. Ghesmati, Comput. Phys. Commun. 181, 772 (2010).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  43. E. Shivanian, Ocean Eng. 89, 173 (2014).

    Article  Google Scholar 

  44. E. Shivanian, Eng. Anal. Bound. Elem. 50, 249 (2015).

    Article  MathSciNet  Google Scholar 

  45. E. Shivanian, H. Khodabandehlo, Eur. Phys. J. Plus 129, 241 (2014).

    Article  Google Scholar 

  46. T. Kaufmann, Y. Yu, C. Engström, Z. Chen, C. Fumeaux, Int. J. Numer. Model. Electron. Netw. Dev. Fields 25, 1099 (2012).

    Google Scholar 

  47. J. Wang, G. Liu, Int. J. Numer. Methods Eng. 54, 1623 (2002).

    Article  MATH  Google Scholar 

  48. J. Wang, G. Liu, Comput. Methods Appl. Math. 191, 2611 (2002).

    ADS  MATH  Google Scholar 

  49. C. Franke, R. Schaback, Appl. Math. Comput. 93, 73 (1997).

    Article  MathSciNet  Google Scholar 

  50. M. Sharan, E. Kansa, S. Gupta, Appl. Math. Comput. 84, 275 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  51. M.J.D. Powell, Theory of radial basis function approximation in 1990, in Advances in Numerical Analysis, edited by W. Light, Vol. 2 (Clarendon Press, Oxford, 1992) pp. 105–210.

  52. H. Wendland, J. Approx. Theory 93, 258 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  53. D. Hu, S. Long, K. Liu, G. Li, Eng. Anal. Bound. Elem. 30, 399 (2006).

    Article  MATH  Google Scholar 

  54. K. Liu, S. Long, G. Li, Eng. Anal. Bound. Elem. 30, 72 (2006).

    Article  Google Scholar 

  55. R. Adams, Sobolev Spaces (Academic Press, New York, 1975).

  56. Y.N. Zhang, Z.Z. Sun, H.W. Wu, SIAM J. Numer. Anal. 49, 2302 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  57. Y.N. Zhang, Z.Z. Sun, X. Zhao, SIAM J. Numer. Anal. 50, 1535 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  58. Z.Z. Sun, X.N. Wu, Appl. Numer. Math. 56, 193 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  59. J. Chen, F. Lin, V. Anh, S. Shen, Q. Liu, C. Liao, Appl. Math. Comput. 219, 1737 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  60. Z. Avazzadeh, V.R. Hosseini, W. Chen, Iran J. Sci. Technol. A 38, 205 (2014).

    MathSciNet  Google Scholar 

  61. V.R. Hosseini, W. Chen, Z. Azazzadeh, Eng. Anal. Bound. Elem. 38, 31 (2014).

    Article  MATH  MathSciNet  Google Scholar 

  62. C. Li, Z. Zhao, Y. Chen, Comput. Math. Appl. 62, 855 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  63. L. Wei, H. Dai, D. Zhang, Z. Si, Calcolo 51, 175 (2014).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, 210098, Nanjing, China

    Vahid Reza Hosseini & Wen Chen

  2. Department of Mathematics, Imam Khomeini International University, 34149-16818, Qazvin, Iran

    Elyas Shivanian

Authors
  1. Vahid Reza Hosseini
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  2. Elyas Shivanian
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  3. Wen Chen
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Correspondence to Elyas Shivanian.

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Hosseini, V.R., Shivanian, E. & Chen, W. Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation. Eur. Phys. J. Plus 130, 33 (2015). https://doi.org/10.1140/epjp/i2015-15033-5

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  • DOI: https://doi.org/10.1140/epjp/i2015-15033-5

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