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A numerical method to solve the 1D and the 2D reaction diffusion equation based on Bessel functions and Jacobian free Newton-Krylov subspace methods

  • K. Parand
  • M. Nikarya
Regular Article

Abstract.

In this paper a novel method will be introduced to solve a nonlinear partial differential equation (PDE). In the proposed method, we use the spectral collocation method based on Bessel functions of the first kind and the Jacobian free Newton-generalized minimum residual (JFNGMRes) method with adaptive preconditioner. In this work a nonlinear PDE has been converted to a nonlinear system of algebraic equations using the collocation method based on Bessel functions without any linearization, discretization or getting the help of any other methods. Finally, by using JFNGMRes, the solution of the nonlinear algebraic system is achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the famous Fisher equation. We compare our results with other methods.

References

  1. 1.
    G. Adomian, J. Math. Anal. Appl. 113, 202 (1986)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A.-M. Wazwaz, A. Gorguis, Appl. Math. Comput. 154, 609 (2004)MathSciNetGoogle Scholar
  3. 3.
    S. Abbasbandy, M. Darvishi, Appl. Math. Comput. 170, 95 (2005)MathSciNetGoogle Scholar
  4. 4.
    M. Dehghan, R. Salehi, Z. Naturforsch. A 66, 259 (2011)ADSGoogle Scholar
  5. 5.
    H. Fatoorehchi, H. Abolghasemi, Acad. Sci. Lett. 38, 67 (2015)CrossRefGoogle Scholar
  6. 6.
    B. Batiha, M. Noorani, I. Hashim, Chaos, Solitons Fractals 36, 660 (2008)ADSCrossRefGoogle Scholar
  7. 7.
    M. Dehghan, M. Najafi, Eng. Anal. Bound. Elem. 72, 111 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J.A. Rad, K. Parand, S. Abbasbandy, Commun. Nonlinear Sci. Numer. Simul. 22, 1178 (2015)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    S. Kazem, A. Hatam, Eng. Anal. Bound. Elem. 76, 90 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A.S. Al-Fhaid, J. Comput. Theor. Nanosci. 13, 3112 (2016)CrossRefGoogle Scholar
  11. 11.
    E. Doha, A. Bhrawy, M. Abdelkawy, ASME J. Comput. Nonlinear Dyn. 10, 21016 (2015)CrossRefGoogle Scholar
  12. 12.
    A. Sahin, I. Dag, B. Saka, Kybernetes 37, 326 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. Macas-Daz, A. Gallegos, H. Vargas-Rodrguez, J. Comput. Appl. Math. 318, 366 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    H.P. Pfeiffer, L.E. Kidder, M.A. Scheel, S.A. Teukolsky, Comput. Phys. Commun. 152, 253 (2003)ADSCrossRefGoogle Scholar
  15. 15.
    A. Asaithambi, Appl. Math. Comput. 216, 2700 (2010)MathSciNetGoogle Scholar
  16. 16.
    A. Soliman, Physica A 361, 394 (2006)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    R. Mittal, R. Rajni, J. Math. Chem. 55, 673 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    H. Shukla, M. Tamsir, Alex. Eng. J. 55, 2871 (2016)CrossRefGoogle Scholar
  19. 19.
    M. Bastani, D.K. Salkuyeh, Pramana 78, 335 (2012)ADSCrossRefGoogle Scholar
  20. 20.
    R.C. Mittal, G. Arora, Int. J. Comput. Math. 87, 3039 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    M. Sari, G. Gurarslan, A. Zeytinoglu, Int. J. Numer. Methods Biomed. Eng. 27, 1296 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    D. He, Comput. Math. Appl. 71, 2594 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    T. Gudi, H.S. Gupta, J. Comput. Appl. Math. 247, 1 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    K. Burrage, N. Hale, D. Kay, SIAM J. Sci. Comput. 34, 2145 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    P. Danumjaya, A.K. Pani, Numer. Methods Partial Differ. Equ. 28, 1227 (2012)CrossRefGoogle Scholar
  26. 26.
    R.E. Alcouffe, A. Brandt, J.E. Dendy, J.W. Painter, SIAM J. Sci. Stat. Comput. 2, 430 (1981)CrossRefGoogle Scholar
  27. 27.
    J. Fish, V. Belsky, Comput. Methods Appl. Mech. Eng. 126, 17 (1995)ADSCrossRefGoogle Scholar
  28. 28.
    K. Parand, M. Nikarya, Appl. Math. Model. 38, 4137 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    K. Parand, J.A. Rad, M. Nikarya, Int. J. Comput. Math. 91, 1239 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    T. Tajvidi, M. Razzaghi, M. Dehghan, Chaos, Solitons Fractals 35, 59 (2008)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    J.P. Yan, B.Y. Guo, Numer. Math. Theor. Methods Appl. 4, 283 (2011)Google Scholar
  32. 32.
    E. Aksan, A. Ozdes, Appl. Math. Comput. 156, 395 (2004)MathSciNetGoogle Scholar
  33. 33.
    N.K. Yamaleev, M.H. Carpenter, J. Comput. Phys. 331, 90 (2017)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    S. Gottlieb, C.W. Shu, E. Tadmor, SIAM Rev. 43, 89 (2001)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    R. Jiwari, Comput. Phys. Commun. 183, 2413 (2012)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    J. Ramos, Appl. Math. Comput. 161, 525 (2005)MathSciNetGoogle Scholar
  37. 37.
    M. Putti, C. Paniconi, Adv. Water Resour. 18, 159 (1995)ADSCrossRefGoogle Scholar
  38. 38.
    J. Shen, T. Tang, L.L. Wang, Spectral Methods: Algorithms, Analysis and Applications (Springer, Berlin, Heidelberg, 2011)Google Scholar
  39. 39.
    J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, New York, 2000)Google Scholar
  40. 40.
    G.M. Shroff, H.B. Keller, SIAM J. Numer. Anal. 30, 1099 (1993)MathSciNetCrossRefGoogle Scholar
  41. 41.
    A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, J. Comput. Appl. Math. 233, 2696 (2010)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    G.H. Nedzhibov, J. Comput. Appl. Math. 222, 244 (2008)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    A. Soulaimani, N.B. Salah, Y. Saad, Int. J. Comput. Fluid Dyn. 16, 1 (2002)CrossRefGoogle Scholar
  44. 44.
    Y. Chen, C. Shen, IEEE Trans. Power Syst. 21, 1096 (2006)CrossRefGoogle Scholar
  45. 45.
    K. Parand, M. Nikarya, J.A. Rad, F. Baharifard, Z. Naturforsch. A 67, 665 (2012)ADSCrossRefGoogle Scholar
  46. 46.
    G. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition (Cambridge University Press, Cambridge, 1967)Google Scholar
  47. 47.
    W.W. Bell, Special Functions For Scientists and Engineers (D. Van Nostrand Company, Ltd., 1967)Google Scholar
  48. 48.
    E. Kreyszig, Introductory Functional Analysis with Applications (John Wiley, New York, 2000)Google Scholar
  49. 49.
    D. Knoll, D. Keyes, J. Comput. Phys. 193, 357 (2004)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    H. Asgharzadeh, I. Borazjani, J. Comput. Phys. 331, 227 (2017)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    A. Hajizadeh, H. Kazeminejad, S. Talebi, Prog. Nucl. Energy 95, 48 (2017)CrossRefGoogle Scholar
  52. 52.
    S. Zhao, G.W. Wei, SIAM J. Sci. Comput. 25, 127 (2003)MathSciNetCrossRefGoogle Scholar
  53. 53.
    M. Rosa, J. Camacho, M. Bruzn, M. Gandarias, J. Comput. Appl. Math. 318, 181 (2017)MathSciNetCrossRefGoogle Scholar
  54. 54.
    V.A. Vijesh, K.H. Kumar, Appl. Math. Comput. 266, 1163 (2015)MathSciNetGoogle Scholar
  55. 55.
    R.E. Mickens, Numer. Methods Partial Differ. Equ. 10, 581 (1994)MathSciNetCrossRefGoogle Scholar
  56. 56.
    S. Succi, Int. J. Mod. Phys. C 25, 1340015 (2014)ADSCrossRefGoogle Scholar
  57. 57.
    Y.-J. Jiao, T.-J. Wang, Q. Zhang, East Asian J. Appl. Math. 6, 400 (2016)MathSciNetCrossRefGoogle Scholar
  58. 58.
    A. Verma, R. Jiwari, M. Koksal, Adv. Differ. Equ. 2014, 229 (2014)CrossRefGoogle Scholar
  59. 59.
    R. Mittal, R. Jiwari, Int. J. Inf. Syst. Sci. 5, 143 (2009)MathSciNetGoogle Scholar
  60. 60.
    R. Rajaraman, G. Hariharan, J. Membrane Biol. 247, 561 (2014)CrossRefGoogle Scholar
  61. 61.
    V. Chandraker, A. Awasthi, S. Jayaraj, Proc. Eng. 127, 1256 (2015)CrossRefGoogle Scholar
  62. 62.
    S.T. Yu, C.R. Jun, G.H. Xia, Chin. Phys. B 22, 1 (2013)ADSGoogle Scholar
  63. 63.
    A.H. Bhrawy, M.A. Alghamdi, Abs. Appl. Anal. 2013, 176730 (2013)Google Scholar
  64. 64.
    A. Habbal, H. Barelli, G. Malandain, Math. Biosci. 252, 45 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Computer SciencesShahid Beheshti University, G.C.TehranIran

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