Advertisement

New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networks

  • C. J. Zúñiga-Aguilar
  • A. Coronel-Escamilla
  • J. F. Gómez-AguilarEmail author
  • V. M. Alvarado-Martínez
  • H. M. Romero-Ugalde
Regular Article

Abstract.

In this paper, we approximate the solution of fractional differential equations with delay using a new approach based on artificial neural networks. We consider fractional differential equations of variable order with the Mittag-Leffler kernel in the Liouville-Caputo sense. With this new neural network approach, an approximate solution of the fractional delay differential equation is obtained. Synaptic weights are optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional delay differential equations, linear systems with delay, nonlinear systems with delay and a system of differential equations, for instance, the Newton-Leipnik oscillator. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network, different performance indices were calculated.

References

  1. 1.
    D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods, in Series on Complexity, Nonlinearity and Chaos (World Scientific, 2012)Google Scholar
  2. 2.
    I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications (Academic Press, San Diego, CA, USA, 1999)Google Scholar
  3. 3.
    X.J. Yang, J.T. Machado, D. Baleanu, Fractals 25, 1740006 (2017)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Y. Zhang, A. Kumar, S. Kumar, D. Baleanu, X.J. Yang, J. Nonlinear Sci. Appl. 9, 5821 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    X.J. Yang, F. Gao, H.M. Srivastava, Comput. Math. Appl. 73, 203 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    X.H. Zhao, Y. Zhang, D. Zhao, X. Yang, Fundam. Inf. 151, 419 (2017)CrossRefGoogle Scholar
  7. 7.
    X.J. Yang, J.T. Machado, D. Baleanu, Rom. Rep. Phys. 69, 115 (2017)Google Scholar
  8. 8.
    X.J. Yang, F. Gao, H.M. Srivastava, Rom. Rep. Phys. 69, 113 (2017)Google Scholar
  9. 9.
    Y.M. Guo, Y. Zhao, Y.M. Zhou, Z.B. Xiao, X.J. Yang, Math. Methods Appl. Sci. 40, 6127 (2015)CrossRefGoogle Scholar
  10. 10.
    X.J. Yang, Therm. Sci. 21, 317 (2017)CrossRefGoogle Scholar
  11. 11.
    M. Ma, D. Baleanu, Y.S. Gasimov, X.J. Yang, Rom. J. Phys. 61, 784 (2016)Google Scholar
  12. 12.
    K.M. Owolabi, A. Atangana, Chaos, Solitons Fractals 99, 171 (2017)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    K.M. Owolabi, A. Atangana, J. Comput. Nonlinear Dyn. 12, 031010 (2017)CrossRefGoogle Scholar
  14. 14.
    J.D. Munkhammar, Riemann-Liouville fractional derivatives and the Taylor-Riemann series, UUDM project report, 7, 1-18 (2004)Google Scholar
  15. 15.
    C. Li, D. Qian, Y. Chen, Discr. Dyn. Nat. Soc. 2011, 562494 (2011)Google Scholar
  16. 16.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)Google Scholar
  17. 17.
    A. Atangana, D. Baleanu, Therm. Sci. 20, 763 (2016)CrossRefGoogle Scholar
  18. 18.
    S.G. Samko, B. Ross, Integral Transform. Spec. Funct. 1, 277 (1993)MathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Atangana, J. Comput. Phys. 293, 104 (2015)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    R. Almeida, Numer. Funct. Anal. Optim. 38, 1 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    D. Valério, J.S. Da Costa, Signal Process. 91, 470 (2011)CrossRefGoogle Scholar
  22. 22.
    C. Li, G. Chen, Physica A 341, 55 (2004)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    A.H. Bhrawy, M.A. Zaky, Comput. Math. Appl. 73, 1100 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    A.H. Bhrawy, M.A. Zaky, Appl. Numer. Math. 111, 197 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    H.G. Sun, W. Chen, W. Wei, Y.Q. Chen, Eur. Phys. J. ST 193, 185 (2011)CrossRefGoogle Scholar
  26. 26.
    G.R.J. Cooper, D.R. Cowan, Comput. Geosci. 30, 455 (2004)ADSCrossRefGoogle Scholar
  27. 27.
    B.P. Moghaddam, J.A.T. Machado, Comput. Math. Appl. 73, 1262 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    A. Atangana, J.F. Botha, Bound. Value Probl. 2013, 53 (2013)CrossRefGoogle Scholar
  29. 29.
    S. Yaghoobi, B.P. Moghaddam, K. Ivaz, Nonlinear Dyn. 87, 815 (2017)CrossRefGoogle Scholar
  30. 30.
    A. Atangana, R.T. Alqahtani, J. Comput. Theor. Nanosci. 13, 2710 (2016)CrossRefGoogle Scholar
  31. 31.
    B.P. Moghaddam, S. Yaghoobi, J.T. Machado, J. Comput. Nonlinear Dyn. 11, 061001 (2016)CrossRefGoogle Scholar
  32. 32.
    J.K. Hale, S.M.V. Lunel, Introduction to functional differential equations (Springer Science & Business Media, 2013)Google Scholar
  33. 33.
    V. Volterra, J. Math. Pures Appl. 7, 249 (1928)Google Scholar
  34. 34.
    K.L. Cooke, J.A. Yorke, Equations modelling population growth, economic growth, and gonorrhea epidemiology, in Ordinary Differential Equations, edited by L. Weiss (Academic Press, New York, 1972)Google Scholar
  35. 35.
    W.C. Chen, Chaos, Solitons Fractals 36, 1305 (2008)ADSCrossRefGoogle Scholar
  36. 36.
    M.P. Lazarevic, Mech. Res. Commun. 33, 269 (2006)CrossRefGoogle Scholar
  37. 37.
    S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, R. Magin, Comput. Math. Appl. 61, 1355 (2011)MathSciNetCrossRefGoogle Scholar
  38. 38.
    A. Lin, Y. Ren, N. Xia, Math. Comput. Model. 51, 413 (2010)CrossRefGoogle Scholar
  39. 39.
    D. Baleanu, T. Maaraba, F. Jarad, J. Phys. 41, 315403 (2008)MathSciNetGoogle Scholar
  40. 40.
    S. Abbas, R.P. Agarwal, M. Benchohra, Nonlinear Anal. Hybrid Syst. 4, 818 (2010)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Z.M. Odibat, S. Momani, J. Appl. Math. Inf. 26, 15 (2008)Google Scholar
  42. 42.
    S. Irandoust-Pakchin, M. Javidi, H. Kheiri, Comput. Math. Math. Phys. 56, 116 (2016)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Y. Zhang, C. Cattani, X.J. Yang, Entropy 17, 6753 (2015)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 59, 1326 (2010)MathSciNetCrossRefGoogle Scholar
  45. 45.
    S. Ma, Y. Xu, W. Yue, J. Appl. Math. 2012, 417942 (2012)Google Scholar
  46. 46.
    U. Saeed, M.U. Rehman, J. Differ. Equ. 2014, 359093 (2014)Google Scholar
  47. 47.
    Y. Yang, Y. Huang, Adv. Math. Phys. 2013, 821327 (2013)Google Scholar
  48. 48.
    M.M. Khader, A.S. Hendy, Int. J. Pure Appl. Math. 74, 287 (2012)Google Scholar
  49. 49.
    Z. Wang, J. Appl. Math. 2013, 256071 (2013)Google Scholar
  50. 50.
    Z. Li, Y. Yan, N.J. Ford, Appl. Numer. Math. 114, 201 (2017)MathSciNetCrossRefGoogle Scholar
  51. 51.
    B.P. Moghaddam, Z.S. Mostaghim, J. Taibah Univ. Sci. 7, 120 (2013)CrossRefGoogle Scholar
  52. 52.
    B.P. Moghaddam, S. Yaghoobi, J.T. Machado, J. Comput. Nonlinear Dyn. 11, 061001 (2016)CrossRefGoogle Scholar
  53. 53.
    H.M. Romero-Ugalde, C. Corbier, J. Dyn. Syst. Meas. Control 138, 051001 (2016)CrossRefGoogle Scholar
  54. 54.
    G. Cybenko, Math. Control, Signals Syst. 2, 303 (1989)CrossRefGoogle Scholar
  55. 55.
    G.B. Huang, Q.Y. Zhu, C.K. Siew, Neurocomputing 70, 489 (2006)CrossRefGoogle Scholar
  56. 56.
    H.M. Romero-Ugalde, J.C. Carmona, J. Reyes-Reyes, V.M. Alvarado, C. Corbier, Neural Comput. Appl. 26, 171 (2015)CrossRefGoogle Scholar
  57. 57.
    H.M. Romero-Ugalde, J.C. Carmona, V.M. Alvarado, J. Reyes-Reyes, Neurocomputing 101, 170 (2013)CrossRefGoogle Scholar
  58. 58.
    T. Das, I.N. Kar, IEEE Trans. Control Syst. Technol. 14, 501 (2006)CrossRefGoogle Scholar
  59. 59.
    M. Chen, S.S. Ge, B.V.E. How, IEEE Trans. Neural Netw. 21, 796 (2010)CrossRefGoogle Scholar
  60. 60.
    A. Krizhevsky, I. Sutskever, G.E. Hinton, Imagenet classification with deep convolutional neural networks, in Advances in neural information processing systems, Vol. 1 (2012) pp. 1097--1105Google Scholar
  61. 61.
    D. Ciregan, U. Meier, J. Schmidhuber, Multi-column deep neural networks for image classification, in Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference (2012) pp. 3642--3649Google Scholar
  62. 62.
    H. Qu, X. Liu, Adv. Math. Phys. 2015, 439526 (2015)CrossRefGoogle Scholar
  63. 63.
    A. Jafarian, M. Mokhtarpour, D. Baleanu, Neural Comput. Appl. 28, 765 (2017)CrossRefGoogle Scholar
  64. 64.
    M.A.Z. Raja, M.A. Manzar, R. Samar, Appl. Math. Model. 39, 3075 (2015)MathSciNetCrossRefGoogle Scholar
  65. 65.
    M. Pakdaman, A. Ahmadian, S. Effati, S. Salahshour, D. Baleanu, Appl. Math. Comput. 293, 81 (2017)MathSciNetCrossRefGoogle Scholar
  66. 66.
    B.P. Moghaddam, Z.S. Mostaghim, J. Taibah Univ. Sci. 7, 120 (2013)CrossRefGoogle Scholar
  67. 67.
    L. Tavernini, Continuous-Time modeling and simulation (Gordon and Breach, Amsterdam, 1996)Google Scholar
  68. 68.
    I. Petras, Fractional-order nonlinear systems: modeling, analysis and simulation (Springer Science & Business Media, 2011)Google Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Tecnológico Nacional de México/CENIDETInterior Internado Palmira S/N, Col. PalmiraCuernavacaMexico
  2. 2.CONACyT-Tecnológico Nacional de México/CENIDETInterior Internado Palmira S/N, Col. PalmiraCuernavacaMexico
  3. 3.Univ. Grenoble AlpesGrenobleFrance
  4. 4.CEA LETI MINATEC CampusGrenobleFrance

Personalised recommendations