Advertisement

Design of cascade artificial neural networks optimized with the memetic computing paradigm for solving the nonlinear Bratu system

  • Aisarul Hassan
  • Siraj-ul-Islam AhmadEmail author
  • Muhammad Kamran
  • Ahsan Illahi
  • Raja Muhammad Asif Zahoor
Regular Article
  • 33 Downloads

Abstract.

A nature-inspired, integrated computational heuristic paradigm is developed for the piecewise solution of the nonlinear Bratu problem arising in fuel ignition model, electrically conducting solids and related fields, by exploiting the strength of Cascade Artificial Neural Networks (CANN) modeling, optimized with the memetic computing procedure based on global search efficacy of genetic algorithms (GAs), aided with the efficient local search of teaching learning based optimization (TLBO). The proposed technique incorporates the log-sigmoid activation function in the CANN model, trained by GAs hybridized with TLBO, i.e., CANN-GA-TLBO. As a first application of CANN-GA-TLBO, 1D nonlinear Bratu's system represented with a boundary value problem of the second-order ordinary different equation has been solved, which is a benchmark for testing new algorithms. Comparison of the results with exact solution and previously reported solutions, including Adomian decomposition method, Laplace transformed decomposition method, B-Spline method and artificial neural network solutions, confirms the superiority of the designed stochastic solver CANN-GA-TLBO in terms of accuracy and convergence measures.

References

  1. 1.
    W.S. McCulloch, W. Pitts, Bull. Math. Biophy. 5, 115 (1943)CrossRefGoogle Scholar
  2. 2.
    V. Papadopoulos, G. Soimiris, D.G. Giovanis, M. Papadrakakis, Comput. Methods Appl. Mech. Eng. 328, 411 (2018)ADSCrossRefGoogle Scholar
  3. 3.
    Z. Sabir et al., Appl. Soft Comput. 65, 152 (2018)CrossRefGoogle Scholar
  4. 4.
    I. Ahmad et al., Springer Plus 5, 1866 (2016)CrossRefGoogle Scholar
  5. 5.
    M.A.Z. Raja, M.A. Manzar, F.H. Shah, F.H. Shah, Appl. Soft Comput. 62, 359 (2018)CrossRefGoogle Scholar
  6. 6.
    I. Ahmad et al., Eur. Phys. J. Plus 133, 184 (2018)CrossRefGoogle Scholar
  7. 7.
    I. Ahmad et al., Neural Comput. Appl. 28, 929 (2017)CrossRefGoogle Scholar
  8. 8.
    J.A. Khan et al., Connect. Sci. 27, 377 (2015)ADSCrossRefGoogle Scholar
  9. 9.
    M.A.Z. Raja, Connect. Sci. 26, 195 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    A. Mehmood et al., J. Taiwan Inst. Chem. Eng. 91, 57 (2018)CrossRefGoogle Scholar
  11. 11.
    M.A.Z. Raja, M. Umar, Z. Sabir, J.A. Khan, D. Baleanu, Eur. Phys. J. Plus 133, 364 (2018)CrossRefGoogle Scholar
  12. 12.
    A. Ara et al., Adv. Differ. Equ. 2018, 8 (2018)CrossRefGoogle Scholar
  13. 13.
    C.J. Zúñiga-Aguilar, A. Coronel-Escamilla, J.F. Gómez-Aguilar, V.M. Alvarado-Martínez, H.M. Romero-Ugalde, Eur. Phys. J. Plus 133, 75 (2018)CrossRefGoogle Scholar
  14. 14.
    M.A.Z. Raja, M.A. Manzar, R. Samar, Appl. Math. Model. 39, 3075 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    S.Y. Chen, F. Zheng, S.Q. Wu, Z.Z. Zhu, Curr. Appl. Phys. 17, 454 (2017)ADSCrossRefGoogle Scholar
  16. 16.
    W. Zang, L. Ren, W. Zhang, X. Liu, Fut. Gen. Comput. Syst. 81, 465 (2018)CrossRefGoogle Scholar
  17. 17.
    A.R. Hosseinzadeh, A.H. Mahmoudi, Mech. Mater. 114, 57 (2017)CrossRefGoogle Scholar
  18. 18.
    N. Shaukat, A. Majeed, N. Ahmad, B. Mohsin, Nucl. Eng. Design 240, 2831 (2010)CrossRefGoogle Scholar
  19. 19.
    M.A.Z. Raja, Z. Shah, M.A. Manzar, I. Ahmad, M. Awais, D. Baleanu, Eur. Phys. J. Plus 133, 254 (2018)CrossRefGoogle Scholar
  20. 20.
    H.H. Chen, Y.C. Lee, C.S. Liu, Phys. Scr. 20, 490 (1979)ADSCrossRefGoogle Scholar
  21. 21.
    B. Chen, R. García-Bolós, L. Jódar, M.D. Roselló, Nonlinear Anal. 63, e629 (2005)CrossRefGoogle Scholar
  22. 22.
    J.H. He, H.Y. Kong, R.X. Chen, M.S. Hu, Q.L. Chen, Carbohydr. Polym. 105, 229 (2014)CrossRefGoogle Scholar
  23. 23.
    A. Hasseine, H.J. Bart, Appl. Math. Model. 39, 1975 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    M.C. Devi, L. Rajendran, A.B. Yousaf, C. Fernandez, Electrochim. Acta 243, 1 (2017)CrossRefGoogle Scholar
  25. 25.
    M. Lakestani, M. Dehghan, Comput. Phys. Commun. 181, 957 (2010)ADSCrossRefGoogle Scholar
  26. 26.
    A. Heydari, M. Mirparizi, F. Shakeriaski, F.S. Samani, M. Keshavarzi, Propulsion Power Res. 6, 223 (2017)CrossRefGoogle Scholar
  27. 27.
    A. Bouharguane, J. Comput. Appl. Math. 328, 497 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    B. Sepehrian, M.K. Radpoor, Appl. Math. Comput. 262, 187 (2015)MathSciNetGoogle Scholar
  29. 29.
    M. Al-Smadi, O.A. Arqub, Appl. Math. Comput. 342, 280 (2019)MathSciNetGoogle Scholar
  30. 30.
    O.A. Arqub, M. Al-Smadi, Chaos, Solitons Fractals 117, 161 (2018)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    O.A. Arqub, Z. Odibat, M. Al-Smadi, Nonlinear Dyn. 94, 1819 (2018)CrossRefGoogle Scholar
  32. 32.
    A.M. Wazwaz, Rom. J. Phys. 61, 774 (2016)Google Scholar
  33. 33.
    R. Saleh, S.M. Mabrouk, M. Kassem, Comput. Math. Appl. 76, 1219 (2018)MathSciNetCrossRefGoogle Scholar
  34. 34.
    M. Grover, A.K. Tomer, Global J. Pure Appl. Math. 13, 5813 (2017)Google Scholar
  35. 35.
    E. Keshavarz, Y. Ordokhani, M. Razzaghi, Appl. Numer. Math. 128, 205 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Z. Yang, S. Liao, Commun. Nonlinear Sci. Numer. Simul. 53, 249 (2017)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    O. Ragb, L.F. Seddek, M.S. Matbuly, Comput. Math. Appl. 74, 249 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    H. Temimi, M. Ben-Romdhane, J. Comput. Appl. Math. 292, 76 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Z. Altawallbeh, M. Al-Smadi, I. Komashynska, A. Ateiwi, Ukr. Math. J. 70, 687 (2018)CrossRefGoogle Scholar
  40. 40.
    N. Das, R. Singh, A.M. Wazwaz, J. Kumar, J. Math. Chem. 54, 527 (2016)MathSciNetCrossRefGoogle Scholar
  41. 41.
    M.A.Z. Raja, Z. Shah, M.A. Manzar, I. Ahmad, M. Awais, D. Baleanu, Eur. Phys. J. Plus 133, 254 (2018)CrossRefGoogle Scholar
  42. 42.
    R.G. Peyvandi, S.I. Rad, Eur. Phys. J. Plus 132, 511 (2017)CrossRefGoogle Scholar
  43. 43.
    A. Mehmood et al., Appl. Soft Comput. 67, 8 (2018)CrossRefGoogle Scholar
  44. 44.
    K. Majeed et al., Appl. Soft Comput. 56, 420 (2017)CrossRefGoogle Scholar
  45. 45.
    I. Ahmad et al., Neural Comput. Appl. 29, 449 (2018)CrossRefGoogle Scholar
  46. 46.
    M.A.Z. Raja, F.H. Shah, M.I. Syam, Neural Comput. Appl. 30, 3651 (2018)CrossRefGoogle Scholar
  47. 47.
    I.A.H. Hassan, V.S. Ertürk, Int. J. Contemp. Math. Sci. 2, 1493 (2007)MathSciNetCrossRefGoogle Scholar
  48. 48.
    M.R. Ali, A.R. Hadhoud, Results Phys. 12, 525 (2019)ADSCrossRefGoogle Scholar
  49. 49.
    P. Roul, K. Thula, Int. J. Comput. Math. 96, 85 (2019)MathSciNetCrossRefGoogle Scholar
  50. 50.
    J.H. He, H.Y. Kong, R.X. Chen, M.S. Hu, Q.L. Chen, Carbohydr. Polym. 105, 229 (2014)CrossRefGoogle Scholar
  51. 51.
    S. Hichar, A. Guerfi, S. Douis, M.T. Meftah, Rep. Math. Phys. 76, 283 (2015)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    M.A.Z. Raja, R. Samar, E.S. Alaidarous, E. Shivanian, Appl. Math. Model. 40, 5964 (2016)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Z. Masood et al., Neurocomputing 221, 1 (2017)CrossRefGoogle Scholar
  54. 54.
    S. Chanillo, M. Kiessling, Commun. Math. Phys. 160, 217 (1994)ADSCrossRefGoogle Scholar
  55. 55.
    M.M. Mousa, Brit. J. Math. Comput. Sci. 5, 515 (2015)CrossRefGoogle Scholar
  56. 56.
    M.A.Z. Raja, S.I. Ahman, R. Samar, Neural Comput. Appl. 25, 1723 (2014)CrossRefGoogle Scholar
  57. 57.
    M.A.Z. Raja, R. Samar, M.M. Rashidi, Neural Comput. Appl. 25, 1585 (2014)CrossRefGoogle Scholar
  58. 58.
    Z. Abo-Hammour, O. Abu Arqub, S. Momani, N. Shawagfeh, Discr. Dyn. Nat. Soc. 2014, 401696 (2014)Google Scholar
  59. 59.
    M.A.Z. Raja, Neural Comput. Appl. 24, 549 (2014)CrossRefGoogle Scholar
  60. 60.
    H. Caglar, N. Caglar, M. Özer, A. Valaristos, A.N. Anagnostopoulos, Int. J. Comput. Math. 87, 1885 (2010)MathSciNetCrossRefGoogle Scholar
  61. 61.
    A.M. Wazwaz, Appl. Math. Comput. 166, 652 (2005)MathSciNetGoogle Scholar
  62. 62.
    E. Deeba, S.A. Khuri, S. Xie, J. Comput. Phys. 159, 125 (2000)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    S.A. Khuri, Appl. Math. Comput. 147, 131 (2004)MathSciNetGoogle Scholar
  64. 64.
    M. Kumar, N. Yadav, Natl. Acad. Sci. Lett. 38, 425 (2015)CrossRefGoogle Scholar
  65. 65.
    A. Başhan, Y. Uçar, N.M. Yağmurlu, A. Esen, Eur. Phys. J. Plus 133, 12 (2018)CrossRefGoogle Scholar
  66. 66.
    N. Ahmed, S. Bibi, U. Khan, S.T. Mohyud-Din, Eur. Phys. J. Plus 133, 45 (2018)CrossRefGoogle Scholar
  67. 67.
    E. Fendzi-Donfack, J.P. Nguenang, L. Nana, Eur. Phys. J. Plus 133, 32 (2018)CrossRefGoogle Scholar
  68. 68.
    M. Inc, A.I. Aliyu, A. Yusuf, D. Baleanu, J. Mod. Opt. 66, 647 (2019)ADSCrossRefGoogle Scholar
  69. 69.
    A. Yusuf, S. Qureshi, M. Inc, A.I. Aliyu, D. Baleanu, A.A. Shaikh, Chaos 28, 123121 (2018)ADSMathSciNetCrossRefGoogle Scholar
  70. 70.
    A.I. Aliyu, A. Yusuf, D. Baleanu, Commun. Theor. Phys. 70, 511 (2018)ADSCrossRefGoogle Scholar
  71. 71.
    H.I. Abdel-Gawad, M. Tantawy, M. Inc, A. Yusuf, Mod. Phys. Lett. B 32, 1850353 (2018)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Aisarul Hassan
    • 1
  • Siraj-ul-Islam Ahmad
    • 1
    Email author
  • Muhammad Kamran
    • 1
  • Ahsan Illahi
    • 1
  • Raja Muhammad Asif Zahoor
    • 2
  1. 1.Research in Modeling and Simulation (RIMS) Group, Department of PhysicsCOMSATS University IslamabadIslamabadPakistan
  2. 2.Department of Electrical and Computer EngineeringCOMSATS University Islamabad, Attock CampusAttockPakistan

Personalised recommendations