Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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The Multi-step Adomian Decomposition Method for Approximating a Fractional Smoking Habit Model

  • Zuriqat, Mohammad (Department of Mathematics, Al al-Bayt University) ;
  • Freihat, Asad (Department of Applied Science, Ajloun College, Al-Balqa Applied University)
  • Received : 2015.06.10
  • Accepted : 2019.06.21
  • Published : 2020.12.31
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Abstract

Smoking is one of the main causes of health problems and continues to be one of the world's most significant health challenges. In this paper, we use the multi-step Adomian decomposition method (MSADM) to obtain approximate analytical solutions for a mathematical fractional model of the evolution of the smoking habit. The proposed MSADM scheme is only a simple modification of the Adomian decomposition method (ADM), in which ADM is treated algorithmically with a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding problems. A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically. The results reveal that the method is effective and convenient for solving linear and nonlinear differential equations of fractional order.

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References

  1. K. Abboui and Y. Cherruault, New ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29(7)(1995), 103-108.
  2. F. M. Allan, Construction of analytic solution to chaotic dynamical systems using the homotopy analysis method, Chaos, Solitons & Fractals, 39(4)(2009), 1744-1752. https://doi.org/10.1016/j.chaos.2007.06.116
  3. G. Barro, O. So, J. M. Ntaganda, B. Mampassi and B. Some, A numerical method for some nonlinear differential equation models in biology, Appl. Math. Comput., 200(1)(2008), 28-33. https://doi.org/10.1016/j.amc.2007.10.041
  4. C. Castillo-Garsow, G. Jordan-Salivia and A. Rodriguez Herrera, Mathematical models for the dynamics of tobacco use, recovery and relapse, Technical Report Series BU-1505-M, Cornell University, Ithaca, NY, USA, 1997.
  5. V. Daftardar-Gejji and H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301(2)(2005), 508-518. https://doi.org/10.1016/j.jmaa.2004.07.039
  6. V. S. Erturk, S. Momani and Z. Odibat, Application of generalized differential transform method to multi-order fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 13(2008), 1642-1654. https://doi.org/10.1016/j.cnsns.2007.02.006
  7. A. Freihat and M. AL-Smadi, A new reliable algorithm using the generalized differential transform method for the numeric-analytic solution of fractional-order Liu chaotic and hyperchaotic systems, Pensee J., 75(9)(2013), 263-276.
  8. A. Freihat and S. Momani, Adaptation of differential transform method for the numeric-analytic solution of fraction-order Rössler chaotic and hyperchaotic systems, Abstr. Appl. Anal., (2012), Art. ID 934219, 13 pp.
  9. R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics (Udine, 1996), 223-276, CISM Courses and Lect., 378, Springer, Vienna, 1997.
  10. F. Guerrero, F. Santonja and R. Villanueva, Analysing the Spanish smoke-free legislation of year 2006: A new method to quantify its impact using a dynamic model, Int. J. Drug Policy, 22(2011), 247-251. https://doi.org/10.1016/j.drugpo.2011.05.003
  11. F. Guerrero, F. Santonja and R. Villanueva, Solving a model for the evolution of smoking habit in spain with homotopy analysis method, Nonlinear Anal. Real World Appl., 14(2013), 549-558. https://doi.org/10.1016/j.nonrwa.2012.07.015
  12. I. Hashim, M. S. H. Chowdhury and S. Mawa, On multistage homotopy-perturbation method applied to nonlinear biochemical reaction model, Chaos, Solitons & Fractals, 36(4)(2008), 823-827. https://doi.org/10.1016/j.chaos.2007.09.009
  13. H. Jafari and V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math., 196(2006), 644-651 https://doi.org/10.1016/j.cam.2005.10.017
  14. S. Kempfle and H. Beyer, Global and causal solutions of fractional differential equations, Proceedings of the 2nd International Workshop on Transform Methods and Special Functions, 210-216, Science Culture Technology Publishing, Varna, Bulgaria, 1996.
  15. A. Lahrouz, L. Omari, D. Kiouach and A. Belmaati, Deterministic and stochastic stability of a mathematical model of smoking, Statist. Probab. Lett., 81(2011), 1276-1284. https://doi.org/10.1016/j.spl.2011.03.029
  16. J. H. Lubin and N. E. Caporaso, Cigarette smoking, and lung cancer: modeling total exposure and intensity, Cancer Epidemiol Biomarkers Prev, 15(3)(2006), 517-523. https://doi.org/10.1158/1055-9965.EPI-05-0863
  17. F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics (Udine, 1996), 291-348, CISM Courses and Lect., 378, Springer, Vienna, 1997.
  18. K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, New York, 1993.
  19. S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A , 365(5-6)(2007), 345-350. https://doi.org/10.1016/j.physleta.2007.01.046
  20. I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
  21. N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131(2002), 517-529. https://doi.org/10.1016/S0096-3003(01)00167-9
  22. X. Y. Shi, H. Gao, V. I. Lazouskaya, Q. Kang, Y. Jin and L. P. Wang, Viscous flow and colloid transport near air-water interface in a microchannel, Comput. Math. Appl., 59(7)(2010), 2290-2304. https://doi.org/10.1016/j.camwa.2009.08.059
  23. H. Vazquez-Leal and F. Guerrero, Application of series method with Pade and Laplace-Pade resummation methods to solve a model for the evolution of smoking habit in Spain, Comput. Appl. Math., 33(1)(2014), 181-192. https://doi.org/10.1007/s40314-013-0054-2
  24. A.-M. Wazwaz, New solutions of distinct physical structures to high-dimensional nonlinear evolution equations, Appl. Math. Comput., 196(1)(2008), 363-370. https://doi.org/10.1016/j.amc.2007.06.002
  25. Y. Yongguang and H. Xiong, Application of the multistage homotopy-perturbation method to solve a class of hyperchaotic systems, Chaos, Solitons & Fractals, 42(4)(2009), 2330-2337. https://doi.org/10.1016/j.chaos.2009.03.154
  26. G. Zaman, Optimal campaign in the smoking dynamics, Comput. Math. Methods Med., (2011), Article ID 163834, 9 pp. https://doi.org/10.1080/17486700801982713
  27. M. Zurigat, S. Momani and A. Alawneh, Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method, Comput. Math. Appl., 59(3)(2010), 1227-1235. https://doi.org/10.1016/j.camwa.2009.07.002
  28. M. Zurigat, S. Momani, Z. odibat and A. Alawneh, The homotopy analysis method for handling systems of fractional differential equations, Appl. Math. Model., 34(1)(2010), 24-35. https://doi.org/10.1016/j.apm.2009.03.024