Abstract
In a recent paper (Zeb et al. in Appl Math Model 37(7):5326–5334, 2013), the authors presented a new model of giving up smoking model. In the present paper, the dynamics of this new model involving the Caputo derivative was studied numerically. For this purpose, generalized Euler method and the multistep generalized differential transform method are employed to compute accurate approximate solutions to this new giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between these two methods and the well-known Runge–Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.
Similar content being viewed by others
References
Alomari AK (2011) A new analytic solution for fractional chaotic dynamical systems using the differential transform method. Comput Math Appl 61(9):2528–2534
Brauer F, Castillo-Chávez C (2001) Mathematical models in population biology and epidemiology. Springer, Berlin
Choi H, Jung I, Kang Y (2011) Giving up smoking dynamic on adolescent nicotine dependence: a mathematical modeling approach. In: KSIAM 2011 spring conference, Daejeon, Korea, 27–28 May 2011
Das S (2011) Functional fractional calculus. Springer, Berlin
Ertürk VS, Momani S, Odibat Z (2008) Application of generalized differential transform method to multi-order fractional differential equations. Commun Nonlinear Sci 13(8):642–1654
Ertürk VS, Zaman G, Momani S (2012) A numeric–analytic method for approximating a giving up smoking model containing fractional derivatives. Comput Math Appl 64(10):3065–3074
Gökdoğan A, Yildirim A, Merdan M (2011) Solving a fractional order model of HIV infection of CD4+ T cells. Math Comput Model 54(9–10):2132–2138
Guerrero F, Santonja FJ, Villanueva RJ (2013) Solving a model for the evolution of smoking habit in Spain with homotopy analysis method. Nonlinear Anal: Real World Appl 14(1):549–558
Guindon GE, Boisclair D (2003) Current and future trends in tobacco use. Health, Nutrition and Population (HNP). Discussion Paper No 6, World Bank, Washington, DC
Jafari H, Daftardar-Gejji V (2006) Solving a system of nonlinear fractional differential equations using Adomian decomposition. J Comput Appl Math 196(2):644–651
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam
Kurulay M, Bayram M (2010) Approximate analytical solution for the fractional modified KdV by differential transform method. Commun Nonlinear Sci 15(7):777–1782
Lahrouz A, Omari L, Kiouach D, Belmaâti A (2011) Deterministic and stochastic stability of a mathematical model of smoking. Stat Probab Lett 81(8):1276–1284
Lin W (2007) Global existence theory and chaos control of fractional differential equations. J Math Anal Appl 332(1):709–726
Magin RL (2006) Fractional calculus in bioengineering. Begell House Publishers, Redding
Momani S, Odibat Z (2008) A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula. J Comput Appl Math 220(1–2):85–95
Ockene IS, Miller NH (1997) Cigarette smoking, cardiovascular disease, and stroke: a statement for healthcare professionals from the American Heart Association. Circulation 96(32):43–47
Odibat Z, Momani S (2008a) An algorithm for the numerical solution of differential equations of fractional order. Appl Math Inf Sci 26(1–2):15–27
Odibat Z, Momani S (2008b) A generalized differential transform method for linear partial differential equations of fractional order. Appl Math Lett 21(2):194–199
Odibat ZM, Shawagfeh NT (2007) Generalized Taylor’s formula. Appl Math Comput 186(1):286–293
Odibat Z, Momani S, Ertürk VS (2008) Generalized differential transform method: application to differential equations of fractional order. Appl Math Comput 197(2):467–477
Podlubny I (1999) Fractional differential equations. Academic Press, New York
Swartz JB (1992) Use of a multistage model to predict time trends in smoking induced lung cancer. J Epidemiol Community Health 46(3):11–31
Yıldırım A, Koçak H (2009) Homotopy perturbation method for solving the space–time fractional advection–dispersion equation. Adv Water Resour 3(12):1711–1716
Zaman G (2011) Qualitative behavior of giving up smoking models. Bull Malays Math Sci Soc 34(2):403–415
Zeb A, Chohan M, Zaman G (2012) The homotopy analysis method for approximating of giving up smoking model in fractional order. Appl Math 3(8):914–919
Zeb A, Zaman G, Momani S (2013) Square-root dynamics of a giving up smoking model. Appl Math Model 37(7):5326–5334
Zhang T (1999) Role of peer pressure of smoking as an epidemic, Thesis
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Erturk, V.S., Zaman, G., Alzalg, B. et al. Comparing Two Numerical Methods for Approximating a New Giving Up Smoking Model Involving Fractional Order Derivatives. Iran J Sci Technol Trans Sci 41, 569–575 (2017). https://doi.org/10.1007/s40995-017-0278-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-017-0278-x