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Comparing Two Numerical Methods for Approximating a New Giving Up Smoking Model Involving Fractional Order Derivatives

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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.

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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

    Article  MATH  MathSciNet  Google Scholar 

  • Brauer F, Castillo-Chávez C (2001) Mathematical models in population biology and epidemiology. Springer, Berlin

    Book  MATH  Google Scholar 

  • 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

    Book  MATH  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Kurulay M, Bayram M (2010) Approximate analytical solution for the fractional modified KdV by differential transform method. Commun Nonlinear Sci 15(7):777–1782

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Lin W (2007) Global existence theory and chaos control of fractional differential equations. J Math Anal Appl 332(1):709–726

    Article  MATH  MathSciNet  Google Scholar 

  • Magin RL (2006) Fractional calculus in bioengineering. Begell House Publishers, Redding

    Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Google Scholar 

  • 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

    MATH  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Odibat ZM, Shawagfeh NT (2007) Generalized Taylor’s formula. Appl Math Comput 186(1):286–293

    MATH  MathSciNet  Google Scholar 

  • 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

    MATH  MathSciNet  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, New York

    MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Yıldırım A, Koçak H (2009) Homotopy perturbation method for solving the space–time fractional advection–dispersion equation. Adv Water Resour 3(12):1711–1716

    Google Scholar 

  • Zaman G (2011) Qualitative behavior of giving up smoking models. Bull Malays Math Sci Soc 34(2):403–415

    MATH  MathSciNet  Google Scholar 

  • 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

    Article  Google Scholar 

  • Zeb A, Zaman G, Momani S (2013) Square-root dynamics of a giving up smoking model. Appl Math Model 37(7):5326–5334

    Article  MathSciNet  Google Scholar 

  • Zhang T (1999) Role of peer pressure of smoking as an epidemic, Thesis

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Authors and Affiliations

  1. Department of Mathematics, Ondokuz Mayis University, Samsun, Turkey

    Vedat Suat Erturk

  2. Department of Mathematics, University of Malakand, Chakdara Dir (Lower), Khyber Pakhtunkhwa, Pakistan

    Gul Zaman

  3. Department of Mathematics, The University of Jordan, Amman, Jordan

    Baha Alzalg & Shaher Momani

  4. Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Khyber Pakhtunkhwa, Pakistan

    Anwar Zeb

Authors
  1. Vedat Suat Erturk
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  2. Gul Zaman
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  3. Baha Alzalg
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  4. Anwar Zeb
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  5. Shaher Momani
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Correspondence to Vedat Suat Erturk.

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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

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  • DOI: https://doi.org/10.1007/s40995-017-0278-x

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