zbMATH — the first resource for mathematics

A numeric-analytic method for approximating a giving up smoking model containing fractional derivatives. (English) Zbl 1268.65107
Summary: 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, the dynamics of a giving up smoking model containing fractional derivatives is studied numerically. The multistep generalized differential transform method (for short MSGDTM) is employed to compute accurate approximate solutions to a giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. 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.

65L99 Numerical methods for ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
92C60 Medical epidemiology
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
Full Text: DOI
[1] World Health Organization report on the global tobacco epidemic, 2009. http://whqlibdoc.who.int/publications/2009/9789241563918_eng_full.pdf.
[2] Lahrouz, A.; Omari, L.; Kiouach, D.; Belmaâti, A., Deterministic and stochastic stability of a mathematical model of smoking, Statistics &probability letters, 81, 8, 1276-1284, (2011) · Zbl 1219.92043
[3] J.S. Leszczynski, An Introduction to Fractional Mechanics, Czestochowa University of Technology, Czestochowa, 2011.
[4] Magin, R.L., Fractional calculus in bioengineering, (2006), Begell House Inc. Redding
[5] Craiem, D.O.; Rojo, F.J.; Atienza, J.M.; Guinea, G.V.; Armentano, R.L., Fractional calculus applied to model arterial viscoelasticity, Latin American applied research, 38, 2, 141-145, (2008)
[6] Abdullah, F.A., Using fractional differentialequations to model the michaelis – menten reaction in a 2-d region containing obstacles, Science Asia, 37, 1, 75-78, (2011)
[7] Zaman, G., Optimal campaign in the smoking dynamics, Computational and mathematical methods in medicine, 2011, (2011), Article ID 163834, 9 pages · Zbl 1207.92051
[8] Zaman, G., Qualitative behavior of giving up smoking models, Bulletin of the Malaysian mathematical sciences society, 34, 2, 403-415, (2011) · Zbl 1221.92067
[9] Lubin, J.H.; Caporaso, N.E., Cigarette smoking, and lung cancer: modeling total exposure and intensity, Cancer epidemiology, biomarkers & prevention, 15, 3, 517-523, (2006)
[10] C. Castillo-Garsow, G. Jordan-Salivia, A. Rodriguez Herrera, Mathematical models for the dynamics of tobacco use, recovery and relapse, Technical Report Series BU-1505-M, Cornell University, 2000.
[11] Sharomi, O.; Gumel, A.B., Curtailing smoking dynamics: a mathematical modeling approach, Applied mathematics and computation, 195, 2, 475-499, (2008) · Zbl 1261.92023
[12] Zaman, G.; Islam, S., A non-standard numerical method for a giving-up smoking model, Nonlinear science letters A, 1, 4, 397-402, (2010)
[13] Odibat, Z.; Momani, S.; Ertürk, V.S., Generalized differential transform method: application to differential equations of fractional order, Applied mathematics and computation, 197, 2, 467-477, (2008) · Zbl 1141.65092
[14] Momani, S.; Odibat, Z., A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized taylor’s formula, Journal of computational and applied mathematics, 220, 1-2, 85-95, (2008) · Zbl 1148.65099
[15] Odibat, Z.; Momani, S., A generalized differential transform method for linear partial differential equations of fractional order, Applied mathematics letters, 21, 2, 194-199, (2008) · Zbl 1132.35302
[16] Ertürk, V.S.; Momani, S.; Odibat, Z., Application of generalized differential transform method to multi-order fractional differential equations, Communications in nonlinear science and numerical simulation, 13, 8, 1642-1654, (2008) · Zbl 1221.34022
[17] Odibat, Z.; Bertelle, C.; Aziz-Alaoui, M.A.; Duchamp, G., A multi-step differential transform method and application to non-chaotic or chaotic systems, Computers & mathematics with applications, 59, 4, 1462-1472, (2010) · Zbl 1189.65170
[18] Erturk, V.; Odibat, Z.; Momani, S., An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells, Computers & mathematics with applications, 62, 3, 992-1002, (2011) · Zbl 1228.92064
[19] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[20] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Amsterdam · Zbl 1092.45003
[21] Das, S., Functional fractional calculus, (2011), Springer · Zbl 1225.26007
[22] Magin, R.L., Fractional calculus in bioengineering, (2006), Begell House Publishers
[23] Odibat, Z.M.; Shawagfeh, N.T., Generalized taylor’s formula, Applied mathematics and computation, 186, 1, 286-293, (2007) · Zbl 1122.26006
[24] Lin, W., Global existence theory and chaos control of fractional differential equations, Journal of mathematical analysis and applications, 332, 1, 709-726, (2007) · Zbl 1113.37016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.