The Homotopy Analysis Method for Approximating of Giving Up Smoking Model in Fractional Order

Affiliation(s)

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

Buraimi University College, Department of Business Administration and Accounting, Al-Buraimi, Oman.

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

Buraimi University College, Department of Business Administration and Accounting, Al-Buraimi, Oman.

ABSTRACT

In this paper, we consider the giving up smoking model. First, we present the giving up smoking model in fractional order. Then the homotopy analysis method (HAM) is employed to compute an approximate and analytical solution of the model in fractional order. The obtained results are compaired with those obtained by forth order Runge-Kutta method and nonstandard numerical method in the integer case. Finally, we present some numerical results.

In this paper, we consider the giving up smoking model. First, we present the giving up smoking model in fractional order. Then the homotopy analysis method (HAM) is employed to compute an approximate and analytical solution of the model in fractional order. The obtained results are compaired with those obtained by forth order Runge-Kutta method and nonstandard numerical method in the integer case. Finally, we present some numerical results.

Cite this paper

A. Zeb, M. Chohan and G. Zaman, "The Homotopy Analysis Method for Approximating of Giving Up Smoking Model in Fractional Order,"*Applied Mathematics*, Vol. 3 No. 8, 2012, pp. 914-919. doi: 10.4236/am.2012.38136.

A. Zeb, M. Chohan and G. Zaman, "The Homotopy Analysis Method for Approximating of Giving Up Smoking Model in Fractional Order,"

References

[1] I. Podlubny, “Fractional Differential Equations,” Academic Press, London, 1999.

[2] L. Debnath, “Recent Applications of Fractional Calculus to Science and Engineering,” International Journal of Mathematics and Mathematical Sciences, Vol. 54, 2003, pp. 3413-3442.

[3] S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” Willey, New York, 1993.

[4] G. Zaman, “Qualitative Behavior of Giving Up Smoking Models,” Bulletin of the Malaysian Mathematical Sciences Society, Vol. 34, 2011, pp. 403-415.

[5] S. J. Liao, “Notes on the Homotopy Analysis Method: Some Definitions and Theorems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 4, 2009, pp. 983-997. doi:10.1016/j.cnsns.2008.04.013

[6] S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[7] M. Zurigat, S. Momani, Z. Odibat and A. Alawneh, “The Homotopy Analysis Method for Handling Systems of Fractional Differential Equations,” Applied Mathematical Modelling, Vol. 34, No. 1, 2010, pp. 24-35. doi:10.1016/j.apm.2009.03.024

[8] M. Zurigat, S. Momani and A. Alawneh, “Analytical Approximate Solutions of Systems of Fractional Algebraic-Differential Equations by Homotopy Analysis Method,” Computers & Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 1227-1235. doi:10.1016/j.camwa.2009.07.002

[9] S. Momani and Z. Odibat, “Homotopy Perturbation Method for Nonlinearl Partial Differential Equations,” Physics Letters A, Vol. 365, No. 5-6, 2007, pp. 345-350. doi:10.1016/j.physleta.2007.01.046

[10] Z. Odibat, S. Momani and H. Xu, “A Reliable Algorithm of Homotopy Analysis Method for Solving Nonlinear Fractional Differential Equations,” Applied Mathematical Modelling, Vol. 34, No. 3, 2010, pp. 593-600. doi:10.1016/j.apm.2009.06.025

[11] S. Momani, V. S. Erturk and G. Zaman, “An Approximate Solution of a Giving Up Smoking Model in Fractional Order,” Computers & Mathematics with Applications, 2012, in press.

[1] I. Podlubny, “Fractional Differential Equations,” Academic Press, London, 1999.

[2] L. Debnath, “Recent Applications of Fractional Calculus to Science and Engineering,” International Journal of Mathematics and Mathematical Sciences, Vol. 54, 2003, pp. 3413-3442.

[3] S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” Willey, New York, 1993.

[4] G. Zaman, “Qualitative Behavior of Giving Up Smoking Models,” Bulletin of the Malaysian Mathematical Sciences Society, Vol. 34, 2011, pp. 403-415.

[5] S. J. Liao, “Notes on the Homotopy Analysis Method: Some Definitions and Theorems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 4, 2009, pp. 983-997. doi:10.1016/j.cnsns.2008.04.013

[6] S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[7] M. Zurigat, S. Momani, Z. Odibat and A. Alawneh, “The Homotopy Analysis Method for Handling Systems of Fractional Differential Equations,” Applied Mathematical Modelling, Vol. 34, No. 1, 2010, pp. 24-35. doi:10.1016/j.apm.2009.03.024

[8] M. Zurigat, S. Momani and A. Alawneh, “Analytical Approximate Solutions of Systems of Fractional Algebraic-Differential Equations by Homotopy Analysis Method,” Computers & Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 1227-1235. doi:10.1016/j.camwa.2009.07.002

[9] S. Momani and Z. Odibat, “Homotopy Perturbation Method for Nonlinearl Partial Differential Equations,” Physics Letters A, Vol. 365, No. 5-6, 2007, pp. 345-350. doi:10.1016/j.physleta.2007.01.046

[10] Z. Odibat, S. Momani and H. Xu, “A Reliable Algorithm of Homotopy Analysis Method for Solving Nonlinear Fractional Differential Equations,” Applied Mathematical Modelling, Vol. 34, No. 3, 2010, pp. 593-600. doi:10.1016/j.apm.2009.06.025

[11] S. Momani, V. S. Erturk and G. Zaman, “An Approximate Solution of a Giving Up Smoking Model in Fractional Order,” Computers & Mathematics with Applications, 2012, in press.