Applied Mathematics, 2012, 3, 914-919 http://dx.doi.org/10.4236/am.2012.38136 Published Online August 2012 (http://www.SciRP.org/journal/am) The Homotopy Analysis Method for Approximating of Giving Up Smoking Model in Fractional Order Anwar Zeb 1 , M. Ikhlaq Chohan 2 , Gul Zaman 1 1 Department of Mathematics, University of Malakand, Khyber Pakhtunkhawa, Pakistan 2 Department of Business Administration and Accounting, Buraimi University College, Al-Buraimi, Oman Email: gzaman@uom.edu.pk Received June 6, 2012; revised July 6, 2012; accepted July 14, 2012 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 compared with those obtained by forth order Runge-Kutta method and nonstandard numerical method in the integer case. Finally, we present some numerical results. Keywords: Fractional Differential Equations; Epidemic Model; Homotopy Analysis Method 1. Introduction The common Calculus has been studied well and its ap- plications can be encountered in several areas of science and engineering. Relating to fractional Calculus, it is not familiar to several researchers. Indeed, fractional Calcu- lus is a three centuries old mathematical tools. But the searching of the theory of differential Equations of frac- tional order has just been began quite recently [1-3]. An expanding of fractional notions in Biomathematics has also been improved. Fairly, no field of standard analysis has been left unconcerned by fractional Calculus. Smok- ing is one of the most important health problems in the world and it infentend different organ of human body which cover many death in all over the world. Smoking is dangerous to people health even only for a short term period. The effects of short smoking are bad breath, stained teeth, smell of smoke in the fingers and hair. Other effects on a temporary basis are also coughing, rapid heart rate, high blood pressure and sore throat. The long term effects of smoking are considered more threa- tening and these are lung cancer, throat cancer, mouth cancer and gum disease, heart disease, stomach ulcers, emphysema and other smoke related conditions. In fact, because of the nature of the long term effects, millions of people around the world have already died from smoking. All of these matters can be stopped if they are treated. Another way is merely to abdicate cigarettes. The aim of this paper is to enhance two numerical schemes for solv- ing a mathematical model describing a giving up smok- ing model and shows the dynamical interaction. There has been some efforts made in the mathematical modeling of giving up smoking since the 2000s. In [4], Zaman proposed a modified model that describes giving up smoking model. In his paper he studied the qualitative behavier of smoking and represented numerical simula- tion by using numerical methods. The homotopy analysis method (HAM) is proposed first by Liao [5,6] for solving linear and nonlinear differential and integral equations. Different from perturbation techniques; the (HAM) doesn’t depend upon any small or large parameter. This method has been successfully applied to solve many types of nonlinear [5-10] differential equations. In this paper, (HAM) is applied to solve nonlinear fractional initial- value problem of the non-fatal epidemic model to obtain symbolic approximate solutions for linear and nonlinear differential Equations of fractional order. (HAM) is dif- ferent from all analytical methods; it provides us with a simple way to adjust and control the convergence region of the series solution by introducing the auxiliary para- meter h and the auxiliary function. In fact, it is the auxiliary parameter h that provides us, for the first time, a simple way to ensure the convergence of the series solu- tion. Due to this reason, it seems reasonable to rename h the convergence-control parameter. It should be empha- sized that, without the use of the convergence-parameter, one had to assume that the homotopy series is convergent. However, with the use of the convergence-parameter h, such an assumption is unnecessary; because it seems that one can always choose a proper value of h to obtain convergent homotopy-series solution. So, the use of the convergence-parameter h in the zeroth-order deformation equation greatly modifies the early homotopy analysis method. Since then, the homotopy analysis method has Copyright © 2012 SciRes. AM