%0 Journal Article
%T Solving Fractional-Order Logistic Equation Using a New Iterative Method
%A Sachin Bhalekar
%A Varsha Daftardar-Gejji
%J International Journal of Differential Equations
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/975829
%X A fractional version of logistic equation is solved using new iterative method proposed by Daftardar-Gejji and Jafari (2006). Convergence of the series solutions obtained is discussed. The solutions obtained are compared with Adomian decomposition method and homotopy perturbation method. 1. Introduction The following model describing growth of population was first studied by Pierre Verhulst in 1938 [1] where is population at time , and is Malthusian parameter describing growth rate and is carrying capacity. Defining gives the following differential equation: which is called as logistic equation. Logistic equation of fractional order has been discussed in the literature [2, 3]. El-Sayed et al. [2] have investigated the equation , where is Caputo fractional derivative of order . Momani and Qaralleh [3] have employed Adomian decomposition method (ADM) for solving fractional population growth model in a closed system. In the present paper we use New Iterative Method (NIM) introduced by Daftardar-Gejji and Jafari [4] to solve fractional version of logistic equation. NIM is useful for solving a general functional equation of the form where is a given function, and linear and nonlinear operators, respectively. The NIM has fairly simple algorithm and does not require any knowledge of involved concepts such as Adomian polynomials, homotopy, or Lagrange multipliers. Rigorous convergence analysis of NIM has been worked out recently [5]. This method has been applied by present authors successfully for solving partial differential equations [6], evolution equations [7], and fractional diffusion-wave equations [8]. NIM has been further explored by many researchers. Several numerical methods with higher order convergence can be generated using NIM. M. A. Noor and K. I. Noor [9, 10] have developed a three-step predictor-corrector method for solving nonlinear equation . Further, they have shown that this method has fourth-order convergence [11]. Some new methods [12, 13] are proposed by these authors using NIM. Mohyud-Din et al. [14] solved Hirota-Satsuma coupled KdV system using NIM. These authors [15] also have applied NIM in solutions of some fifth order boundary value problems. Noor and Mohyud-Din [16] have used NIM to solve Helmholtz equations. NIM is applied to solve homogeneous and inhomogeneous advection problems [17], diffusion equations [18], Schr£żdinger equations [19], time fractional partial differential equations [20], and so on. Yaseen and Samraiz [21] proposed modified NIM and used it to solve Klein-Gordon equations. Srivastava and Rai [22] have proposed
%U http://www.hindawi.com/journals/ijde/2012/975829/