%0 Journal Article
%T On the Solutions Fractional Riccati Differential Equation with Modified Riemann-Liouville Derivative
%A Mehmet Merdan
%J International Journal of Differential Equations
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/346089
%X Fractional variational iteration method (FVIM) is performed to give an approximate analytical solution of nonlinear fractional Riccati differential equation. Fractional derivatives are described in the Riemann-Liouville derivative. A new application of fractional variational iteration method (FVIM) was extended to derive analytical solutions in the form of a series for these equations. The behavior of the solutions and the effects of different values of fractional order are indicated graphically. The results obtained by the FVIM reveal that the method is very reliable, convenient, and effective method for nonlinear differential equations with modified Riemann-Liouville derivative 1. Introduction In recent years, fractional calculus used in many areas such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics, engineering, acoustics, viscoelasticity, material science and signal processing can be successfully modelled by linear or nonlinear fractional order differential equations [1每10]. As it is well known, Riccati differential equations concerned with applications in pattern formation in dynamic games, linear systems with Markovian jumps, river flows, econometric models, stochastic control, theory, diffusion problems, and invariant embedding [11每17]. Many studies have been conducted on solutions of the Riccati differential equations. Some of them, the approximate solution of ordinary Riccati differential equation obtained from homotopy perturbation method (HPM) [18每20], homotopy analysis method (HAM) [21], and variational iteration method proposed by He [22]. The He＊s homotopy perturbation method proposed by He [23每25] the variational iteration method [26] and Adomian decomposition method (ADM) [27] to solve quadratic Riccati differential equation of fractional order. The variational iteration method (VIM), which proposed by He [28, 29], was successfully applied to autonomous ordinary and partial differential equations and other fields. He [30] was the first to apply the variational iteration method to fractional differential equations. In recent years, a new modified Riemann-Liouville left derivative is suggested by Jumarie [31每35]. Recently, the fractional Riccati differential equation is solved with help of new homotopy perturbation method (HPM) [23]. In this paper, we extend the application of the VIM in order to derive analytical approximate solutions to nonlinear fractional Riccati differential equation: subject to the initial conditions where is fractional
%U http://www.hindawi.com/journals/ijde/2012/346089/