Homotopy Analysis Method for Nonlinear Differential Equations with Fractional Orders

Most problems in science and engineering are nonlinear. Thus, it is important to develop efficient methods to solve them. In the past decades, with the fast development of high-quality symbolic computing software, such as Maple, Mathematica and Matlab, analytic as well as numerical techniques for nonlinear differential equations have been developed quickly. The homotopy analysis method (HAM) [1 – 5] is one of the most effectivemethods to construct analytically approximate solutions of nonlinear differential equations. This method has been applied to a wide range of nonlinear differential equations. Compared with the traditional analytic approximation tools, such as the perturbation method [6 – 9], the δ -expansion method [10], and the Adomian decomposition method [11 – 13], the HAM provides a convenient way to control and adjust the convergence range and the rate of approximation. Also, the HAM is valid even if a nonlinear problem does not contain a small or large parameter. In addition, it can be employed to approximate a nonlinear problem by choosing different sets of base functions. In recent years, considerable interest in fractional differential equations has been stimulated due to their numerous applications in physics and engineering [14]. For instance for the propagation of waves through a fractal medium or diffusion in a disordered system it is reasonable to formulate the structure of the nonlinear evolution equations in terms of fractional derivatives rather than in the classical form. Furthermore, we known that many nonlinear differential equations exhibit strange attractors and their solutions have