Chin. Phys. B Vol. 22, No. 1 (2013) 010201 Analytical approximate solution for nonlinear space–time fractional Klein Gordon equation Khaled A. Gepreel a)b) † and Mohamed S. Mohamed b)c) a) Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt b) Mathematics Department, Faculty of Science, Taif University, Taif, Saudi Arabia c) Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt (Received 29 May 2012; revised manuscript received 26 June 2012) The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approxi- mate solution for the nonlinear space–time fractional derivatives Klein–Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space–time fractional derivatives Klein– Gordon equation. This method introduces a promising tool for solving many space–time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations. Keywords: homotopy analysis method, nonlinear space–time fractional Klein–Gordon equation, Caputo derivative PACS: 02.30.Jr DOI: 10.1088/1674-1056/22/1/010201 1. Introduction The seeds of fractional calculus (that is, the theory of integrals and derivatives of arbitrary real or complex orders) were planted over 300 years ago. Fractional differential equations (FDEs) have found applications in many problems in physics and engineering. [1–4] Since most of the nonlin- ear FDEs cannot be solved exactly, approximate and numer- ical methods must be used. Some of the analytical methods for solving nonlinear problems include the Adomian decom- position method (ADM), [5–7] the variational iteration method (VIM), [8] the homotopy-perturbation method (HPM), [9–11] and the homotopy analysis method (HAM). [12–17] The HAM, first proposed in 1992 by Liao, [12] has been successfully applied to solve many problems in physics and science. Deng et al. [18,19] used the finite difference methods and the finite ele- ment method to solve the fractional Klein–Kramers equations and the fractional Fokker–Planck equation, respectively. Re- cently Gepreel et al. [20] used the fractional complex transfor- mation to obtain the exact solutions for some nonlinear partial fractional differential equations. There are many methods for obtaining analytic approximate solutions for nonlinear frac- tional differential equations, such as the Adomian decomposi- tion method, the variational iteration method, the homotopy- perturbation method, and the homotopy analysis method. The homotopy anaylsis method is the generalized method and con- tains a certain auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region of the solution series. In this article, we use the HAM to calculate the analytic approximate solution to the nonlinear space–time fractional derivative Klein–Gordon equation [21] ∂ α u ∂ t α − ∂ β u ∂ x β + u 3 = Γ(α + 1)x β − Γ(β + 1)t α + x 3β t 3α , (t > 0, 1 < α , β ≤ 2). (1) We study the behavior of the approximate solution to the non- linear space–time fractional derivative Klein–Gordon equation (1) and we determine the convergence region of the approxi- mate solutions. 2. Preliminaries and notations In this section, we give some definitions and properties of the fractional calculus. Several definitions of fractional calcu- lus have been proposed in the last two centuries. The fractional calculus is the theory of integrals and derivatives of arbitrary order. There are many researchers [1–4] who have developed the fractional calculus and given various definitions of frac- tional integration and differentiation, such as the Grunwald– Letnikov definition, Riemann–Liouville definition, Caputo’s definition, and the generalized function approach. Definition 1 A real function h(t ), t > 0 is said to be in the space C μ , μ ∈ R if there exists a real number p > μ such that h(t )= t p h 1 (t ), where h 1 (t ) ∈ C[0, ∞), and it is said to be in the space C n μ if and only if h (n) ∈ C μ , n ∈ N. Definition 2 The Riemann–Liouville fractional integral operator J α of order α ≥ 0 of function h ∈ C μ , μ ≥−1 is de- fined as J α h(t )= 1 Γ(α )  t 0 (t − τ ) α−1 h(τ ) d τ , (α > 0), J 0 h(t )= h(t ), (2) † Corresponding author. E-mail: kagepreel@yahoo.com © 2013 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn 010201-1