A Reliable Treatment of Homotopy Perturbation Method for the Sine-gordon Equation of Arbitrary (fractional) Order

In this paper, the reliable treatment of homotopy perturbation method (HPM) [19] is applied to obtain the solution of the sine-Gordon partial di¤erential equation of arbitrary (fractional) order. The advantage of this algorithm is its ability to provide the analytical or approximate solutions to nonlinear equations with the capability to overcome the di¢ culty that arises in calculating complicated integrals. The numerical results are presented to show the e¢ ciency of this method. 1. Introduction The sine-Gordon equation which …rst appeared in the study of the di¤erential geometry of surfaces with Gaussian curvature K = 1 found wide applications in the propagation of ‡uxons in Josephson junctions between two superconductors [1], the motion of a rigid pendulum attached to a stretched wire [2], solid state physics, nonlinear optics, stability of ‡uid motions, dislocations in crystals [2] and other scienti…c …elds. Due to its wide applications and important mathematical properties, a great deal of e¤ort has been devoted to studying the di¤erent solutions and physical phenomena related to this equation [3]-[11]. In 1998, J. H. He proposed the homotopy perturbation method (HPM) for addressing nonlinear problems in [13] and [14]. This method has been the subject of extensive studies, and applied to di¤erent linear and nonlinear problems in [14][20]. The advantage of this method is solving nonlinear equations without invoking unrealistic assumptions, discretization or linearization. The HPM has the advantage of dealing directly with the problem without transformations, linearization, discretizations or any unrealistic assumption. The method yields a rapidly convergent series solution, and usually a few iterations lead to accurate approximation of the exact solution [18] and [21]. Recently, Momani and Odibat suggested a reliable algorithm for the HPM for dealing with nonlinear terms [19]. The advantage of this algorithm is its ability to provide the analytical or approximate solutions to nonlinear equations and overcome 2010 Mathematics Subject Classi…cation. 35R11, 35J60, 35C10 . Key words and phrases. Homotopy perturbation method; sine-Gordon equation; Fractional partial di¤erential equation. Submitted Aug. 21, 2011. Published Jan. 1, 2012. 1 2 A. ELSAID, D. HAMMAD JFCA-2012/2 the di¢ culty that arising in calculating complicated integrals. Our aim here, is to apply the reliable treatment of HPM to obtain the solution of the initial value problem of the sine-Gordon equation of fractional order D t u(x; t) = auxx(x; t) + b sin ( u (x; t)) ; x 2 R; t > 0; 2 (1; 2] ; (1) subjected to the initial conditions u(x; 0) = gk(x); x 2 R; k = 0; 1: (2) The article begins by presenting some basic de…nitions of fractional derivatives in section two. The HPM and the reliable treatment of HPM are introduced in section three. In section four, some case studies of the nonlinear sine-Gordon equations of arbitrary (fractional) orders are presented to illustrate the validity of this approach and to show the e¤ects of fractional order parameters involved on solution accuracy and behavior. 2. Basic definitions De…nition 1. A real function f(t), t > 0, is said to be in the space C , 2 R, if there exists a real number p > , such that f(t) = tf1(t), where f1(t) 2 C(0;1), and it is said to be in the space C if f (m) 2 C ; m 2 N: De…nition 2. The Riemann-Liouville fractional integral operator of order 0 of a function f(t) 2 C ; 1 is de…ned as [22] 8><>>: J f(t) = 1 ( ) t Z 0 (t ) f( )d ; > 0; t > 0; Jf(t) = f(t): (3) The operator J satisfy the following properties. For f 2 C , 1; ; 0 and > 1: 1: J J f(t) = J + f(t); 2: J J f(t) = J J f(t); 3: J t = ( +1) ( + +1) t + : De…nition 3. The fractional derivative in Caputo sense of f(t) 2 C 1; m 2 N; t > 0 is de…ned as D t f(t) = J d m dtm f(t); m 1 < < m; d dtm f(t); = m: (4) The operator D satisfy the following properties. For f 2 C , 1, ; 0: 1: D t [J f(t)] = f(t); 2: J [ D t f(t)] = f(t) m 1 X k=0 f (0) t k k! ; t > 0; 3: D t t = ( +1) ( +1) t : JFCA-2012/2 HPM FOR FRACTIONAL SINE-GORDON EQUATION 3 3. The homotopy perturbation method (HPM) Consider the following equation A (u (x; t)) f(r) = 0; r 2 ; (5) with boundary conditions B(u; @u=@n) = 0; r 2 ; (6) where A is a general di¤erential operator, u(x; t) is the unknown function and x and t denote spatial and temporal independent variables, respectively. B is a boundary operator, f(r) is a known analytic function, and is the boundary of the domain : The operator A can be generally divided into linear and nonlinear parts, say L and N . Therefore (5) can be written as L (u) +N(u) f(r) = 0: (7) In [12], He constructed a homotopy v(r; p) : [0; 1]! R which satis…es H(v; p) = (1 p) [L(v) L(u0)] + p [L(v) +N(v) f(r)] = 0; r 2 ; (8) or H(v; p) = L(v) L(u0) + pL(u0) + p[N(v) f(r)] = 0; r 2 ; (9) where p 2 [0; 1] is an embedding parameter, u0 is an initial guess of u(x; t) which satis…es the boundary conditions. Obviously, from (8) and (9) one has H(v; 0) = L(v) L(u0); (10) H(v; 1) = L (u) +N(u) f(r) = 0: (11) Changing p from zero to unity is just that change of v(r; p) from u0(r) to u (r) : Expanding v(r; p) in Taylor series with respect to p, one has v = v0 + pv1 + p v2 + . (12) Setting p = 1; results in the approximate solution of (5) u = lim p!1 v = v0 + v1 + v2 + ::: . (13) The reliable treatment of the classical HPM suggested by Momani and Odibat [19] is presented for nonlinear function N(u) which is assumed to be an analytic function and has the following Taylor series expansion