A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders

The reliable treatment of homotopy perturbation method (cid:2) HPM (cid:3) is applied to solve the Klein-Gordon partial di ﬀ erential equation of arbitrary (cid:2) fractional (cid:3) orders. This algorithm overcomes the di ﬃ culty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the e ﬃ ciency of this technique.


Introduction
The Klein-Gordon equation plays a significant role in mathematical physics and many scientific applications such as solid-state physics, nonlinear optics, and quantum field theory 1, 2 . The equation has attracted much attention in studying solitons 3-6 and condensed matter physics, in investigating the interaction of solitons in a collisionless plasma, the recurrence of initial states, and in examining the nonlinear wave equations 7 . The HPM, proposed by He in 1998, has been the subject of extensive studies and was applied to different linear and nonlinear problems [8][9][10][11][12][13] . This method has the advantage of dealing directly with the problem without transformations, linearization, discretization, or any unrealistic assumption, and usually a few iterations lead to an accurate approximation of the exact solution 13 . The HPM has been used to solve nonlinear partial differential equations of fractional order see, e.g., [14][15][16] . Some other methods for series solution that are used to solve nonlinear partial differential equations of fractional order include the

The Homotopy Perturbation Method (HPM)
Consider the following equation: where A is a general differential 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 Γ is the boundary of the domain Ω. The operator A can be generally divided into linear and nonlinear parts, say L and N. Therefore, 3.1 can be written as In 9 , He constructed a homotopy v r, p : where p ∈ 0, 1 is an embedding parameter,and u 0 is an initial guess of u x, t which satisfies the boundary conditions. Obviously, from 3.4 and 3.5 , one has

3.6
Changing p from zero to unity is just that change of v r, p from u 0 r to u r . Expanding v r, p in Taylor series with respect to p, one has v v 0 pv 1 p 2 v 2 · · · . 3.7 4

Journal of Applied Mathematics
Setting p 1 results in the approximate solution of 3.1 The reliable treatment of the classical HPM suggested by Odibat and Momani 26 is presented for nonlinear function N u which is assumed to be an analytic function and has the following Taylor series expansion: According to 26 , the following homotopy is constructed for 1.1 : The basic assumption is that the solution of 3.10 can be written as a power series in p, Substituting 3.11 into 3.10 and equating the terms with identical powers of p, we obtain a series of linear equations in u 0 , u 1, u 2 , . . ., which can be solved by symbolic computation software. Finally, we approximate the solution u x, t ∞ n 0 u n x, t by the truncated series

Numerical Implementation
In this section, some numerical examples are presented to validate the solution scheme. Symbolic computations are carried out using Mathematica.

Example 4.1. Consider the fractional-order cubically nonlinear Klein-Gordon problem
with the exact solution u x, t x β t α .

Journal of Applied Mathematics 5
According to the homotopy 3.10 , we obtain the following set of linear partial differential equations of fractional order:

4.6
The corresponding integer-order problem has the exact solution u 2,2 −sech x t/2 29 .
According to the homotopy 3.10 , we obtain the following set of linear partial differential equations of fractional order: Case 1 α ∈ 1, 2 and β 2 . Solving 4.7 , we have . . .

4.8
and the solution is obtained as u u 0 u 1 u 2 · · · . 4.9 Figure 4 gives the comparison between the HPM 4th-order approximate solution of problem 4.6 in Case 1 with β 2, α 1.99, 1.95, 1.90, and 1.85 and the solution of corresponding problem of integer order denoted by u 2,2 at t 0.3.

Case 2 α
2 and β ∈ 1, 2 . As the attempt to evaluate Caputo fractional derivative of the functions sech x and tanh x yields hypergeometric function, we substitute sech x and tanh x by some terms of its Taylor series. Substituting the initial conditions and solving 4.7 for u 0 , u 1 , u 2 , . . ., the components of the homotopy perturbation solution for 4.6 are derived as follows: . . .

4.10
As the Caputo fractional derivative can not be evaluated for negative powers of the variable at hand, and noting that β ∈ 1, 2 , we can only evaluate the first two components of the series as illustrated. Thus, we suggest to generalize not only the derivatives in the integer-order problem to its fractional form, but also to generalize the conditions as well. For example, a generalized expansion of sech x in a fractional form can be written as for which we have lim β → 2 sech β x sech x . Substituting the generalized form of the initial conditions and solving 4.7 for u 0 , u 1 , u 2 , . . ., the components of the homotopy perturbation solution for this case are derived as follows: . . .

4.12
and the solution is obtained as u u 0 u 1 u 2 · · · . 4.13 Figure 5 gives the comparison between the HPM 4th-order approximate solution of problem 4.6 in Case 2 with α 2, β 1.99, 1.95, 1.90, and 1.85 and the solution of corresponding problem of integer order denoted by u 2,2 at t 0.3.
Case 3 both α and β ∈ 1, 2 . Carrying out the same procedure as in Case 2, we get . . .

Conclusion
The reliable treatment HPM is applied to obtain the solution of the Klien-Gordon partial differential equation of arbitrary fractional orders with spatial and temporal fractional derivatives. The main advantage of this algorithm is the capability to overcome the difficulty arising in calculating complicated integrals when dealing with nonlinear problems.
The numerical examples carried out show good results, and their graphs illustrate the continuation of the solution of fractional-order Klien-Gordon equation to the solution of the corresponding second-order problem when the fractional-order parameters approach their integer limits.