New Solution of the Sine-Gordon Equation by the Daftardar-Gejji and Jafari Method

: In this article, the Daftardar-Gejji and Jafari method (DJM) is used to obtain an approximate analytical solution of the sine-Gordon equation. Some examples are solved to demonstrate the accuracy of DJM. A comparison study between DJM, the variational iteration method (VIM), and the exact solution are presented. The comparison of the present symmetrical results with the existing literature is satisfactory.


Introduction
Differential equations are highly effective tools used to describe real-life phenomena; in most cases, the numerical or theoretical solutions are difficult to find. In recent years, searching for new methods to solve nonlinear differential equations has received increased attention, see [1][2][3].
In this paper, we will apply the Daftardar-Gejji and Jafari method (DJM) to find the solution of the sine-Gordon nonlinear equation: In [4] Herbst et al. used an explicit symplectic method to find numerical results of the sine-Gordon equation. (Wazwaz, 2005) [5] Found new exact solutions to the sine-Gordon equation by using the tanh method. (Kaya, 2003) [6] presented the approximate analytical solution of the sine-Gordon equation by means of the modified decomposition method. (Ray, 2006) [7] applied the modified decomposition method to obtain the solution of coupled sine-Gordon equations. (Batiha et al., 2007) [8] applied the variational iteration method (VIM) to obtain an approximate analytical solution of the sine-Gordon equation.
In this article, the analytical solution of the sine-Gordon Equation (1) is found by using the DJM. Comparisons with the variational iteration method (VIM) and the exact solution will be presented. The results obtained are symmetrical with VIM and exact solution results.

The Daftardar-Gejji and Jafari Method
In this paper, we will discuss the Daftardar-Gejji and Jafari method, and how to use it for solving nonlinear differential equations in the form: where L, N are linear and non-linear operators, f is an arbitrary function. The above equation has a solution in the form: Suppose we have, Then we obtain, Thus, N(y) is decomposed as: So, the recurrence relation is as the following form: Since L is linear, then: So, Thus, We can form the solution in the k-term as follows:

Proof.
You can find the detailed proof in [14].
Proof. You can find the detailed proof in [14].

Numerical Applications
Here, we will use DJM to find the solution of the sine-Gordon Equation (1).

Thus,
Which is exactly the same result as [8] found by VIM.
The exact solution is: For solving Equation (1) by DJM, we integrate Equation (1) and use Equation (25) to obtain: By using algorithm (11) we find: . . .
where s = sech (x), thus: This is exactly the same result as Batiha et al. [8] obtained by VIM. In Figure 2 we show the comparisons between one iteration DJM and the exact solution (26). Figure 2 clearly shows the excellent accuracy of DJM compared to the exact solution.

Conclusions
In this article, the DJM was used to obtain the solutions of the sine-Gordon equation with remarkable success. Comparisons with the variation iteration method (VIM) and the exact solution show that DJM is a very promising technique for solving nonlinear partial differential equations. In future research, DJM should be used to solve fractional differential equations.