A New Homotopy Analysis Method for Approximating the Analytic Solution of KdV Equation

In this study a new technique of the Homotopy Analysis Method (nHAM) is applied to obtain an approximate analytic solution of the well-known Korteweg-de Vries (KdV) equation. This method removes the extra terms and decreases the time taken in the original HAM by converting the KdV equation to a system of first order differential equations. The resulted nHAM solution at third order approximation is then compared with that of the exact soliton solution of the KdV equation and found to be in excellent agreement.

The well-known Korteweg-deVries equation (KdV) is given by: 6 0, , with initial condition: an approximate analytic solution of KdV equation by HAM is given by Nazari et al. (2012a, b).In this study, instead we use a new technique of Homotopy Analysis Method (nHAM) to generate an approximate analytic solution of Eq. (1).

Important idea of the Homotopy Analysis Method (HAM):
To explain the main ideas of HAM, we consider a nonlinear equation in general form: where N is a nonlinear operator ( , ) u r t is an unknown function, 0 ( , ) u r t denotes an initial guess of the exact solution ( , ), u r t 0 ≠ h an auxiliary parameter, ( , ) H r t auxiliary function and L an auxiliary linear operator, p∈ (0, 1) as an embedding parameter, we construct the socalled zeroth-order deformation equation by means of HAM: We have a great freedom to choose auxiliary parameters in HAM.Obviously, if we let p = 0, 1 then respectively we have 0 ( , ;0) ( , ), ( , ;1) ( , ) , r t u r t r t u r t i.e., when p grows from 0 to 1, the solution ( , ; ) r t p ϕ changes from initial guess u 0 (r, t) to exact solution u (r, t).
According to Liao (1992), ( , ; ) r t p ϕ can be rewritten in a power series of the form below: The convergence of the series (5) depends upon the auxiliary parameter ħ, auxiliary function H (r, t), initial guess u 0 (r, t) and auxiliary linear operator L. If they were chosen properly, the series ( 5) is convergence at p = 1, one has: According to definition (6), the governing equation can be inferred from the zeroth-order deformation Eq. ( 4).We define the vector: 0 1 ( , ) { ( , ), ( , ),..., ( , )}, n n u r t u r t u r t u r t = uuuuuuu r and if we take m-times differentiating with respect to p from zeroth-order deformation Eq. ( 4) and dividing them by ݉ǃ then setting p = 1 we obtain the so-called m th -order deformation equation as below: where, Theorem 1: As long as the series ( 7) is convergent, it is convergent to the exact solution of (3).Note that the HAM contains the auxiliary parameter ħ, that we can control and adjustment the convergence of the series solution (7).

Exact solution of KdV:
The Korteweg-de Vries equation (KdV equation) describes the theory of water wave in shallow channels, such as a canal.It is a nonlinear equation which governed by ( 1) and ( 2).We assume that the exact solution and its derivatives tend to zero (Ablowitz and Segur, 1981;Ablowitz and Clarkson, 1991) when ‫|ݔ|‬ → ∞ .
An exact solution of KdV equation as below (Wazwaz, 2001): we will compare our final results with (11): Analysis of HAM: We reformatted (3) in the form as below: with initial conditions: where, L = డ௨ (௫,௧) డ௧ , ( , ), ( , ) Au x t Bu x t are linear and nonlinear parts of equation respectively.The zero-order deformation equation is: The m th -order deformation equation is obtained by taking m times derivative of ( 14), i.e.: and then: We consider ( , ) 1 H r t = and L -1 is an integral operator and u 0 (r, t) is an initial guess of the approximation of exact solution ( 11) and ( 16) then becomes: For m = 1, χ 1 = 0 and u 0 (x, 0) u 0 (x, t) f 0 (x) from (17), we obtain: For m > 1, χ m = 1 and u m-1 (x, 0) = 0: New technique of HAM: We rewrite Eq. (3) in a system of first order differential equation as below: ( , ) ( , ) 0 we consider the initial approximation and an auxiliary linear operator, respectively in the form below: From ( 18) and ( 20) we have, respectively as: In ( 19) for m>1, χ m = 1 and u m (x, 0) = 0, v m (x, 0) = 0, we obtain the following results:

RESULTS AND DISCUSSION
Using new technique of HAM for KdV: We rewrite KdV equation in a system of ( 24) and ( 25) as below: ( , ) ( , ) , From ( 23) we obtain: and for m>1, χ m = 1 and: The results of new technique of HAM for KdV: The results are generated as below: (1 ) x . We can control the rate of convergence of this approximation by the auxiliary parameter ħ.If we let x = t = 0.01, then it is obvious from Fig. 1 that the best reliable region for this analytical solution is in the interval 1.2 0 − < < h .According to theorem 1 the series solution (32) must be the exact solution, as long as the series is convergent.In this case, when 1 1 t − < < and 0.4 , = − h the exact solution and nHAM are similar in Fig. 2. The numerical results are shown in Table 1.In Fig. 3 we obtained the third-order approximation and we drew the shapes of u (x, 0.05) and u (x, 0.15) for 0.4, 1, 0.75 − − − = h and compared them with the exact solution.We can see that the best result is relevant to an approximation that has 0.4 .= − h

CONCLUSION
In this study, a new technique of Homotopy Analysis Method (nHAM) is applied to obtain an approximated analytic solution of KdV equation.The resulted nHAM solution at the third order approximation is compared with the exact soliton solution of the KdV equation and it is shown that the result is a good approximation in comparison with the exact solution.Clearly this technique provides a way for eliminating some extra terms of the original homotopy analysis method and then decreases the time consumed for obtaining the final result.We have used the auxiliary parameter ħ for controlling the convergence of the approximation series which is a fundamental qualitative difference in analysis between nHAM and other methods.

Table 1 :
Comparison between results of exact solution and results of nHAM