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The possible solutions for the two KdV-type equations using a semi-analytical Kamal-iteration method

. In this paper, we used a semi-analytical method that combines the Kamal transform and a modified iteration method to solve the nonlinear homogeneous and nonhomogeneous modified Kerteweg-de Vries and the fifth-order Kerteweg-de Vries equations. The modified iteration method has been used to with a condition that guarantees fast convergence of the approximate solution. The analysis and convergence of the combined method have been discussed. Furthermore, the numerical simulations are presented to illustrate the effectiveness of the proposed semi-analytical method.


Introduction
The Kerteweg-de Vries equation (KdV eq.) is a nonlinear integrable system that has various applications in fluid mechanics, hydromagnetics plasma, electrical transmission lines, and others, see for example [8]. Thus, numerical and semi-analytical methods are designed to solve the nonlinear partial differential equations (PDEs), where; several mechanism methods were employed to study and find out a suitable solution for the PDEs. Then, some of these methods are analytical and others are semi-analytical (numerical with analytic technique) and numerical, for instance, homotopy analysis, differential transformation methods, and many others. These methods are used in different branches of sciences such as; physics, chemistry, and engineering which is why the numerical and semi-analytical methods have become more common in use than pure analytical methods [1,7,10,16]. For example, the integral transforms; by Laplace, Fourier, Hilbert, and Sumida, have been smoothly used to solve linear homogeneous or nonhomogeneous ordinary and partial differential equations (PDEs) [4,25]. For instance, heat, wave, telegraph equations, advection problems. Kamal transform (KT) is used to solve various types of differential and integral equations, see [2,5,15,17,26]. However, all these transforms cannot be used directly if they are employed for the nonlinear homogeneous or nonhomogeneous ordinary and PDEs, because there are no transforms for the nonlinear part. For that reason, some researchers have used the integral transform with the help of decomposition, variational methods and others to solve this problem [6,23,24].
The problem in this project is based on the nonlinear partial KdV eq. [3,9,11,28] of the form where, the initial condition w(x, 0) = F (x), F (x) is a known function. A differentiable solution of eq. (1) is w = w(x, t) which is travelling wave solution tends to zero as |x| → ∞. This kind of equation is classified with the group of integrable equations such as; the nonlinear Schrödinger, Sine-Gordon, Kadomtsev-Petviashvili and Bogoyavlensky-Konopelchenko equations [12,13,19,27]. In addition, the family of the nonlinear PDEs of the KdV eq. comes from the theory of dispersive wave motions. The solutions of these equations have been used to describe several physical phenomena like; "widespread class, elastic scattering property" and others (see [26]).
Here, we are interested in employing the modified iteration method (MI) [21], which was used for the homogeneous and nonhomogeneous nonlinear ordinary differential equations (ODEs). In this paper, we used a combination of the KT [5,17] and MI [21] method which we denote for as K-MI. Using this method, we have explored the acceptable solution for two types of the family of KdV eq.: the modified KdV eq. (mKdV eq.) and the fifth order of KdV eq. (FKdV eq.).
The outline of this paper is arranged as follows. The mathematical model has been presented in Section 2, where, we displayed the KT and some of its related theorems. Next, we have shown the combined method K-MI and applied it to the general description of nonlinear PDEs. The limit of iteration of this method has been discussed. In section 3 we have applied the K-MI method to several examples to show the success of the method in reaching accuracy of the results. Section 4 is related to comparing the classical iterations method to the modified one, both combined with the Kamal transform. Finally, our discussion and conclusion are given in section 5 and 6 respectively.

Kamal transform
This transformation is one of the integral transforms that is employed to find out the solution to differential equations [22]. We have presented some of its properties. We have denoted the integral transform of the Kamal by K and it is specified as: where, l1 ≤ υ ≤ l2. This transform defined for functions on the set A Theorem 1. Assume G(υ) is the KT of S(t) then [17]: where (S (m) ) is the m th derivative. For m = 1, Theorem 2. Let a be a constant and a ≥ 0 then [17]: Theorem 3. For an integer number m ≥ 0, the K [t m ] is [17]: For more information about the properties of this transformation see [5,22] and many others. In the next section, we use the above theorems to calculate the appropriate solution for the nonlinear PDEs with the help of the MI method.

Kamal-Modified Iteration method for the KdV equation types
Consider the nonlinear PDEs of the forṁ where,L is the time dependent operator, M is containing only the dependants of the variable x and w(x, 0) = F (x) is the initial condition (IC) of eq. (7). Here, F (x) and G(x, t) are known functions. First, we apply KT to eq. (7) next, we need to use theorems (1 and 2) and the IC in eq. (8) as To find w(x, t), we need to take the inverse of Kamal transform (K −1 ) of eq. (9), which yields Second, we use the MI method on eq. (10) to deduce the solution of eq. (7). Let w(x, 0) = w 0 (x, t), be the first approximate solution then, eq. (10) becomes Remark 1. During the calculation of classical iteration method, some terms are created, we need to cancel the un necessary terms which make noise to our calculation (see section (4). For that reason, we have defined the condition of the MI method as where, O(t n+1 ) are higher order terms to be neglected. Therefore, eq. (11) becomes: Suppose that G(x, t) is smooth enough, so it has Taylor's expansion in the variable t where, On the other hand, eq. (16) can be written as set n = n − 1 in (16), then we have used it in eq. (17), yields, here, Q −1 = 0, and if we assume η n+1 = T [η 0 + η 1 + η 2 + · · · + η n ], such that then, eq. (18) will be where, Therefore, we have obtained an approximate solution w n+1 (x, t) as a series solution. Hence, w(x, t) is the sum of the iterative solutions, i.e:

Numerical Examples
In this section we clearly show the effectiveness of the K-MI method to solve the certain PDEs mKdV eq. and FKdV eq. The technique of semi-analytical (numerical with analytic ) K-MI method is used on the homogeneous and nonhomogeneous mKdV eq. and FKdV eq. This method started by the initial condition w(x, 0) as a zero iteration solution w 0 (x, t), then, we applied the KT and MI method on the PDE.
Example 1. Consider the third order homogeneous mKdV eq. [26] w t (x, t) using the IC w(x, 0) = 2 λ e λx 1 + e 2λx . Compering eq. (24) to eq. (7), It is clear thaṫ using the K-MI method and following the procedure in section (2.2), we obtain the first iteration w 1 (x, t) of the homogeneous mKdV eq. can be found from substituting w 0 (x, t) in eq. (22). When n = 0, Here, Q 0 in (28) is time independent, so, substitute the value of Q 0 in eq. (19) to find η 1 is time independent, which means it is constant with respect to t, (1 + e 2λx ) 2 )], Theorem 3, Then, the first iteration w 1 is When n = 1, we have substituted w 1 in eq. (26) to find Q 1 . In this case, we neglected the coefficients of t with indices grater than or equal 2 (O(t 2 )). Here, Q 1 is t independent, where, A = 2λ 4 e λx and B = (1 + e 2λx ), and using equation (19), we have So, the second iteration w 2 is When n → ∞, the n th iteration (approximate solution) will be the series solution (34) of the mKdV eq. is the Taylor's expansion in the variable t of the function which is exactly the same as the one obtained in [26]. Figure (1) shows the solution deduced from the convergent Taylor's series in eq. (35). Example 2. Consider the third order nonhomogeneous mKdV eq. [20] w using the IC w(x, 0) = 0.
It is clear that, when we compare eq. (36) to eq. (7), we havė and Similarly as in example (1), we have used K-MI method and following the procedure in section (2.2), we have here, the function G n is a polynomial of degree (9) in the variable t. When n = 0, then η 0 = w 0 = 0, using (39) and (40) to find Hence, substituting Q 0 and G 0 in eq (19), we have therefore, the first iteration w 1 is When n = 1, then, Q 1 (x, t) = 0, and G 1 (x, t) = −x + xt 3 neglect , so, therefore, the second iteration w 2 is When n = 2 therefore, the third iteration w 3 is Finally, when n → ∞, the series solution (48) of the nonhomogeneous mKdV equation (36) has the sum:  In the next example, we apply the K-MI method on the nonlinear homogeneous and nonhomogeneous FKdV eq.
That is clear from equation (50), we havė , t), and G(x, t) = 0, (51) following the same structures as above examples, the Q n is when n = 0, then, therefore, the first iteration w 1 is When n = 1, therefore, the second iteration w 2 is When n = 2 so, Then, the third iteration w 3 is Finally, when n → ∞, the series solution (61) of the homogeneous FKdV eq. (50) when we take 1 x as a common factor from eq. (62), we have the second part of eq.(63) is the Taylor's expansion of, , then, the Taylor's expansion of the exact solution is Equation (50) was solved in [14,20], we have reached the same result with the K-IM method. According to Taylor's series, Figure (3) display the exact solution of the homogeneous FKdV eq. (50). The function in eq. (64) has singularities at (t = x).
(66) Similarly, as above examples, we have and when n = 0, Using the Taylor's expansion in the function G 0 (x, t), for e −t = 1 − t + t 2 2! − t 3 3! + t 4 4! + . . . , then, Substituting Q 0 and G 0 in eq (19), we have therefore, the first iteration w 1 is, When n = 1, and G 1 = −2 cos (x) + 2 sin (x) t after cancelling the neglected terms. So, therefore, the second iteration w 2 is, When n = 2 and so, then, the third iteration w 3 is, Finally, when n → ∞, the series solution (79) of the homogeneous FKdV equation (65) w(x, t) = lim n→∞ n k=0 η k  is far from our expectation, so we need to find the next iteration to see whether the next iteration will be better. When n = 2, the third iteration is unnecessary (noise) terms which makes the approximate solution looks divergent from the exact solution. This case will happen when we try to use the CI method instead of the MI method for the above examples (1,2) and (4).

Discussion
Kamal transform has been used recently in a combination with another method to solve nonlinear differential equations [18,22]. In this work, we use KT in combination with the IT method to solve the KdV eq. Our analysis shows that using the KT and the classical iteration method can lead to divergent series of approximate solutions due to the high order terms induced during the iterations causing high noise and error in the calculation as shown in section (4). However, modifying the iteration method by removing the noise terms and combining them with the KT leads to better approximation results since the series of the solutions converges fast to the accurate solution as shown in the examples in section (3). The semi-analytical K-MI method is used to solve the homogeneous and nonhomogeneous mKdV eq. and FKdV eq. This method begins with the initial condition w(x, 0) as the first approximate solution w 0 (x, t), then, we apply the KT and MI method to construct the other iterations. Each iteration is an approximate solution of the PDE. The approximate solution is a polynomial in the variable t of a degree equal to the order of the iteration, hence, it is not bounded as |t| → ∞. However, in many cases, one can deduce an infinite series by taking the limit of the iteration. When the deduced series is comparable to the Taylor expansion of a known function, then the exact solution is inferred from our semi-analytic method, as we have shown in our examples. In example (1), the exact solution of the homogeneous KdV equation is shown in Figure (1). The solution is called soliton which is a traveling wave that maintains its amplitude for a long distance while vanishing as |x| → ∞. In example (2), the exact solution of the nonhomogeneous mKdV equation is an odd polynomial that is not bounded as shown in Figure (2). In example (3) the solution of the homogeneous FKdV equation has a discontinuity at x = t while decaying to zero as |x| → ∞, hence, the equation in example (3) admits a singular solution. The solution of the nonhomogenous FKdV eq. in example (4) is a periodic sine wave as shown in Figure (4). Figure (5), illustrates the approximate solution of equation (87) obtained after 2 iterations (n = 0, 1) using the CI method. Both approximate solutions w 1 and w 2 are polynomials in the variable t, therefore, they are unbounded along time progress. However, the error of the approximate solution of the classical iterations method is higher than the error of the modified iterations method. The error induced by the polynomial terms of degree higher than the iteration order n = 3 which cause high noise in the calculation, see Figure (6). Removing the noise terms in the modified iterations method leads to better convergence towards the exact solution, see Figure(3).

Conclusion
In this work, the semi-analytical Kamal transform combined to the modified iteration method (K-MI method) has been used to extract approximate solutions for various examples of the non-homogeneous KdV equation. In addition, by considering the limit series of the approximate solutions as Taylors' expansions of known functions, we inferred the exact solutions of the considered equations. The exact solutions of the considered equations varied from polynomial, periodic and even discontinuous singular solution. Moreover, we showed the outweigh of the modified iterations method over the classical iterations method since the latter method can suffer from slow convergence of the approximations to the exact solution. Due to the generated noise terms, those are eliminated in the MI method.