Efficient Iterative Method for Solving Korteweg-de Vries Equations

The Korteweg-de Vries equation plays an important role in fluid physics and applied mathematics. This equation is a fundamental within study of shallow water waves. Since these equations arise in many applications and physical phenomena, it is officially showed that this equation has solitary waves as solutions, The Korteweg-de Vries equation is utilized to characterize a long waves travelling in channels. The goal of this paper is to construct the new effective frequent relation to resolve these problems where the semi analytic iterative technique presents new enforcement to solve Korteweg-de Vries equations. The distinctive feature of this method is, it can be utilized to get approximate solutions for travelling waves of non-linear partial differential equations with small amount of computations does not require to calculate restrictive assumptions or transformation like other conventional methods. In addition, several examples clarify the relevant features of this presented method, so the results of this study are debated to show that this method is a powerful tool and promising to illustrate the accuracy and efficiency for solving these problems. To evaluate the results in the iterative process we used the Matlab symbolic manipulator.


Introduction
In the previous few decades, Non-linear phenomena play an important tool in physics, the solitary wave theory and applied mathematics. On the other hand, the study of non-linear partial differential equations in modelling physical phenomena become a significant gadget in a large class and vastly utilized in assorted fields of the nonlinear natural sciences, As for the behavior and properties of nonlinear partial differential equations can be determined by exact solution ; therefore, non-linear equations play significant role in specifying such problems. Thus, their exact solution is more complicated to find as compared to the other solutions of linear equations. The Korteweg-de Vries equation(KdV) equation has been arisen in the study of shallow water waves, [1]. Particularly, it is utilized to characterize the diffusion of plasma waves in a dispersive medium and furthermore describes long waves traveling in canals too. The KdV equation has been widely studied and has gained a lot of interest. Many analytical and numerical techniques were used to study the solitary waves which outcome of such equation. Kruskal and Zabusky have researched in the interaction of repetition of the initial cases and solitary waves . They detected the solitary waves undergo nonlinear interaction subsequent KdV equation. It is worth mentioning, waves emanate of this interaction retaining its amplitude and shape, also solitary waves maintain its character and identities resembles such as particle conduct, they stimulated to recall solitary wave solitons, [2]. Many research works have arisen in various of the scientific realm in worldwide to study the solution concept. It is renowned now that solutions show as the outcome of an equilibrium among dispersion and weak nonlinearity and it has enticed formidable volume of studies due to its important part within different scientific fields like magneto-acoustic waves, plasma physics, fluid dynamics , astrophysics and others. The inquiry of the solutions of travelling wave perform an essential part in non-linear sciences, like Hirota's Bilinear Method, [3]. Inverse Scattering Transform, [4]. B ̈cklund Transformation, [5]. Sine-Cosine Method, [6]. Tanh-Function Method, [7]. Homotopy Perturbation Method, [8]. Homotopy Analysis Method, [9][10]. Variational Iteration Method(VIM), [11][12]. Exp-Function Method, [13][14]. Differential Transform Method, [15]. Tanh-Coth Method, [16][17]. (´/ )-Expansion Method, [18][19]. Laplace Adomian Decomposition Method, [20]. Generalized Tanh-Coth Method and (´/ )-Expansion Method, [21]. Recently, considerable attention has been given to some efficient analytic iterative methods, Temimi and Ansari have suggested a new iterative method, i.e Semi Analytical Iterative Method(SAIM) to resolve linear and non-linear functional equations, [22]. The SAIM has been successfully applied by many research works to solve some linear and non-linear partial, ordinary differential equations and higher order integro-differential equations, [23][24][25][26][27]. To the best of our knowledge, SAIM is not yet implemented to resolve the KdV equations. This method can be used to solve and get the analytical solutions of these problems. KdV equations were solved by Adomian Decomposition Method (ADM) and VIM, [ 28]. ADM had be utilized to solve the KdV equation to get a solitary wave solution, also the Modified Adomian Decomposition Method(MADM) used to solve non-linear dispersive waves via Boussinesq equation, [29]. The basic concept of this reasech is as follows. Section 2 An explanation for KdV equations is presented by the fundamental idea of SAIM. Several analytical outcomes are supplied in section 3 to clarify the method. Section 4 concludes the research.

Korteweg -deVries equation problems by the Basic idea of (SAIM):
The simplest form of the KdV equation, [30][31], for two independent variables and can be written as follows: (2.1) And the Initial Value Problems (IVPs) such that ( ) at the some domain, the non-linear idiom and is a parameter . the basic idea of the reliable SAIM to solve Eq. (2.1) with IVPs (2. 2) The general form equation as: So, thus ( ) the unknown function, x and t indicate to the independent variables, while L, N represent linear and the non-linear operators, respectively. ( ) represents the inhomogeneous term, which is a known function and C the conditions operator for the problem. Initial approximation is a primary step in the SAIM, by assuming that the initial guess ( ) is a solution of the problem ( ), the solution of the equation can be solving To generate the next iteration of the solution can be resolve the problem as follows After several simple iteration steps of the solution, the general form of this equation as Evidently each iteration of the function ( ) represent effectively alone solution for Eq. (2.3). We will implement the method steps at the Eq.(2.1), so Eq.(2.1) can be express as for: The differential operator ( ( ))is the highest order derivative in the Eq.(2.8), by using the given IVPs in Eq.(2.2) and integrating both sides of Eq.(2.8) from 0 to t, we obtain the following equation: Where the function is arising by integrating the source term from applying the given IVPs in Eq.(2.2) which are prescribed. To find out how this method works, the following steps are as follows: Step 1: to get ( ) solving ( ( )) ( ) with IVPs in Eq.(2.2) (2.10) and integrating both sides of Eq.(2.10) from 0 to t, we obtain ( ) ∫( ( )) ( ) Step 2 : The next iterate is L( ( )) -h(x,t) = 0 with IVPs in Eq.(2.2) (2.11) Solving this equation and integrating both sides of Eq.(2.11) from 0 to t, leads to obtain Step (3)  Evidently each iteration of the function ( ) represent effectively alone solution for Eq. (2.8). In the above steps, this method has merit to solve and apply KdV equation.

Numerical Examples:
We will be apply SAIM for solving several examples of the KdV equations.

Example 1:
We consider the KdV equation with the following IVP [28] : By implementing the same steps as described in the previous section, i.e. in section (2), we first begin by solving the following IVP to find the initial guess ( ) SAIM start as follows Applying the same steps as in the previous example, we first begin by solving the following initial problem in order to find the initial guess ( ) SAIM start by the same step in Eq.

Solution:
Implementing the same procedure as in the previous examples, we begin by solving the following initial problem in order to find the initial guess ( ) SAIM start as follows So, the primary step is L ( ) = 0, with ( ) ( ) (3.18) Then, the general relation is as follows By solving the primary problem in Eq.  with the IVP ( ) (3.24) Solution: By implementing the same steps as described in the previous section, the SAIM algorithm will be applied at the following equations (3.23) and (3.24). We first begin by solving the following IVP to find the initial guess ( ) as follows

Solution:
By implementing the same steps, we first begin by solving the following IVPs in order to find the initial guess ( ) SAIM apply accordingly the same step in Eq. (3.31) and (3.32)  We can be computed more iteration steps to get perfect results, but we stop at the first iteration since it is too long, so the few processes can represent the solution of Boussinesq equation. The essence of this method, the SAIM in comparison with the other analytical methods does not need large computations such as Lagrange multiplier in the VIM or any complex assumptions like nonlinear Adomian's polynomials in the ADM. It also does not need a long transformation or constructive homotopy polynomials in HPM. Furthermore, this method proved that it is efficient in overcoming the difficulties in calculating and solving KdV equation, a solitary wave for KdV equation and Boussinesq equation with easy steps.

4.Conclusion
This paper presents a technique method for solving the Korteweg-de Vries equations that gives faster and easier solutions. These solutions come accurate and in agreement with the exact solution

Yassein and Aswhad
Iraqi Journal of Science, 2019, Vol. 60, No.7, pp: 1575-1583 6711 provided by analytical results. This method is characterized with significant analytical work that effectively reduced the amount of computational work compared with the classical methods, Adomian Decomposition Method, Modified Adomian Decomposition Method and Variation Iteration Method which require longer times. Hence, all examples showed that the results are reliable, efficient, compatible with the exact solution, and much better than those obtained by other methods. We hope that this work is a step towards using applications of the Semi Analytical Iterative Method to resolve non-linear problems emerging various physical phenomena, which may be debated in further work.