Extension of Optimal Homotopy Asymptotic Method with Use of Daftardar–Jeffery Polynomials to Coupled Nonlinear-Korteweg- De-Vries System

Department of Mathematics, Abdul Wali Khan University, Mardan, KP, Pakistan Department of Electrical Power Engineering, U.S.-Pakistan Center for Advanced Studies in Energy (USPCAS-E), National University of Sciences and Technology, Islamabad (44000), Pakistan Department for Management of Science and Technology Development, Ton Duc/ang University, Ho Chi Minh City, Vietnam Faculty of Electrical & Electronics Engineering, Ton Duc/ang University, Ho Chi Minh City, Vietnam


Introduction
Differential equations play a vital role in engineering and applied sciences. e nonlinear coupled Partial Differential Equations (PDEs) have a variety of applications in physics, acoustics, optics, elasticity, hydrodynamics, aerodynamics, electromagnetism, chemical kinetics, economics, computer science, and financial mathematics. e exact solutions of nonlinear PDEs cannot be found easily. Different approaches have been adopted by researchers for the approximate solutions of these equations. e well-known approaches are Runge-Kutta method (R-K) [1,2], Shooting method (SM) [3,4], Finite Difference Method (FDM) [5][6][7], Finite Element Method (FEM) [8,9], Collocation Method [10][11][12][13], and Homotopy Analysis Method (HAM) [14,15]. Recently, Marinca et al. introduced the Optimal Homotopy Asymptotic Method (OHAM) for the solution of nonlinear problems [16][17][18] which made the perturbation methods independent of the hypothesis of small parameters and huge computational work. ey used more flexible function called the auxiliary function which controls the convergence of the proposed method. To improve the accuracy and to ensure the faster convergence, one can use an increased number of convergence control parameters in the first order of approximation. A more general formulation, which emphasizes the above features, was presented in [19,20]. Later on, Ali et al. introduced a new method based on adaptation of the Optimal Homotopy Asymptotic Method (OHAM) with Daftardar-Jeffery polynomials, called OHAM-DJ, to solve nonlinear problems [21,22]. Shah et al. applied OHAM-DJ for the solution of linear and nonlinear Klein-Gordon equations [23]. Our main goal in this work is to extend the applications of OHAM-DJ to a coupled nonlinear KdV system.

(2)
In equation (2), L is a linear operator, N is a nonlinear operator, ζ is an unknown function, g is a known function, and B is a boundary operator. According to OHAM-DJ, the optimal homotopy H(φ(η, t; s)): Ω × [0, 1] ⟶ R satisfies the following equation: In equation (3), s ∈ [0, 1] is an embedding parameter, H(s) is a nonzero auxiliary function for s ≠ 0, and its value is zero for s � 0. e unknown function ζ(η, t; s) starts from ζ(η, t; 0) � ζ 0 (η, t) to ζ(η, t; 1) � ζ(η, t) as s approaches from 0 to 1. e auxiliary function H(s) is chosen in the form: Here, C 1 , C 2 , C 3 , . . . are constants. Next, we use the Taylor series to expand the function φ(η, t; s) about s: e nonlinear function N(ζ(η, t; s)) is decomposed as e expressions on the right-hand side of (6) are the DJ polynomials given as follows:

(7)
In fact, the above polynomials are the terms of Taylor's series of the nonlinear term. e convergence of these polynomials was determined by Bhalekar and Daftardar-Gejji [24]. For simplification we expressed the polynomials as Now, we can also express Substituting equations (8)-(10) in equation (3) and comparing the like terms of s, we get the different order problems given as follows.
e zero-th order problem is e first-order problem is e second-order problem is e general governing equation for ζ j (η, t) is given by using 2 Complexity e solution of higher order problems can be easily estimated; however, the second-order solution gives encouraging results.
For s � 1, equation (5) reduces to Replacing equation (15) into equation (2), we get the residual as If R(η, t; C i ) � 0, then we get the exact solution. Different methods can be used to estimate the values of constants C 1 , C 2 , C 3 , . . . , C j ; but the method of least square is the most common method. In the method of least square, we minimize the errors by taking the square of the residuals over the given domain to get the following functional: Differentiating J with respect to C 1 , C 2 , C 3 , . . . , C j , we get the following system of equations containing Solving the above system, we get the values of C 1 , C 2 , C 3 , . . . , C j . Replacing the values of C 1 , C 2 , C 3 , . . . , C j in equation (14), we get the approximate solution.
e method of least squares is a powerful technique and has been used in many other methods such as Optimal Homotopy Perturbation Method (OHPM) and Optimal Auxiliary Functions Method (OAFM) for calculating the optimum values of arbitrary constants [25,26].

Implementation of OHAM-DJ to a Coupled System of Nonlinear KdV Equations
Problem 1. Consider system (1) with initial conditions [27]: Exact solution of equation (1) is where t � 1, ω � 1.5, l � 0.1, C 0 � 1.5, and C 2 � 0.1 are different parameters. Applying the proposed method, we have the following. e zero-th order problem is Its solution is e first-order problem is

Results and Discussion
OHAM-DJ has been extended for the solution of coupled system of KdV equations. Tables 1-3 show the comparison of absolute errors of first-order approximate solutions Table 1: Absolute errors obtained by first-order OHAM-DJ and comparison is made with MOHAM of ζ(η, t) at t � 1, ω � 1.5, l � 0.1, C 0 � 1.5, and C 2 � 0.1.    Figures 1-3 show the residual graphs for ζ(η, t), ξ(η, t), and w(η, t) at t � 0.1. It is worth pointing out that OHAM-DJ grants fast convergence solutions than OHAM.

Conclusions
OHAM-DJ has been successfully applied for the solution of a coupled nonlinear KdV system. e proposed method contains an adaptable auxiliary function that is used to control the convergence of the solution and grants alteration inside the convergence region wherever it is required. is strategy is free from small parameter assumption and does not need any initial guess. e proposed method does not use discretization, and the convergence is controlled by selfassertive constants. Results revealed that OHAM-DJ is very consistent in comparison with MOHAM. e proposed method is an essential analytical method and well-organized in finding the solutions for an extensive class of coupled systems of PDEs. e accuracy of the proposed method can be further improved by taking higher order approximations. Extension of OHAM-DJ to the coupled nonlinear Kortewegde-Vries system is more accurate and as a result, it will be more appealing for researchers to apply this method to the coupled system of partial differential equations arising in different fields of engineering sciences.

Data Availability
All the data and meta-data concerning the findings of the manuscript are given in the manuscript.

Conflicts of Interest
e authors declared that they have no conflicts of interest.

Authors' Contributions
Zawar Hussain carried out the simulations. Rashid Nawaz linguistically edited and sequenced the paper. Abraiz Khattak made the corrections in the paper, and Adam Khan drafted the paper.