Abstract
In this paper, Daftardar–Jeffery Polynomials are introduced in the Optimal Homotopy Asymptotic Method for solution of a coupled system of nonlinear partial differential equations. The coupled nonlinear KdV system is taken as test example. The results obtained by the proposed method are compared with the multistage Optimal Homotopy Asymptotic Method. The results show the efficiency and consistency of the proposed method over the Optimal Homotopy Asymptotic Method. In addition, accuracy of the proposed method can be improved by taking higher order approximations.
1. Introduction
Differential equations play a vital role in engineering and applied sciences. The 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. The exact solutions of nonlinear PDEs cannot be found easily. Different approaches have been adopted by researchers for the approximate solutions of these equations. The well-known approaches are Runge–Kutta method (R-K) [1, 2], Shooting method (SM) [3, 4], Finite Difference Method (FDM) [5–7], Finite Element Method (FEM) [8, 9], Collocation Method [10–13], and Homotopy Analysis Method (HAM) [14, 15]. Recently, Marinca et al. introduced the Optimal Homotopy Asymptotic Method (OHAM) for the solution of nonlinear problems [16–18] which made the perturbation methods independent of the hypothesis of small parameters and huge computational work. They 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.
The coupled nonlinear KdV system has the following form:
2. Basic Idea of OHAM-DJ
Consider the nonlinear differential equation:
In equation (2), is a linear operator, is a nonlinear operator, is an unknown function, is a known function, and is a boundary operator. According to OHAM-DJ, the optimal homotopy satisfies the following equation:
In equation (3), is an embedding parameter, is a nonzero auxiliary function for , and its value is zero for . The unknown function starts from to as approaches from 0 to 1.
The auxiliary function is chosen in the form:
Here, are constants. Next, we use the Taylor series to expand the function about :
The nonlinear function is decomposed as
The expressions on the right-hand side of (6) are the DJ polynomials given as follows:In fact, the above polynomials are the terms of Taylor’s series of the nonlinear term. The 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 , we get the different order problems given as follows.
The zero-th order problem is
The first-order problem is
The second-order problem is
The general governing equation for is given by using
The solution of higher order problems can be easily estimated; however, the second-order solution gives encouraging results.
For equation (5) reduces to
Replacing equation (15) into equation (2), we get the residual as
If , then we get the exact solution. Different methods can be used to estimate the values of constants ; 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 with respect to , we get the following system of equations containing :
Solving the above system, we get the values of . Replacing the values of in equation (14), we get the approximate solution.
The 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].
3. 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) iswhere are different parameters. Applying the proposed method, we have the following.
The zero-th order problem isIts solution isThe first-order problem isSolution of the first-order problem is given asAdding equations (22) and (24), we get first-order approximate solution by OHAM-DJ asThe values of convergence control constants are calculated using the method of least squares whose values are given as follows:
4. 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 by OHAM-DJ with MOHAM. Figures 1–3 show the residual graphs for at . It is worth pointing out that OHAM-DJ grants fast convergence solutions than OHAM.
5. Conclusions
OHAM-DJ has been successfully applied for the solution of a coupled nonlinear KdV system. The 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. This strategy is free from small parameter assumption and does not need any initial guess. The proposed method does not use discretization, and the convergence is controlled by self-assertive constants. Results revealed that OHAM-DJ is very consistent in comparison with MOHAM. The proposed method is an essential analytical method and well-organized in finding the solutions for an extensive class of coupled systems of PDEs. The accuracy of the proposed method can be further improved by taking higher order approximations. Extension of OHAM-DJ to the coupled nonlinear Korteweg-de-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
The 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.