Improved Finite Difference Technique via Adomian Polynomial to Solve the Coupled Drinfeld’s–Sokolov–Wilson System

Tis study presents a new algorithm for efectively solving the nonlinear coupled Drinfeld’s–Sokolov–Wilson (DSW) system using a hybrid explicit fnite diference technique with the Adomian polynomial (EFD-AP). Te suggested approach addresses the problem of accurately solving the DSW system. Numerical results are obtained by comparing the exact solution with absolute and mean square errors using a test problem to assess the EFD-AP accuracy against the exact solution and the conventional explicit fnite diference (EFD) method. Te results exhibit excellent agreement between the approximate and exact solutions at diferent time values, and the results showed that the proposed EFD-AP method achieves superior accuracy and efciency compared to the EFD method, which makes it a promising method for solving nonlinear partial diferential systems of higher order.


Introduction
Te exploration of nonlinear equations has a long and storied history, dating back to ancient civilizations.Mathematicians of the past were driven to address challenges involving curved shapes and nonlinear phenomena.However, it was not until the 17th century when Sir Isaac Newton and Gottfried Wilhelm Leibniz developed calculus that the fundamental groundwork for solving nonlinear equations began to take shape.Troughout the centuries, renowned fgures in the world of mathematics and science, such as Lagrange, Gauss, and Poincaré, made substantial contributions to the theory and methods for solving nonlinear equations [1].
In the 18th and 19th centuries, the fnite diference method emerged as a numerical technique for approximating solutions to diferential equations [2].Initially, it was applied to linear equations, but its popularity surged with the advent of digital computers in the mid-20th century.Te fnite diference method involves discretizing the continuous domain by dividing it into a grid of discrete points and approximating derivatives using fnite diference formulas.Tis approach efectively transforms diferential equations into systems of algebraic equations, enabling numerical solutions.Te scope of the fnite diference method expanded to encompass nonlinear equations, rendering it a potent tool for addressing challenges in felds such as physics, engineering [3], and fnance.Various felds, including mathematical physics, theoretical physics, and integrable systems, heavily rely on the coupled Drinfeld's-Sokolov-Wilson (DSW) system.Its applications encompass understanding soliton solutions in nonlinear wave equations, shedding light on quantum felds and string theories, and exploring nonlinear dynamics, chaos theory, and nonlinear optical phenomena, particularly in optical fber technology.Moreover, it plays a vital role in information theory, quantum computing, and mathematical modeling across numerous physical domains.In the academic realm, the DSW system serves as an invaluable resource for advanced studies and instruction in mathematics and physics [4].
In summary, nonlinear equations boast a rich historical background and the development of the fnite diference method has been pivotal in enabling numerical solutions.Tese advancements have signifcantly enhanced our grasp of intricate nonlinear phenomena and our capacity to address real-world challenges.For instance, methods such as the Adomian decomposition method with modifed Bernstein polynomials [5], Bernstein polynomials in conjunction with artifcial neural networks [6], and the Elzaki decomposition method for tackling higher-order integrodiferential equations have all contributed to this progress [7][8][9][10][11][12][13][14].Te formula for the Drinfeld's-Sokolov-Wilson (DSW) system also plays a critical role in this context [14][15][16][17][18].
In the system of ( 1) and ( 2), p, q, r, and s are nonzero constants, originally introduced by Drinfeld, Sokolov [8], and Wilson [1], and they represent a model for water waves and fnd extensive applications in the feld of fuid dynamics.Over the years, various numerical and analytical methods have been developed to address these equations, including the Adomian decomposition method [16], the exp-function method [10], the improved F-expansion method [19], the bifurcation method [20], and qualitative theory [21].Tese methods ofer diverse approaches to solve systems (1) and ( 2), contributing to a better understanding of their behavior and properties.Traditionally, techniques such as the inverse scattering transform, Hirota's bilinear method, and other integrable methods have been employed to analyze the DSW system.
Te motivation behind this research is to introduce a novel hybrid approach that combines the fnite diference method with the Adomian polynomial.Tis hybrid method aims to address the nonlinear term in the DSW system and handle cases where points fall outside the solution region.By doing so, this innovative technique enhances the accuracy and efciency of the solution while dealing with the nonlinearities present in the system.In this research, we cover fundamental concepts, the mathematical formulation of the EFD and EFD-AP methods, their respective algorithms, the convergence analysis of EFD-AP, and their practical application, see [22][23][24].

Main Concepts
2.1.Explicit Finite Diference (EFD) Method.Te equations in question contain a third derivative term, which presents a distinctive challenge that calls for the adoption of a specialized numerical solution method.Te peculiar impact of the third derivative on these equations and their behavior makes it necessary to employ a diferent approach or method to ensure accurate and efcient problem solving.To address this, we will utilize the following set of discrete equations [25].

Te Adomian Polynomial (AP).
Te method assumes that the nonlinear term can be approximated by an infnite series of polynomials expressed in a specifc form.

International Journal of Mathematics and Mathematical Sciences
Now, Te Adomian decomposition method's stability was examined in the research conducted by Azim and Hosseini [27].Furthermore, the analysis of its convergence was established by Ahmed [28] and Zhang [19].

Methodology
In this section, the fnite diference method involves solving the DSW system by substituting the derivatives in ( 1) and ( 2) with central fnite diference derivatives as given in (7) and (9).Tis transformation converts the system into a set of regular algebraic equations that can be solved using standard methods to determine the values of certain variables.On the other hand, the hybrid method, which is the EFD-AP method, relies on replacing the nonlinear component in the fnite diferences with central derivatives.After this substitution, we simplify the system, resulting in a set of algebraic equations that are solved to obtain the values of u and v.

Mathematical Formulation of the EFD Method for the DSW System.
When applying the EFD method, solutions appear outside the region and, therefore, this case is dealt with in the second equation of the system in the following manner.
Divide the rectangle R � (x, t): a ≤ x ≤ b, 0 ≤ t ≤ c { } into n− 1 and m− 1 of rectangles length of each side (∆x � H) and (∆t � k).We start from t � t 0 � 0, where the solution values are calculated from the initial conditions as follows: Ten, the calculation of approximations is explained for u(x, t) and v(x, t), and the grid points of the other rows u(x i , t j ), v(x i , t j ), i � 1, 2, . . ., n− 1, j � 1, 2, . . ., m   and the solution method will be as follows.
To calculate the approximate value of the solution u(x i , t j ), we use (3) and the central fnite diference (7), and substituting them into (1), we get Let α � k/2H and by rearranging the above equation, we get Tis equation computes the value of the solution u at level j + 1 from the known values of level j at i � 1, 2, . . ., n − 1 from (1), and to calculate approximate values for the solution v(x i , t j ) when i � 1, we use forward diference equations ( 3), ( 4), and ( 6), and substituting them into (2), we get Suppose that β � k/2H 3 , μ � k/2H and by rearranging the above equation, we get International Journal of Mathematics and Mathematical Sciences Equation ( 19) denotes the solution v at point x 1 on the grid.
To determine the remaining approximate values v(x i , t j ), where i is in the range of 2, 3, . . ., n − 2 within the same row, we substitute the central ( 7) and ( 9) into (2) as follows: By compensating for values of α, β, and µ and rearranging equation ( 20), we get Equation ( 21) computes the value of the solution v at level j + 1 using the known values of level j at i � 3, 4, . . ., n − 2.

Subsequently, we can calculate the values of v at level
Similarly, by substituting the backward equations ( 10) and ( 12) into ( 2) and rearranging, we obtain Tis equation calculates the solution v at the point x n− 1 of the grid.Upon examining the central numerical approximation of the third derivative as defned in (22), we observe that the two approximate solutions computed at grid node locations include values for points situated beyond the boundaries of the solution domain and these values remain unknown.Furthermore, the boundary conditions cannot be applied to these points as they do not fall within the boundary constraints.It is important to highlight that the local truncation error for these points is signifcant that the local truncation error for the method is (i, j) � O(k) + O(H 2 ); the method is stable and its stability condition is 0 ≤ α ≤ 1/2 (Algorithm 1).

Mathematical Formulation of the EFD-AP Method.
To calculate the approximate value of the solution u(x i , t j ), we substitute the central fnite diference (7) into (1), and in the second step, we substitute the Adomian polynomial  ∞ n�0 An into (22), i.e., u Now, by replacing the above derivative by the fnite diferences formula 4 International Journal of Mathematics and Mathematical Sciences and by compensating equation ( 24) in (1), we obtain Tis equation computes the value of the solution u at level j + 1 from the known values of level j at i � 1, 2, . . ., n − 1 from (1).
To calculate approximate values for the solution v(x i , t j ) when i � 1, we use forward diference equations (3), (4), and (6), and substituting them into (2), we get where  ∞ n�0 An and  ∞ n�0 Bn are Adomian polynomials as follows: Now, by replacing the above derivative by the fnite diferences formula Input: the values of a, b, h, and confections p, q, r, and s, number of space fragmentation n, and for time m.
Step 1: evaluate H � (b − a)/n space step size and k � t/m time step size.
Step 2: fnd the values of α � T/2H, β � T/H 2 , and µ � k/2H 2 Step 3: compute the initial condition and boundary conditions Step 4: compute the numerical solution Step 6: compute the numerical solution Step 7: print the numerical solutions u(x, t) and v(x, t).
ALGORITHM 1: Algorithm of the EFD method.
International Journal of Mathematics and Mathematical Sciences ), and so on, we obtain where To determine the remaining approximate values v(x i ,t j ), where i is in the range of 2, 3, . . ., n − 2 within the same level, we substitute the central equations into equation ( 2) as follows: Subsequently, we can calculate the values of v at the level j + 1 when i � n − 1.
Similarly, by substituting the backward equations ( 10) and ( 12) into (2) and rearranging, we obtain By expressing nonlinearities through Adomian polynomials, we can enhance the precision in approximating nonlinear components.Tis has the potential to result in a more precise overall solution, particularly when extending the polynomial expansion to higher degrees.In addition, augmenting the number of terms in the Adomian polynomial can boost the efciency of the method proposed.However, choosing the optimal number of terms in the Adomian expansion to strike a balance between accuracy and computational efciency can be a challenging task.
Similar to other numerical methods, errors have the potential to propagate and amplify, especially if not applied correctly.

Convergence Analysis of the EFD-AP Method.
Consider the approximate solution for the DSW system from (25) and the exact solutions of this system  U,  V and all the necessary derivatives of  U,  V exist and from (25), we have where T i,j is the local truncation error.

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Input: the values of a, b, h, and confections p, q, r, and s, number of space fragmentation n, and for time m.
Step 1: evaluate H � (b − a)/n space step size and k � t/m time step size.
ALGORITHM 2: Algorithm of the EFD-AP method.International Journal of Mathematics and Mathematical Sciences 7 Defning the function and then subtracting ( 25) from (32), we get Te initial and boundary conditions are c 0 j � 0 , c j 0 � 0, and where Tis is trivially true for the cases i � 0 and i � n + 1.
and from the initial condition ∈ 0 � 0 and by induction, we can show that ∈ j ≤ jKT for all j ≥ 0. Since t j � t 0+ jK, then t j � jK and We have convergence if keeping x i , t j fxed and letting which gives the convergence.By the same approach, we have the approximate solution from (30); similarly, if we take the approximate solution from ( 29) and (31), we obtain (39) In these three cases, we have |v It is worth mentioning that the proposed method is stable and its stability condition is 0 ≤ R ≤ 1/2 (Algorithm 2).

Numerical Application
Te analytic solution of the DSW system given by [2,29] To assess the accuracy and convergence of the numerical solution in comparison to the exact solution (as outlined in (41) and ( 42)), we will employ two error criteria, namely, the absolute error (ABSE) and the mean square error (MSE).Tese error metrics enable us to quantitatively evaluate how closely the numerical solution aligns with the exact solution.Te application of these error measures yields valuable insights into the efectiveness and reliability of the numerical method being used.
Comparing Tables 1 and 2, it becomes evident that the MSE values obtained through the proposed technique surpass those achieved by the explicit approach.Tis implies that better results can be achieved using the proposed method.In addition, Figures 1 and 2  solution alongside the numerical solutions generated by the EFD and EFD-AP methods for both u and v. Tese visual representations correspond to specifc parameter values, namely, a � 0, c � 2, t � 0.01, H � 0.4, k � 0.002, and s � 5, where "s" denotes the number of iterations in 2D and 3D, respectively.
Tese results underscore the efciency of the EFD-AP method in adeptly managing the nonlinearity within the equation, leading to accurate and efcient solutions.In addition, the comparison of outcomes derived from the EFD-AP and EFD methods against the exact solution further reinforces this point.

Conclusion
In this study, we introduce a novel approach called EFD-AP, which integrates the fnite diference method with the Adomian polynomial to address the nonlinearity within the DSW system.Tis innovative method ofers a fresh perspective on handling the nonlinear aspects of the equations, resulting in improved accuracy and efciency of the solutions.By combining the strengths of these two methods, our goal is to overcome the limitations of conventional numerical techniques, achieve more precise approximations for the DSW system, and address situations where points fall outside the solution region.Te fnite diference method provides a robust framework for discretizing the equations in space, while the Adomian polynomial ofers a systematic and efcient method for solving the resulting nonlinear subproblems.We conducted a convergence analysis of the proposed method and found that EFD-AP exhibits greater stability compared to the EFD method, as shown in Figure 3, as evidenced by the results presented in Tables 1 and 2 and in Figures 1-4.Te EFD-AP approach ofers enhanced computational efciency, higher precision, reduced mean square error, improved stability with larger time step sizes, and faster convergence.Tis approach also proves to be highly efcient and suitable for approximating solutions.In the future, this method can be extended to tackle nonlinear fractional systems and nonlinear fuzzy systems.In addition, there is a potential to develop new techniques by combining the homotopy method with the fnite diference method.

Figure 1 :Figure 2 :
Figure 1: 2D solutions of the exact, EFD, and EFD-AP methods for each of u and v at a � 0 and c � 2.

Figure 3 :
Figure 3: Comparison EFD, EFD-AP, and exact solutions for u with diferent times.

Figure 4 :
Figure 4: Comparison EFD, EFD-AP, and exact solutions for v with diferent times.

Table 1 :
Comparing the EFD and EFD-AP methods with the analytical solution for the DSW system for u.

Table 2 :
Comparing the EFD and EFD-AP methods with the analytical solution for the DSW system for v.Bold values represent the overall mean for MSE.