Efficient Approximate Analytical Methods to Solve Some Partial Differential Equations

The goal of this research is to solve several one-dimensional partial differential equations in linear and nonlinear forms using a powerful approximate analytical approach. Many of these equations are difficult to find the exact solutions due to their governing equations. Therefore, examining and analyzing efficient approximate analytical approaches to treat these problems are required. In this work, the homotopy analysis method (HAM) is proposed. We use convergence control parameters to optimize the approximate solution. This method relay on choosing with complete freedom an auxiliary function linear operator and initial guess to generate the series solution. Moreover, the method gives a convenient way to guarantee the convergence of series solutions via the control parameter curve graphical method to rate the convergence and obtain the best solution. Decoding and analyzing potential Korteweg-de-Vries, Benjamin, and Airy equations, followed by convergence analysis to demonstrate the applicability of the method. By using the programs Mapel and Mathematica, the obtained results are very effective, accurate, and convergent to the


‫طرق‬ ‫الجزئية‬ ‫التفاضلية‬ ‫المعادالت‬ ‫بعض‬ ‫لحل‬ ‫فعالة‬ ‫تقريبية‬ ‫تحليلية‬ ‫الق‬ ‫عبد‬ ‫عيد‬ ‫محمد‬ ‫سعدون‬ ‫ا‬ ‫در‬
1. Introduction several real problems can be formulated by using mathematical models, these models can take the form of linear or nonlinear partial differential equations (PDEs), which represent an indispensable tool for modeling several physical and engineering problems.In this paper, three typical issues are examined, namely, the potential KdV equation, which is expected to repeat tsunami waves, the Benjamin equation, which is used to study long waves in shallow water, and Airy's partial differential equation, which belongs to the category of linear partial differential equations, which is used in a group variety of realistic physical applications, and it is one of the oldest models of water waves.Small wave "trains" in deep water.Unfortunately, The PDEs remain impractical to be solved to provide the physical or engineering describtion of the specified problems.Furthermore, the solutions give an overview of the features and properties of the physical and engineering problems.Although the analytical methods provide an exact solution to the problems, these methods apply only to some linear problems.At the same time, nonlinearity represent the govering enviroument of real-world phenomena.It will be urgent to provide alternative approximate methods to resolve these equations with acceptable accuracy.These methods are the main entrance to numerical analysis.The primary purpose of delivering approximate solutions is that most problems are too complicated to be solved exactly, or sometimes it is impossible to find analytical solutions [1].Several of the approximate analytical methods were formulated to solve PDEs, such as the homotopy perturbation method (HPM) [2], variational iteration method (VIM) [3], Adomian decomposition method (ADM) [4], the local meshless method (LMM) [5][6][7][8][9][10], the fractional iterative algorithm [11], modified variational iteration algorithm-I (mVIA-I) [12], modified variational iteration algorithm-II (MVIA-II) [13], and other methods.The significant gap in these proposed methods is their inability to control and adjust the convergence region of the approximate solutions, especially for the nonlinear cases, therefore such methods are not practical for solving nonlinear PDEs.Liao proposed the homotopy analysis method (HAM) in 1992 by employing the concept of the homotopy from topology to deform the nonlinear equations to a system of linear equations, making the complicated nonlinear equations easier to solve [14].The difference between HAM and other approximation methods is the auxiliary convergence control parameter, which can optimize and rate the convergence of the method per order of solution.The operator and auxiliary function with the optimal value of the convergence parameter allows for solving the deformation equations and developing a solution series to obtain series solutions for differential equations.
The HAM has been used to find solutions to problems with various differential equations [15][16][17][18][19][20].In this work, we will illustrate and formulate an approximate analytical method with the ability to control the convergence of solutions through applications by adopting an auxiliary parameter which is called the Homotopy Analysis Method (HAM) to optimize the approximate analytical solution of the linear and nonlinear PDEs.This research is arranged as follows: In part 2, we offer the outline of the HAM.In part 3, we will explain the applications of the method, and we will plot the obtained results.Part 3 includes details Parts 3, 4 and 5.The method is applied to the KdV equations, Benjamin equation and Airy partial differential equation, and approximate solutions from the exact solutions are shown.The results are compared with other analytical methods as shown in the tables and graphs.Finally, part 6 contains the conclusions of the research.

Outline of the Method
To describe the basic idea of the homotopy analysis method, we will impose the following nonlinear differential equation [19]: where  is a nonlinear operator, (, ) is an unknown function,  and t denote the spatial and the temporal independent variables, respectively.Let  0 (, ) be the initial guess of the exact solution (, ), ℎ ≠ 0 is the auxiliary convergence parameter, (, )) ≠ 0 is an auxiliary function, ℒ is an auxiliary linear operator, and  ∈ [0,1] represents the embedding parameter by means of the homotopy analysis method, according to [16], we construct the so-called zeroth-order deformation equation as below It is very significant that one has great freedom to choose the auxiliary functions of HAM.
Clearly, for  = 0, it holds the initial approximation of Eq. ( 1)  (, ; 0) =  0 (, ) While for  = 1, since ℎ ≠ 0 and (, t) ≠ 0 then we get the exact solution of Eq. ( 1) (, ; 1) = (, ) On the other hand, when  increases from 0 to 1, the approximate solution  (, t; ) deforms from the initial guess  0 (, ) to the exact solution (, ) According to [18], by utilizing the Taylor series theorem, the approximate solution (, t; ) expanded in a power series of  as follows: Where If the auxiliary linear operator, the initial guess, the auxiliary parameter, and the auxiliary function are so properly chosen, the series (3) converges to the exact solution at  = 1, then we have According to Eq. ( 4), the governing equation will be inferred from the zeroth-order deformation equation (2) .Define the vector Now, by differentiating Eq. ( 2) m-times with respect to the embedding parameter , then setting  = 0, and finally dividing them by m! we obtain the mth-order deformation equation as follows If the series ( 5) is convergent, that yields the approximate solution of Eq. ( 1) will converge to the exact solution at  = 1.Note that the homotopy analysis method contains the auxiliary parameter ħ, which provides us with that control and adjustment of the convergence of the series solution (5).

HOMOTOPY ANALYSIS METHOD FOR (P-KDV) EQUATION:
We use the homotopy analysis method to analyse the p-KdV equation, which is crucial in and of itself because it is thought to repeat tsunami waves.It is frequently observed while investigating water waves where the first term is the evolution term, (  ) 2 nonlinear term, and   scattering term are all present.
According to [16] and based on the above curve in Fig. 1, it is clear that the valid region of the series solution via forth order HAM for Eq. ( 9) corresponds to the line segment nearly parallel to the horizontal axis such that the series solution is convergent when −1.5 ≤ ℎ ≤ −0.8, and the optimum value of ℎ = −1.0005556227424213.The fourth order approximate series solution via HAM for the (p-KdV) equation compared with the exact solution are summarized in Table 1 for different values of ,  ∈ [0,1] as below Table 1: Comparison between the solutions of the approximate obtained by HAM and exact solutions of Eq. ( 9), also a comparison between absolute errors in HAM and RDTM of Eq. ( 9), at  =  = 1   = 0.5.We can also summarize the solutions of fourth order HAM series solution over all  ∈ [0,1] corresponding with the best value of ℎ = −1.0005556227424213for Eq. ( 9) compared with the exact solution (, ) in the following figures.

𝑢(𝑥, 𝑡)
4 (, ) Figure 2: The exact solution and fourth order HAM approximate solution of p-KdV equation at  =  = 1,   = 0.5 From Table 1 and Figure 2, we conclude that the fourth order HAM provide an accurate series solution of Eq. ( 9) compared with exact solution.

HOMOTOPY ANALYSIS METHOD FOR BENJAMIN EQUATION:
We use the homotopy analysis of Benjamin's equation, which is one of the most significant non-linear partial differential equations used to study long waves in shallow water.It simulates the single propagation of long internal waves of small amplitude along the interface of two fluid layers under the influence of gravity and surface tension.
we can identify the convergence region for the ℎ −curve to optimize the series solution of Eq. ( 16) we will plot the ℎ-curves of the second order HAM series solution (0.1,0.1;ℎ) as shown in Figure 3 below

Figure 3:
The h-curve of the solution in Eq.( 16). with n = 2.
According to [20], and because the valid region of the series solution via second order HAM for Eq. ( 16) corresponds to the line segment nearly parallel to the horizontal axis, then based on Figure 3, above, the series solution of Eq. ( 16) is convergent when ℎ = −1 Hence, the series solution convergent when the convergence parameter ℎ = −1 for solving Eq. ( 16) we have the following new convergence HAM series solution (, ) ≈  0 (, )+  1 (, )+  2 (, ) for different values of x, and t∈ [0,0.5], the second order approximate series solution via HAM for the Benjamin equation in Eq. ( 16) compared with the exact solution are summarized in Table 2.We can also summarize the solutions of the second order HAM series solution over all  ∈ [0, 5] corresponding with best value of ℎ = −1 for Eq. ( 16) compared with the exact solution (, ) in the Figure 4.

𝑢(𝑥, 𝑡)
2 (, )  According to Table 2 and Figures 3-4, we conclude that the second order HAM approximate series solution satisfies the initial condition of the Benjamin equation for  = −1 ,  = −4 and  = 0.25 with sufficient accuracy compared with the exact solution of Benjamin equation.Furthermore, from Table 2, we conclude that the accuracy of the approximate solution solved by the second order HAM is better than the fourth order HAM for solving RDTM approximate solution at ℎ = -1, and  = 0.1.Finally, we will show the 3-D accuracy of the second order HAM series for solving the Benjamin equation at  = -1, β = -3, and  = 0.25 for all ,  ∈ [0,5] in Figure 5.

APPLYING HOMOTOPY ANALYSIS METHOD FOR AIRY EQUATION:
In this part, we apply the homotopy analysis method of Airy's equation, which is one of the first models of water waves and is one of the linear partial differential equations which is used in many practical physical applications.

Homotopy analysis method for
We will construct the zeroth-order deformation equation as follows: ( ) Now, by applying ℒ −1 on both sides of Eq. ( 27) we have: Finally, According to HAM series in Section 2, and since m ≥ 1, χ  = 1 , (r, t) =1, we have the following approximate series solution of the Airy equation  −1 (, ) =  −1 (, 0) + + ℎℒ −1 [  ( −1 , , )] Then, we have the following HAM series terms for solving the Airy equation starting with the initial approximation as follows: we can identify the convergence region for the ℎ −curve to optimize the series solution of the Airy equation by plotting the ℎ-curves of the tenth order HAM series solution (0.1,0.1;ℎ) as shown in Figure 6.According to [20] and based on Figure 6 above the h-curve that provides the convergence region of tenth order HAM for solving the Airy equation is parallel to the horizontal line.Therefore, it is straightforward to choose an appropriate range for h which ensure the convergence of the solution series.We identify the h-curve of u'(0.01;0.01) in Figure 7, which shows that the solution series is convergent when −0.4 < ℎ < −1.5.with in the valid region of −0.4 < ℎ < −1.5 we have three convergence control parameters; in the following figure we will show the accuracy of the Airy equation solving by tenth order HAM series solution based on each convergence control parameter of ℎ:   According to Table 3 and Figure 7, we conclude that the tenth-order HAM approximate series solution satisfies the initial condition of the Airy equation with sufficient accuracy compared with the exact solution.Finally the comparison between the tenth order HAM series and the exact solution of Airy equation explained below in

CONCLUSION
An approximate analytical approach for solving partial differential equations in parabolic form was formed as a result of this research.It is discussed and studied here how to use a method based on HAM to approximate the solution to the Korteweg-de Vries equation, the Benjamin equation, and the Airy equation.The research illustrates that HAM can be implemented by managing the convergence of the series solution through the benefit of the convergence control parameter being convergent to the exact solution.An engineer or scientist can benefit from this method to obtain a better knowledge of a physical problem.It is expected that it will contribute to improving future methods and designs employed to confront their obstacles, which plays an essential role in solving various mathematical problems.The test problems demonstrated good accuracy when tested against the right solution.It tends to be a suitable method for solving parabolic partial differential equations because of its efficiency and conformity with the HIM.This approach will be utilised in the future when we need to address problems involving elliptic and hyperbolic models.

Figure 6 :
Figure 6: The h-curve of the solution in example 5.3 with  = 10.

Figure 7 :
Figure 7: The absolute error of Airy equation solving by tenth order HAM series Based on the Figure 7, one can say the tenth order HAM series provided various convergence control parameters, on the other, the convergence parameter ℎ = −0.9732306585978938 is the best convergence control parameter to optimize the approximate solution of Airy equation via tenth order HAM series as shown in next Table.

Figure 8 :
Figure 8: The 3D graph of the exact and approximate solution via tenth order HAM based on the best convergence control parameter h for solving the Airy equation for all 0 ≤  ≤ 1, and 0 ≤  ≤ 0.5

Table 3 :
The numerical results for the approximate solutions obtained by tenth order HAM compared with the exact solution of Eq. (23), when  = 1, ℎ = −0.9732306585978938.