Utilizing the Optimal Auxiliary Function Method for the Approximation of a Nonlinear Long Wave System considering Caputo Fractional Order

In this article


Introduction
A variation of classical calculus known as fractional calculus (FC) deals with noninteger (fractional)-order integration and diferentiation procedures.Fractional operator theory was introduced nearly simultaneously with the creation of classical ones.Te subject of the semiderivative meaning was brought up in a correspondence between G. W. Leibniz and Marquis de l'Hospital in 1695, and here is where the earliest instance of this may be discovered [1][2][3].As a result, this problem attracted the attention of several eminent mathematicians, including Euler, Liouville, Laplace, Riemann, Grünwald, Letnikov, and many more [4].Te rapid development of the theory of fractional calculus throughout the eighteenth century has been extremely benefcial to fractional diferential equations (FDEs), fractional dynamics, and other practical domains.FC is employed in many diferent applications these days.It is accurate to state that fractional calculus's techniques and tools have an infuence on almost every area of modern engineering and science in general.
Te versatility of fractional calculus is evident in its applications across diverse felds such as bioengineering, rheology, viscoelasticity, acoustics, optics, robotics, control theory, chemical and statistical physics, and electrical and mechanical engineering [5][6][7][8].One may even claim that fractional-order systems in general explain real-world occurrences.Te major reason for the success of FC applications is that these new fractional-order models are often more accurate than integer-order models; that is, the fractional-order model has more degrees of freedom than the similar classical one.Fractional derivatives (and integrals) are not local (or point) variables, which is one of the subject's fascinating aspects.In order to simulate the nonlocal and dispersed efects frequent in technological and natural events, all fractional operators take into account the whole history of the process being studied.
In practice, fractional calculus serves as a valuable tool for understanding the memory and hereditary characteristics of materials and processes.Tere are several available methods in the literature for approximating problem-related diferential equations, both linear and nonlinear.A linear problem's solution is easier to approach than a nonlinear one.For these issues, several numerical and analytical methods have been proposed, including the control volume scheme, the Laplace transformation method, the fnite element method (FVM), the Adomain decomposition method (ADM), the variation iteration method (VIM), and the homotopy analysis method (HAM) [9][10][11][12].Although these techniques ofer many advantages, not all issues can be solved with them.Te strategy put out by Vasile Marinca uses a potent method called optimum auxiliary function.For fractional-order equations containing the Caputo operator, we introduce OAFM in this study as a unique variation of the recently created semianalytical technique known as the optimal auxiliary function method (OAFM).It is explained how OAFM works mathematically, and its efcacy is demonstrated by applying it to the well-known ALW.To show the OAFM's validity, tables and charts are used to contrast the results of the OAFM with those of other approaches and their precise answers.A quick convergence series solution from OAFM is verifed by contrasting it with other outputs.
Te study shows that our approach is straightforward to use, needs minimal computing efort, and quickly converges to the precise solution within the frst iteration.A variety of solutions to the issue are found using OAFM.By contrasting the OAFM results with those from the literature, the validity of the results is confrmed.OAFM is discovered to be quickly convergent, less computationally intensive, and easily adaptable.OAFM involves less computing work than other approaches, and even a low-spec machine can easily complete it.Te Optimal Auxiliary Function Method, despite its advantages, has certain limitations.One key limitation is its applicability to specifc types of problems.Te method may not be suitable for all types of fractional nonlinear long wave equations or for problems with certain boundary conditions.Additionally, the method's efectiveness can depend on the choice of the auxiliary function, which may not always be straightforward to determine, especially for complex problems.Overall, while the Optimal Auxiliary Function Method is a powerful tool, its limitations should be considered when applying it to solve fractional nonlinear long wave equations.
Te future direction of this work could involve further exploring the capabilities of the adapted Optimal Auxiliary Function Method (OAFM) for solving a wider range of complex fractional diferential equations.One direction could be to investigate its applicability to systems of equations or to problems with more intricate boundary conditions.Additionally, the method could be refned or extended to handle problems in diferent domains or with diferent types of fractional orders.
Te novelty of this work lies in its successful application of the adapted OAFM to a wide range of linear and nonlinear fractional diferential equations, showing its potential for solving real-world problems with simplicity, speed, and efciency.Tis study's contribution is its innovative approach to improving an existing method and its demonstration of the method's applicability to various complex models, highlighting its usefulness in solving fractional-order integro-diferential equations arising from physical processes.It is strongly advised that you carry out the suggested system research in order to understand continuous quantum measurement and estimation.We are dealing with the ALW system, which is given as follows: where 0 < η, ω ≤ 1. Subject to the subsidiary conditions, ( Tis article's sections are organised as follows.Basic terms are given in Section 2. In Section 3, the proposed method for solving the current model is covered in detail.Several challenges are tried in Section 4, and the results and conclusions of the tests are provided in Section 5.

Basic Terminologies
To understand the OAFM concept, the following is a list of basic terms.
where the special function symbolised by Γ is the gamma function.
Defnition 2. Te Riemann-Liouville order function f ′ s fractional derivative is defned as In this case, p is a positive integer that satisfed Defnition 3. Caputo states that a fractional derivative of order is as follows [16].For Defnition 4. A numerical technique for resolving an integral, partial, or ordinary diferential equation is the collocation method.A fnite-dimensional space of polynomials up to a specifed degree and a specifc number of points (collocation points) in the domain is chosen, and the solution that solves the provided equation is then chosen at the collocation points.
Defnition 5. Auxiliary functions are not predetermined types of functions; instead, they are functions that are either expressly established or at the very least proved to exist, present a contradiction to some presumptive notion, or otherwise prove the desired outcome.Defnition 6.A mistake often manifests as a diference between an estimated value and an exact mathematical value.Terefore, absolute error refers to the size of the discrepancy between the precise value and the approximate value.
where a is exact solution and  a is approximate solution.

Formulation of Mathematical Models
A partial diferential equation of fractional order is expressed in general form as Subject to the boundary conditions, where (z η /zβ η ) denotes the Caputo or R-L operator, an unknown function is denoted by ϕ(α, β), whereas a known statistical function is denoted by θ(α, β).
Step 7. Two-component form of equation ( 8) will be taken into account in order to get the estimated answer of the equation and is presented as Step 8. We obtain the zero-and frst-order solution by substituting equation ( 10) into (8), which is given as Step 9.Because the nonlinear equation is complicated and has a difcult time being solved, we utilise the linear equation to generate an initial approximation of the kind shown below.Lastly, we use the result to inform our frst forecast.
With the help of the inverse operator, we arrive to ϕ 0 (α, β) as follows: Step 10.Te expanding version of the nonlinear component in equation ( 11) is Step 11.Let us suggest an equation to simplify equation ( 14), smooth its convergence, and accelerate the frst-order approximation.Te expression is shown below  ϕ(α, β): where G 1 and G 2 are the auxiliary functions depending upon ϕ 0 (α, β) and convergence control parameter C i and in the combination of both ϕ 0 (α, β) and L 2 [θ 0 (α, β)] but they are not particular.
Step 14. Tere are many methods in the literature such as Galerkins method, Ritz method, and collocation method, for the values of C i and C j , using which one must compute the square of the residual error.
In this context, the residual R is defned.
Te following system will function as the convergence control parameter:

Applications
In this part of the article, a few instances are given to demonstrate the precision and intensity of the method that was previously explained.

Complexity
Using equations ( 26) and ( 28) into (27), we obtained the frst approximation as We derive an approximation of the frst-order solution as equations ( 28) and (29).8 Complexity

Discussion
OAFM was used to resolve the nonlinear ALW system's fractional-order equations.Section 4, tables, and fgures for the ALW system give the results of OAFM for the fractionalorder equation using ADM, VIM, and OHAM.
In the problem, the absolute errors of the variational iteration method (VIM) solution, the Adomian decomposition method (ADM) solution, the optimum homotopy asymptotic method (OHAM) solution, and the second-order new iterative method (NIM) solution for the fractional-order approximate long wave (ALW) equation's ϕ(α, β) and θ(α, β) variables are compared with η � ω � 1. Table 1 displays the various values of C 1 and C 2 used in the calculations.Te values of C 1 and C 2 are crucial parameters in the solutions obtained by the diferent methods.Tables 2-5 provide insight into how the methods perform with diferent choices of these parameters, showing which combinations lead to more accurate solutions for ϕ(α, β) and θ(α, β).Te tables likely contain numerical values showing the errors for each method and parameter combination, allowing for a detailed comparison of their performance.
Figures 1 and 2 compare the precise and Optimal Auxiliary Function Method (OAFM) solutions in 2D plots of ϕ(α, β) and θ(α, β) at β � 0.1.Tese fgures likely illustrate how well the OAFM approximates the precise solution for diferent values of α.In Figures 3 and 4, the frst-order OAFM solution of ϕ(α, β) and θ(α, β) is shown for various values of the parameter ω.Tese fgures likely demonstrate how the OAFM solution changes with diferent values of ω and how it compares to the precise solution.Figures 5-8 display the accurate answer and the OAFM 3D graphic of ϕ(α, β) and θ(α, β) for the problem at β � 1. Tese fgures likely provide a detailed comparison between the accurate solution and the OAFM solution in a 3D graphical format, showing the behavior of ϕ and θ in the α-β plane and highlighting any discrepancies between the two solutions.

Conclusion
In this work, we instituted a systematic adaptation in employing the OAFM to approximate the fractional nonlinear long wave equation with the application of Caputo fractional order.Several problems, both linear and nonlinear, that entail fractional diferential equations were investigated, and it is shown that the suggested adjustment to the Optimal Auxiliary Function Method has increased the method's efectiveness compared to it's the previous iteration.Te OAFM takes less computational work than previous methods, and even a machine with smaller space may successfully fnish the operation.Tis method is currently unrestricted, allowing us to utilize it in the future for more intricate models drawn from real-world difculties.Moreover, it is observed in this paper that OAFM is simple, quick, and efcient.Tus, fractional-order integro-diferential equations that arise from physical processes can be solved using the suggested approach based on our mathematical fndings.However, certain limitations, such as scope, assumptions, and numerical stability, should be considered when applying the method to real-world problems.Future research can address these limitations to further improve the method's applicability and accuracy.

Figure 4 :
Figure 4: Efect of ω on the solution for OAFM to the problem.

Figure 3 :
Figure 3: Efect of η on the solution for OAFM to the problem.

Table 1 :
Confguring convergence control settings for varying η values in given problem.