Abstract
This paper presents the concept of a new strategy that examines the analytical solution of the Harry Dym equation (HDEq) with fractional derivative in the Caputo sense. This new approach is called the Mohand homotopy perturbation transform scheme (MHPTS) which is constructed on the basis of the Mohand transform (MT) and the homotopy perturbation method (HPM). The implementation of MT produces the recurrence relation without any assumption and hypothesis theory whereas HPM is additionally used to overcome the nonlinearity in differential problems. Our primary focus is to handle the error analysis in the recurrence relation and generates the solution in the order of series. These obtained results yield the exact solution very rapidly due to its fast convergence. Some graphical representations are demonstrated to show the high efficiency and performance of this approach.
1. Introduction
Fractional calculus is a different study of examining the properties of integral and derivatives of nonintegers order. This theory of FC has a significant advantage due to its involvement of a differential equation with fractional derivatives of unknown functions called the fractional differential equation. In the current decades, numerous aspects of science have been studied with the help of fractional differential equation. FC presents the study of general behavior more than classical calculus, and it has gained more attention due to its wide applications in physical science. Several partial differential equations (PDEs) have been demonstrated with the help of FC in the description of some physical phenomena such as fluid dynamics, signal processing, astronomy, calculus, genetics, and kinetics [1–4]. Many researches have studied these PDEs with various analytical and numerical schemes such as Exp-function scheme [5], homotopy perturbation method [6], subequation technique [7], homotopy analysis method [8], rational function strategy [9], the first integral approach [10], quasi-wavelet process [11], auxiliary equation procedure [12], fractional reduced differential transform method [13], and fractional complex transform [14]. Kruskal and Moser [15] studied the Harry Dym equation for the first time in 1974. This article presents the study of the nonlinear HDEq with fractional derivatives such as
With the initial condition,where and are arbitrary constants, and is representing the order of the fractional derivative whereas is a function of and . In the case of , (1) brings down to the classical nonlinear HDEq. This Harry Dym equation is used to identify applications in several physical systems, and it does not possess the Painleve property. The HDEq has strong links to the Korteweg-de-Vries problem, and applications of this equation were found to the problems of hydrodynamics. Kumar et al. [16] studied the fractional model of the Harry Dym equation and obtained the approximate solution was obtained by using HPM. This equation is coupled with a system of variation and modulation so that HDEq becomes entirely an integrable and nonlinear evolution problem. The HDEq has a great significance in that it follows the conversion laws. Mokhtari [17] studied the exact travelling wave solutions of the Harry Dym equation. Huanga and Zhdanovb [18] used Lie symmetry approach to analyze the exact solutions in explicit forms of the time fractional HDEq. Yokus and Glbahar [19] used finite difference method to achieve approximate results of HDEq with fractional derivative.
He [20, 21] developed an idea of the homotopy perturbation method (HPM) for the analytical solution of linear and nonlinear problems. Later, many authors [22] have focused their interest to examine the solutions of such challenges by HPM. Nadeem and Li [23, 24] used HPM coupled with Laplace transform to obtain the solution of nonlinear vibration systems. Lu and Sun [25] used fractional complex transform with the HPM to obtain the solution of fractional Boussinesq–burger equations. Kharrat and Toma [26] applied a new hybrid approach where HP is coupled with Sumudu transform for solving boundary value problems.
In this article, we develop Mohand homotopy perturbation transform scheme (MHPTS) built on the concept of Mohand transform and HPM to solve the nonlinear HDEq with fractional derivative. The key purpose of this approach is to destruct the nonlinear terms and obtain the analytic solution without any restrictive factors in terms of convergent series. This approach has the capability to reduce the heavy computational work during the implementation of fractional order . The rest of the paper is arranged as follows: we review some basic concepts and descriptions of MT in Section (2). In Section (3), we describe the idea of HPM to destruct nonlinear terms. We construct the strategy of MHPTS in Section (4). We implement our suggested approach with a numerical example in Section (5), and finally, we give the detail of obtained results with discussion in Section (6), and the conclusion is in Section (7).
2. Preliminaries Concepts of Mohand Transform
In this section, we introduce some basic definitions and preliminary concepts of the Mohand transform which reveals the idea of its implementations to functions.
Definition 1. Consider is the function of [4], thenwhich is called Laplace transform, where is function (i.e., a function of time domain), defined on to a function of (i.e., of frequency domain).
Definition 2. If symbolizes the Laplace transform of , thenwhich is known as inverse Laplace transform of .
Definition 3. Mohand and Mahgoub [27, 28] presented the idea of MT to deal with some differential problems which are defined aswhere represents the Mohand transform, and is the independent variable of the transformed function . Conversely, since is MT of , therefore, function is called the inverse of , and thus,
Definition 4. The MT in the presence of fractional order is defined as [29]
Definition 5. Assume so that MT of have some supporting factors(a)(b)(c)
Definition 6. The fractional derivative in the Caputo sense is defined as [30]
3. Basic Idea of HPM
This segment presents the basic concept of HPM. We start this concept with a nonlinear differential problem such that
With conditionswhere and are known as general functional operator and boundary operator, respectively, and is known function with as an interval of the domain . We now divide into two units such as represents a linear and represents a nonlinear operator. As a result, we can express (9) as
Assume a homotopy in such a way that it is appropriate for
Orwhere is the embedding parameter, and is an initial guess of (9), which is suitable for the boundary conditions. The theory of HPM states that is considered as a slight variable and the solution of (9) in the resulting form of
Let , then the particular of (14) is written as
The nonlinear terms can be calculated as
Then, He’s polynomials can be obtained using the following expression:
The series solution in (16) is mostly convergent due to the convergence rate of the series depending on the nonlinear operator .
4. Formulation of MHPTS
This segment presents the formulation of MHPTS which is significantly used to obtain the analytical results of HDEq with time-fractional derivative in the order of series. We start the formulation of this scheme with the consideration of a nonlinear differential problem with fractional order
With initial conditionwhere is fractional derivative of order in Caputo sense, and are linear and nonlinear operator with as a identified function. Employing MT on (18)
Applying the definition of MT on (20), we get
It yields
Using (19), we get
Applying inverse MT, we get
This (24) is known as the relation of recurrence for , where
Let us consider the analytical solution of (18) such as
We now implement the following formula for obtaining He’s polynomial results
Using (26) and 27 we obtain (24) such as
Evaluating the familiar elements of , we get
Finally, the estimated results can be arraigned in the form of a series
5. Numerical Examples
In this part, we apply MHPTS for the analytical solution of HDEq with time-fractional derivative and obtained the findings in the order of a series. We demonstrate the 3D solution graphs to show the behavior of this paper in different fractional order.
5.1. Example
Let us consider the following form of HDEq with fractional order
Subject to the initial condition
Taking MT of (32), we get
Applying the properties of MT, we gain
The inverse MT yields
That is the recurrence relation of (32), we now implement the idea of HPM; therefore,
On comparing, the following iterations can be obtained:which gives the solution
Proceeding the same procedure, we can get the other elements of , and thus, the results are in form of a series that converges to the exact solution by using (31).
For ,which is in full agreement with [31, 32].
6. Results and Discussion
In Figure 1, we present Table 1 the approximate solution and the exact solution of with 3D surface plots at and for (40) and (41), respectively. Figure 1(a) shows that the obtained results in the form of 3D surface plot are very close to the exact solution of 3D surface plot of Figure 1(b). Figure 2 presents the graphical distribution of . In Figure 2(a), we consider the plot distribution of at different fractional order with and . In Figure 2(b), we demonstrate the error distribution of between the approximate solution and the exact solution at fractional order with and . This graphical representation shows that the increase in fractional order gives the approximate solution very close to the exact solution which means that the rate of convergence depends on the fractional order . Table 1 demonstrates the approximate values of at fractional order and with and . The absolute error of between the approximate solution and the exact solution is also presented at fractional order which shows that MHPTS is very authentic and reliable approach. This approach produces the error distribution with less computation for obtaining the approximate solution HDEq with Caputo fractional derivative.
(a)
(b)
(a)
(b)
7. Conclusion
In this article, we effectively used MHPTS to achieve the analytical solution of HDEq with fractional derivative. The implementation of the Mohand transform is very simple and direct for order differential problems with fractional. Due to this direct approach, it is very easy to tackle the nonlinear components with the help of the homotopy perturbation method. The graphical results and solution behavior in different fractional order demonstrate that the approximate solution converges to the exact solution very rapidly. This strategy is incredibly simple to use and suitable for finding the approximate solution of nonlinear problems without using any assumption theory. This analysis presented the evaluation of the series solution and the procedure for handling the fractional derivative. This approach demonstrates a strong performance with minimal calculations than other schemes. We used Mathematica software 11 to calculate the values of iterations in the recurrence relation. This scheme is also appropriate for a wide range of nonlinear differential problems with a fractional derivative for future applications.
Data Availability
This study provides all the data within the article.
Conflicts of Interest
The authors confirm that they have no conflicts of interest.