# Analytical Approach for the Approximate Solution of Harry Dym Equation with Caputo Fractional Derivative

Authors: Muhammad Nadeem; Zitian Li; Yahya Alsayyad
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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(1)Dξαϑθ,ξ=ϑ3θ,ξ∂ϑ3θ,ξ∂θ3.With the initial condition,(2)ϑθ,0=a−3b2θ2/3,where a and b are arbitrary constants, and α is representing the order of the fractional derivative whereas ϑθ,ξ is a function of θ and ξ. In the case of α=1, (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 orderα. 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.
Considerϑ is the function of ξ≥0 [4], then(3)ℒϑξ=Vr=∫0∞ϑξe−rξdξ,which is called Laplace transform, where ξ is function (i.e., a function of time domain), defined on 0,∞ to a function of θ (i.e., of frequency domain).Definition 2.
IfVr symbolizes the Laplace transform of ϑξ, then(4)ϑξ=ℒ−1Vr,which is known as inverse Laplace transform of Vr.Definition 3.
Mohand and Mahgoub [27, 28] presented the idea of MT to deal with some differential problems which are defined as(5)Θϑξ=Sr=r2∫0∞ϑξe−rξdξ,k1≤r≤k2,where Θ represents the Mohand transform, and r is the independent variable of the transformed function ξ. Conversely, since Sr is MT of ϑξ, therefore, function ϑξ is called the inverse of Sr, and thus,(6)Θ−1Sr=ϑξ,Θ−1is inverse MT.Definition 4.
The MT in the presence of fractional orderα is defined as [29](7)ΘSαξ=rαSr−∑k=0n−1ϑk0rk−α+1,0<α≤n.Definition 5.
AssumeΘϑξ=Sr so that MT of ϑ′ξ have some supporting factors(a)
Θϑ′ξ=rSr−r2ϑ0(b)
Θϑ″ξ=r2Sr−r3ϑ0−ϑ2ϑ′0(c)
Θϑnξ=rnSr−rn+1ϑ0−rnϑ′0−⋯−r2ϑn−10Definition 6.
The fractional derivative in the Caputo sense is defined as [30](8)Dταϑθ,ξ=∂nϑθ,ξ∂ξn,α∈N,1Γn−α,∫0ξξ−φn−α−1ϑnφ∂φn−1<α<n.


## 3. Basic Idea of HPM

This segment presents the basic concept of HPM. We start this concept with a nonlinear differential problem such that(9)Yϑ−fh=0,q∈Ω.With conditions(10)Sϑ,∂ϑ∂n=0,q∈Γ,where Y and S are known as general functional operator and boundary operator, respectively, and fq is known function with Γ as an interval of the domain Ω. We now divide Y into two units such as Y1 represents a linear and Y2 represents a nonlinear operator. As a result, we can express (9) as(11)Y1ϑ+Y2ϑ−fq=0.Assume a homotopyϑh,p:Ω×0,1⟶ℍ in such a way that it is appropriate for(12)Hϑ,p=1−pY1ϑ−Y1ϑ0+pY1ϑ−Y2ϑ−fq.Or(13)Hϑ,p=Y1ϑ−Y1ϑ0+qLϑ0+pY2ϑ−fh=0,where p∈0,1 is the embedding parameter, and ϑ0 is an initial guess of (9), which is suitable for the boundary conditions. The theory of HPM states that p is considered as a slight variable and the solution of (9) in the resulting form of p(14)ϑ=ϑ0+pϑ1+p2ϑ2+p3ϑ3+⋯=∑i=0∞piϑi.Letp=1, then the particular of (14) is written as(15)ϑ=limp⟶1ϑ=ϑ0+ϑ1+ϑ2+ϑ3+⋯=∑i=0∞ϑi.The nonlinear terms can be calculated as(16)Y2ϑθ,Y=∑n=0∞pnHnϑ.Then, He’s polynomialsHnϑ can be obtained using the following expression:(17)Hnϑ0+ϑ1+⋯+ϑn=1n!∂n∂pnY2∑i=0∞piϑip=0,n=0,1,2.The series solution in (16) is mostly convergent due to the convergence rate of the series depending on the nonlinear operator Y2.


## 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(18)Dξαϑθ,ξ+Ruθ,ξ+Nuθ,ξ=gθ,ξ.With initial condition(19)ϑθ,0=hθ,where Dξαϑθ,ξ is fractional derivative of order alpha in Caputo sense, R and N are linear and nonlinear operator with gθ,ξ as a identified function. Employing MT on (18)(20)ΘDξαϑθ,ξ+Rϑθ,ξ+Nϑθ,ξ=Θgθ,ξ.Applying the definition of MT on (20), we get(21)rαRr−rϑ0=−ΘRϑθ,ξ+Nϑθ,ξ+Θgθ,ξ.It yields(22)Rr=rϑ0−1rαΘRϑθ,ξ+Nϑθ,ξ+gθ,ξ.Using (19), we get(23)Rr=rhθ−1rαΘRϑθ,ξ+Nϑθ,ξ+gθ,ξ.Applying inverse MT, we get(24)ϑθ,ξ=Gθ,ξ−Θ−11rαΘRϑθ,ξ+Nϑθ,ξ.This (24) is known as the relation of recurrence for ϑθ,ξ, where(25)Gθ,ξ=Θ−1rhθ+Θgθ,ξ.Let us consider the analytical solution of (18) such as(26)ϑθ,ξ=∑n=0∞pnunθ,ξ,(27)Nuθ,ξ=∑n=0∞pnHnϑθ,ξ.We now implement the following formula for obtaining He’s polynomial results(28)Hnϑ0+ϑ1+⋯+ϑn=1n!∂n∂pnN∑i=0∞piϑip=0,n=0,1,2.Using (26) and 27 we obtain (24) such as(29)∑n=0∞pnϑnθ,ξ=Gθ,ξ−pΘ−11rαΘR∑n=0∞pnϑnθ,ξ+∑n=0∞pnHnϑnθ,ξ.Evaluating the familiar elements ofp, we get(30)p0:ϑ0θ,ξ=Gθ,ξ,p1:ϑ1θ,ξ=−Θ−11rαΘRϑ0θ,ξ+H0,p2:ϑ2θ,ξ=−Θ−11rαΘRϑ1θ,ξ+H1,p3:ϑ3θ,ξ=−Θ−11rαΘRϑ2θ,ξ+H2.Finally, the estimated results can be arraigned in the form of a series(31)ϑθ,ξ=ϑ0θ,ξ+ϑ1θ,ξ+ϑ2θ,ξ+⋯ϑθ,ξ=limN⟶∞∑n=0Nϑnθ,ξ.


## 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 orderα(32)∂ϑαθ,ξ∂ξα=ϑ3θ,ξ∂ϑ3θ,ξ∂θ3.Subject to the initial condition(33)ϑθ,0=a−3b2θ2/3.Taking MT of (32), we get(34)Θ∂ϑα∂ξα=Θϑ3∂ϑ3∂θ3.Applying the properties of MT, we gain(35)Rv=vu0+1vαΘϑ3∂ϑ3∂θ3.The inverse MT yields(36)ϑθ,ξ=ϑθ,0−Θ−11vαΘϑ3∂ϑ3∂θ3.That is the recurrence relation of (32), we now implement the idea of HPM; therefore,(37)∑n=0∞pnϑnθ,ξ=ϑθ,0−pΘ−11vαΘ∑n=0∞pnϑn3∂ϑn3∂θ.On comparing, the following iterations can be obtained:(38)p0:ϑ0θ,0=ϑθ,0,p1:ϑ1θ,0=−Θ−11vαΘϑ03∂ϑ03∂θ3,p2:ϑ2θ,0=−Θ−11vαΘϑ03∂ϑ13∂θ3+3ϑ02ϑ1∂ϑ03∂θ3,p3:ϑ3θ,0=−Θ−11vαΘϑ03∂ϑ23∂θ3+3ϑ02ϑ1∂ϑ13∂θ3+3ϑ0ϑ12+3ϑ02ϑ2∂ϑ03∂θ3,which gives the solution(39)ϑ0θ,0=a−3b2θ2/3,ϑ1θ,0=−b32a−3b2θ1/3ξαα!,ϑ2θ,0=−b32a−3b2θ4/3ξ2α2α!,ϑ3θ,0=b9/2a−3b2θ7/3152α+12α+12−16ξ3α3α!.Proceeding the same procedure, we can get the other elements ofϑn, and thus, the results are in form of a series that converges to the exact solution by using (31).(40)ϑθ,ξ=ϑ0θ,ξ+ϑ1θ,ξ+ϑ2θ,ξ+⋯ϑθ,ξ=a−3b2θ2/3−b3/2a−3b2θ1/3ξαα!−b32a−3b2θ4/3ξ2α2α!+b9/2a−3b2θ7/3152α+12α+12−16ξ3α3α!+.Forα=1,(41)ϑθ,ξ=a−3b2θ+cξ2/3,which is in full agreement with [31, 32].


## 5.1. Example

Let us consider the following form of HDEq with fractional orderα(32)∂ϑαθ,ξ∂ξα=ϑ3θ,ξ∂ϑ3θ,ξ∂θ3.Subject to the initial condition(33)ϑθ,0=a−3b2θ2/3.Taking MT of (32), we get(34)Θ∂ϑα∂ξα=Θϑ3∂ϑ3∂θ3.Applying the properties of MT, we gain(35)Rv=vu0+1vαΘϑ3∂ϑ3∂θ3.The inverse MT yields(36)ϑθ,ξ=ϑθ,0−Θ−11vαΘϑ3∂ϑ3∂θ3.That is the recurrence relation of (32), we now implement the idea of HPM; therefore,(37)∑n=0∞pnϑnθ,ξ=ϑθ,0−pΘ−11vαΘ∑n=0∞pnϑn3∂ϑn3∂θ.On comparing, the following iterations can be obtained:(38)p0:ϑ0θ,0=ϑθ,0,p1:ϑ1θ,0=−Θ−11vαΘϑ03∂ϑ03∂θ3,p2:ϑ2θ,0=−Θ−11vαΘϑ03∂ϑ13∂θ3+3ϑ02ϑ1∂ϑ03∂θ3,p3:ϑ3θ,0=−Θ−11vαΘϑ03∂ϑ23∂θ3+3ϑ02ϑ1∂ϑ13∂θ3+3ϑ0ϑ12+3ϑ02ϑ2∂ϑ03∂θ3,which gives the solution(39)ϑ0θ,0=a−3b2θ2/3,ϑ1θ,0=−b32a−3b2θ1/3ξαα!,ϑ2θ,0=−b32a−3b2θ4/3ξ2α2α!,ϑ3θ,0=b9/2a−3b2θ7/3152α+12α+12−16ξ3α3α!.Proceeding the same procedure, we can get the other elements ofϑn, and thus, the results are in form of a series that converges to the exact solution by using (31).(40)ϑθ,ξ=ϑ0θ,ξ+ϑ1θ,ξ+ϑ2θ,ξ+⋯ϑθ,ξ=a−3b2θ2/3−b3/2a−3b2θ1/3ξαα!−b32a−3b2θ4/3ξ2α2α!+b9/2a−3b2θ7/3152α+12α+12−16ξ3α3α!+.Forα=1,(41)ϑθ,ξ=a−3b2θ+cξ2/3,which is in full agreement with [31, 32].


## 6. Results and Discussion

In Figure1, we present Table 1 the approximate solution and the exact solution of ϑθ,ξ with 3D surface plots at 0≤θ≤1 and 0≤ξ≤0.01 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 ϑθ,ξ. In Figure 2(a), we consider the plot distribution of ϑθ,ξ at different fractional order α=0.25,0.50,0.75 with 0≤θ≤2 and 0≤ξ≤0.01. In Figure 2(b), we demonstrate the error distribution of ϑθ,ξ between the approximate solution and the exact solution at fractional order α=1 with 0≤θ≤2 and 0≤ξ≤0.01. This graphical representation shows that the increase in fractional order α gives the approximate solution very close to the exact solution which means that the rate of convergence depends on the fractional order α. Table 1 demonstrates the approximate values of ϑθ,ξ at fractional order α=0.50 and α=1 with 0.25≤θ≤1 and 0.1≤ξ≤0.3. The absolute error of ϑθ,ξ between the approximate solution and the exact solution is also presented at fractional order α=1 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.Figure 1
3D surfaces between the analytical and exact solutions atα=1. (a) 3D surface solution of (1). (b) 3D surface solution of (1).
(a)(b)Table 1
Absolute error between the approximate solution and the exact solution at different fractional order.
θξα=0.5α=1Exact solutionAbsolute error0.250.12.107522.294152.294230.000080.21.982222.227112.227720.000610.31.873372.158142.160210.002070.500.11.928622.125962.126050.000090.21.792512.056112.056910.00080.31.671961.983911.986580.002670.750.11.739831.950811.950940.000130.21.589121.877621.878670.001050.31.452401.801441.804990.003551.00.11.538191.767391.767580.000190.21.366701.690081.691540.001460.31.206431.608841.613750.00491Figure 2
Graphical distribution of HDEq at different fractional order. (a) Plot distribution atα=0.25,0.50,0.75. (b) Plot distribution between the approximate and exact solutions.
(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.



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