Approximate solutions of the fractional harry DYM equation

Abstract

Keywords: Approximate Solution; Fractional Partial Differential Equation; Caputo Derivative; Harry Dym Equation; Residual Power Series Method

Introduction

was first studied by Kruskal and Moser and is referred to an unpublished work of Harry Dym. This equation is entirely integrable nonlinear evolution equation linked to the traditional string problems [1]. More detailed information about these problems can be seen in [2-5]. The HarryDym equation is also closely related to the Korteweg-de Vries equation [6]. In the literature, numerous methods have been utilized to solve this equation. The solution methods for the Harry Dym equation are moving frame [7], Adomian decomposition [8], He’s variational iteration

[8], direct integration [8], power series [8], residual power series [8], Bäcklund transform [9], new iterative method [10], haar wavelet [11], homotopy perturbation [12], reconstruction of variational iteration [12], Darboux transformation [13], and nonlinear steepest decent [14]. Recently, it has become very popular for scientists to obtain solutions of the fractional differential equations. These equations are widely used to model problems in viscoelasticity, turbulence, electrical networks, nonlinear biological systems, control theory, thermodynamics, fluid dynamics, signal processing, and so on [15-20]. The time fractional Harry Dym equation is one of the most important of them. So far, many researchers have used various analytical and numerical methods to obtain the time fractional Harry Dym equation. These methods are Adomian decomposition [21,22], homotopy perturbation Sumudu

*Corresponding author. *E-mail address: sevilunal@sdu.edu.tr This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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transform [22], Elzaki transform technique [23], Lie symmetry group analysis [24], similarity [25,26], homotopy analysis [27,28], Lie classical [29], homotopy perturbation [30], Mohand homotopy perturbation transform scheme [31], reduced differential transform [32], finite difference [33], q-homotopy analysis [34], and optimal system [35]. However, it is seen that the time fractional Harry Dym equation has not yet been solved with the RPSM. The RPSM, proposed by Abu Arqub in 2013, is an efficient approach to obtain the approximate solutions of the different differential equations. These solutions are gained without the need for linearization, discretization, or perturbation. The RPSM does not require comparing the coefficients of the corresponding terms and does not need a recursion relation. By selecting an appropriate value for the initial guesses approximations, the proposed method can be also directly applied to the equations. Besides, with this method, high precision is achieved by utilizing less time and small calculations. Moreover, by minimizing the residual error, the suggested method provides an easy way to achieve the convergence of the series solution. Furthermore, the RPSM relies on derivation, which is more accurate and much easier than integration. This is the basis of most other solution methods. In addition to all these, the proposed method suggests obtaining infinite series solutions with iterated operations. In the present paper, the RPSM is used to get the approximate solutions of the time fractional Harry Dym equation of the form

part, Caputo’s definition is utilized since the derivative of a constant is zero and the initial conditions for the fractional differential equations with Caputo derivative take the familiar manner of integer order differential equations. The definition of Caputo derivative is defined as follows: Definition 1. [36] The time fractional derivative of u(x, t) in Caputo form is described as

The definition and theorems for the fractional power series are given below. Details of them can be found in [37]. Definition 2. [37] A power series expansion of the manner

is called the fractional power series about t0. Here, t is a variable and the cm’s are constants. Theorem 1. [37] Assume that g is a fractional power series representation at t0 of the manner

(1) by the initial condition (2) where is the Caputo fractional derivative of order β with respect to the time variable t. When β = 1, Eq. (1) turns into the standard Harry Dym equation. The exact solution for the Harry Dym is

The plan of this paper is as follows. In Section 2, the definitions and theorems of the Caputo derivative and the fractional power series are mentioned. In Section 3, the basic idea of the RPSM is expressed. In Section 4, the RPS solutions for the time fractional Harry Dym equation are obtained by suggested method. Besides, the efficiency and the reliability of this method are demonstrated by table and figures. In Section 5, the Conclusions are given. Preliminaries There are numerous definitions of fractional operators, such as Grunwald-Letnikov, Caputo, Riemann-Liouville, Hadamard, Wely, and Marchaud in the literature. In this

Theorem 2. [37] Assume that u(x, t) has a multiple fractional power series representation at t0 of the manner

Sigma J Eng Nat Sci, Vol. 42, No. 5, pp. 1604−1611, October, 2024

Here, , and that Rc is radius of convergence of the fractional power series .

Substituting the ul(x, t) in Eq. (6) and calculating the of for l = 1, 2..., the suggested method is clearly expressed. Then, applying the relation (7), the following equation

Basic Idea of the RPSM In this section, to demonstrate the basic idea of the RPSM, we examine a general nonlinear fractional differential equation by the initial condition of the manner

(3) represents the fractional derivative in the where Caputo sense, N is nonlinear differential operator and R is linear differential operator. This method suggests the solution for Eq. (3) as a fractional power series for t = 0. Assume the solution takes the following form:

is solved to obtain the gm(x) with m = 1, 2..., l in Eq. (5). Approximate Solutions of the Fractional Harry Dym Equation By RPSM In this segment of the study, we utilize the RPSM to gain the RPS solutions for Eq. (1) by the initial condition (2). Let us consider the residual function for Eq. (1) as

To determine the g1(x), we write l = 1 in Eq. (9) and we have The ul(x, t) is also expressed as (4)

Then, the 0-th RPS approximate solution of u(x, t) is given as

Some significant relations of the suggested method are as follows and it can be seen in [38-42].

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In Table 1, the u4(x, t) solution is gained for β = 0.25, β = 0.50, β = 0.75, and β = 1 with the different values of t and x. Besides, the exact solution is compared with the u4(x, t) solution for β = 1 in this table. From Table 1, it can be seen that the absolute error gets smaller as the value of t decreases. For 0 ≤ t ≤ 1 and -30 ≤ x ≤ 0 at β = 1, the comparison of the u4(x, t) and the exact solution is illustrated in Figure 1. When equal parameters are used, it is seen that the u4(x, t) solution has nearly the same shape as the exact solution in this figure. In Figure 2, the geometrical behavior of the u4(x, t) with 3D plot for 0 ≤ t ≤ 5, 0 ≤ x ≤ 1, and the different values

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Table 1. Comparing the u4(x, t) solution and the exact solution with the different values of t and x. x

1.35745x10-10 4.39464x10-9 3.37672x10-8 1.44003x10-7 4.44810x10-7 1.20481x10-9 3.93112x10-8 3.04515x10-7 1.30959x10-6 4.08059x10-6 1.21435x10-7 4.12452x10-6 3.33870x10-5 1.50729x10-4 4.95747x10-4 1.92919x10-7 5.83621x10-6 4.20345x10-5 1.68489x10-4 4.90352x10-4 1.40554x10-9 4.41275x10-8 3.28892x10-7 1.36081x10-6 4.07902x10-6

Figure 2. 3D plot of the u4(x, t): (a) u4(x, t) for β = 0.25, (b) u4(x, t) for β = 0.50, (c) u4(x, t) for β = 0.75, (d) u4(x, t) for β = 1.

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Figure 3. 2D plot of the u4(x, 5) for the different values of β.

Table 2. Comparison of HPSTM, ADM, RPSM, and exact solution for β = 1. x

of β is illustrated by suggested method. Besides, the same solution with 2D plot for t = 5 and -10 ≤ x ≤ 10 is demonstrated in Figure 3. The solution at β = 0.25 is showed with the blue line, the solution at β = 0.50 is showed with the orange line, the solution at β = 0.75 is showed with the green line, and the solution at β = 1 is showed with the red line in this figure. All plots in figures are illustrated by the aid of Mathematica 11.3. For β = 1, the third order term solution u3(x, t) of the RPSM, homotopy perturbation Sumudu transform method (HPSTM) [22], Adomian decomposition method (ADM) [22], and exact solution are compared in Table 2. It is observed from this table that the RPSM solution performed a high accuracy agreement with the ADM and HPSM solution. It is also seen that the accuracy increases as the order of the approximation increases.

Conclusion

In this study, the RPSM was utilized for obtaining the approximate solutions of Eq. (1). These solutions were illustrated by numerically and graphically for the different values of β, t and x. By comparing the approximate solution and the exact solution, the accuracy and efficiency of the suggested method were demonstrated. When equal

parameters were selected, it was observed that the approximate solution had almost the same shape as the exact solution. The proposed method was compared numerically with the HPSTM and the ADM by table. It was seen from this table that the RPSM made a good agreement with this methods. It is seen from the approximate solutions that only a few iterates were used by the proposed method. With these iterates, an infinite series solutions can be found. The accuracy of the RPSM increases as the order of these solutions increases. Besides, this method does not need a lot of time and computer memory. The RPSM indicates strong performance with less computation than other methods in the literature. Moreover, the RPSM does not require transformation, linearization, discretization, or perturbation. Furthermore, the suggested method can be used to get approximate solutions of different kinds of fractional partial differential equations.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw

Sigma J Eng Nat Sci, Vol. 42, No. 5, pp. 1604−1611, October, 2024

data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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