Next Article in Journal
Adaptive Wavelet Methods for Earth Systems Modelling
Next Article in Special Issue
Modelling of Ocean Waves with the Alber Equation: Application to Non-Parametric Spectra and Generalisation to Crossing Seas
Previous Article in Journal
Insight of Numerical Simulation for Current Circulation on the Steep Slopes of Bathymetry and Topography in Palu Bay, Indonesia
Previous Article in Special Issue
Generation of Gravity Waves by Pedal-Wavemakers
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation

1
Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang 212013, China
2
Department of Mathematics, Obour High Institute for Engineering and Technology, Cairo 11828, Egypt
*
Author to whom correspondence should be addressed.
The authors did all this work equally.
Fluids 2021, 6(7), 235; https://doi.org/10.3390/fluids6070235
Submission received: 5 March 2021 / Revised: 1 April 2021 / Accepted: 4 April 2021 / Published: 29 June 2021
(This article belongs to the Special Issue Mathematical and Numerical Modeling of Water Waves)

Abstract

:
This research paper targets the fractional Hirota’s analytical solutions–Satsuma ( HS ) equations. The conformable fractional derivative is employed to convert the fractional system into a system with an integer–order. The extended simplest equation (ESE) and modified Kudryashov (MKud) methods are used to construct novel solutions of the considered model. The solutions’ accuracy is investigated by handling the computational solutions with the Adomian decomposition method. The solutions are explained in some different sketches to demonstrate more novel properties of the considered model.

1. Introduction

Recently, the integrable nonlinear partial differential equations (INLPDEs) have been used in many typical applications [1,2,3]. These applications depend on the solitary wave solutions’ features, which are considered a fundamental tool for discovering more properties of the formulated phenomena by INLPDEs [4,5,6,7]. These solutions are handled to investigate the elastic and inelastic interaction in waves’ pulse through the transmission [8,9,10,11]. Studying these solutions has attracted the focus of much research, forcing them to formulate many analytical techniques for obtaining this kind of wave that gives them to investigate more properties of these solutions [12,13,14]. Such techniques are used to construct the analytical solutions for example extended tanh–expansion method, rational–expansion method, sine-Gordon expansion method, inverse Scattering transformation, Darbous transformation, Riccati, expansion method, multiple exp-function method, etc., [13,15,16,17,18,19,20,21,22,23,24,25,26].
In this article, we study a well-known model in INLPDEs presented by Hirota et al. [27,28,29]. This model is known by HS shallow water wave equation which is given by
D τ ϱ E = D τ ϱ E Ξ Ξ + 3 E D τ ϱ E 3 E Ξ D τ ϱ E + E Ξ , E Ξ = E .
where E = E ( Ξ , τ ) , E = E ( Ξ , τ ) , 0 < ϱ 1 . System (1) describes the dynamical behavior of the solitary wave in the shallow water. Applying the next wave transformation E = S ( Γ ) , E = S ( Γ ) , Γ = Ξ + λ t ϱ ϱ where λ is an arbitrary constant, then substituting the second equation in the system into the first, convert the above–fractional system into the following equation with an-integer order
λ S 3 λ S 2 + ( λ 1 ) S = 0 .
Handling Equation (2) by the homogeneous balance principles and the following auxiliary equation method of ESE and MKud method f ( Γ ) = h 3 f ( Γ ) 2 + h 2 f ( Γ ) + h 1 & Q ( Γ ) = log ( a ) ( Q ( Γ ) 2 Q ( Γ ) ) , gets the value of balance equal two. Consequently, the general solutions of Equation (2) are given by
S ( Γ ) = { a 2 f ( Γ ) 2 + a 1 f ( Γ ) + a 2 f ( Γ ) 2 + a 1 f ( Γ ) + a 0 , a 2 Q ( Γ ) 2 + a 1 Q ( Γ ) + a 0 }
where a 2 , a 1 , a 0 , a 1 , a 2 are arbitrary constants to be calculated later.
The rest sections are ordered as follows, we test, by means of two suggested analytical techniques [30,31,32], the analytical solutions to the nonlinear HS fractional equation. We search for the accuracy of the solutions we obtain in conjunction with the semi-analytical AD schema [33,34] in Section 2 part. In Section 3 we clarify the innovation of our approach and its physical interpretation. In theSection 4 portion, the outcome of a paper is summed up.

2. Computational Solutions vs. Accuracy

Here, we employ two recent analytical schemes (ESE and MKud methods) to formulate some novel computational wave solutions of the considered model. Additionally, the evaluated solutions are used to calculate the initial and boundary conditions. These conditions allow applying the AD method to test the accuracy of the obtained solutions and used schemes. This investigation takes the following steps:

2.1. Analytical Solutions

Applying the ESE and MKud methods’ framework gets the values of the above–mentioned parameters as following:
  • Through the ESE method’s steps gets the next values:
    Set I
    a 2 0 , a 1 0 , a 0 1 3 ( h 2 2 2 h 1 h 3 ) , a 1 2 h 2 h 3 , a 2 2 h 3 2 , λ 1 h 2 2 4 h 1 h 3 + 1 .
    Set II
    a 2 2 h 1 2 , a 1 2 h 1 h 2 , a 0 1 3 h 2 2 2 h 1 h 3 , a 1 0 , a 2 0 , λ 1 h 2 2 4 h 1 h 3 + 1 .
    Set III
    a 2 0 , a 1 0 , a 0 2 h 1 h 3 , a 1 2 h 2 h 3 , a 2 2 h 3 2 , λ 1 h 2 2 + 4 h 1 h 3 + 1 .
    Set IV
    a 2 2 h 1 2 , a 1 2 h 1 h 2 , a 0 2 h 1 h 3 , a 1 0 , a 2 0 , λ 1 h 2 2 + 4 h 1 h 3 + 1 .
Consequently, the exact solutions of the fractional nonlinear HS equation are constructed in the following. For h 2 = 0 , h 1 h 3 > 0 , we find
E I , 1 ( Ξ , τ ) = 1 3 ( 2 ) h 1 h 3 3 tan 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ + ϑ + 1 ,
E I , 2 ( Ξ , τ ) = 1 3 ( 2 ) h 1 h 3 3 cot 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ + ϑ + 1 ,
E II , 1 ( Ξ , τ ) = 1 3 ( 2 ) h 1 h 3 3 cot 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ + ϑ + 1 ,
E II , 2 ( Ξ , τ ) = 1 3 ( 2 ) h 1 h 3 3 tan 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ + ϑ + 1 ,
E III , 1 ( Ξ , τ ) = 2 h 1 h 3 sec 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ + ϑ ,
E III , 2 ( Ξ , τ ) = 2 h 1 h 3 csc 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ + ϑ ,
E IV , 1 ( Ξ , τ ) = 2 h 1 h 3 csc 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ + ϑ ,
E IV , 2 ( Ξ , τ ) = 2 h 1 h 3 sec 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ + ϑ .
For h 2 = 0 , h 1 h 3 < 0 , we find
E I , 3 ( Ξ , τ ) = 2 3 h 1 h 3 3 tanh 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ log ( ϑ ) 2 1 ,
E I , 4 ( Ξ , τ ) = 2 3 h 1 h 3 3 csc h 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ log ( ϑ ) 2 + 2 ,
E II , 3 ( Ξ , τ ) = 2 3 h 1 h 3 3 csc h 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ log ( ϑ ) 2 + 2 ,
E II , 4 ( Ξ , τ ) = 2 3 h 1 h 3 3 tanh 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ log ( ϑ ) 2 1 ,
E III , 3 ( Ξ , τ ) = 2 h 1 h 3 sec h 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ log ( ϑ ) 2 ,
E III , 4 ( Ξ , τ ) = 2 h 1 h 3 csc h 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ log ( ϑ ) 2 ,
E IV , 3 ( Ξ , τ ) = 2 h 1 h 3 csc h 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ log ( ϑ ) 2 ,
E IV , 4 ( Ξ , τ ) = 2 h 1 h 3 sec h 2 h 1 h 3 τ ϱ 1 4 h 1 h 3 ϱ + Ξ log ( ϑ ) 2 .
For h 1 = 0 , h 2 > 0 , we find
E I , 5 ( Ξ , τ ) = h 2 2 2 h 3 e h 2 τ ϱ h 2 2 + 1 ϱ + Ξ + ϑ 1 2 2 h 3 e h 2 τ ϱ h 2 2 + 1 ϱ + Ξ + ϑ 1 1 3 ,
E III , 5 ( Ξ , τ ) = 2 h 2 2 h 3 e h 2 τ ϱ h 2 2 + 1 ϱ + Ξ + ϑ h 3 e h 2 τ ϱ h 2 2 + 1 ϱ + Ξ + ϑ 1 2 .
For h 1 = 0 , h 2 < 0 , we find
E I , 6 ( Ξ , τ ) = 1 3 6 h 3 4 e 2 h 2 τ ϱ h 2 2 + 1 ϱ + Ξ + ϑ h 3 e h 2 τ ϱ h 2 2 + 1 ϱ + Ξ + ϑ + 1 2 + 6 h 2 h 3 1 1 h 3 e h 2 τ ϱ h 2 2 + 1 ϱ + Ξ + ϑ + 1 h 2 2 ,
E III , 6 ( Ξ , τ ) = 2 h 3 h 3 1 h 3 e h 2 τ ϱ h 2 2 + 1 ϱ + Ξ + ϑ + 1 2 1 h 2 2 h 3 h 3 e h 2 τ ϱ h 2 2 + 1 ϱ + Ξ + ϑ + 1 + h 2 .
For 4 h 1 h 3 > h 2 2 , we find
E I , 7 ( Ξ , τ ) = 1 6 h 2 2 4 h 1 h 3 3 sec 2 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ 2 ,
E I , 8 ( Ξ , τ ) = 1 6 h 2 2 4 h 1 h 3 3 csc 2 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ 2 ,
E II , 5 ( Ξ , τ ) = h 2 2 3 + 4 h 1 h 3 h 2 h 2 4 h 1 h 3 h 2 2 tan 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ + 2 3 h 1 h 3 12 h 1 h 3 h 2 4 h 1 h 3 h 2 2 tan 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ 2 1 ,
E II , 6 ( Ξ , τ ) = h 2 2 3 + 4 h 1 h 3 h 2 h 2 4 h 1 h 3 h 2 2 cot 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ + 2 3 h 1 h 3 12 h 1 h 3 h 2 4 h 1 h 3 h 2 2 cot 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ 2 1 ,
E III , 7 ( Ξ , τ ) = h 2 2 4 h 1 h 3 cos 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ + 1 ,
E III , 8 ( Ξ , τ ) = 1 2 h 2 2 4 h 1 h 3 csc 2 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ ,
E IV , 5 ( Ξ , τ ) = 2 h 1 h 3 h 2 2 4 h 1 h 3 / ( ( h 2 cos 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ 4 h 1 h 3 h 2 2 sin 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ ) 2 ,
E IV , 6 ( Ξ , τ ) = 2 h 1 h 3 h 2 2 4 h 1 h 3 / ( h 2 sin 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ 4 h 1 h 3 h 2 2 cos 1 2 4 h 1 h 3 h 2 2 τ ϱ h 2 2 4 h 1 h 3 + 1 ϱ + Ξ + ϑ ) 2 .
2.
Through the MKud method’s steps gets the next values
Set I
a 0 0 , a 1 2 log 2 ( a ) , a 2 2 log 2 ( a ) , λ 1 1 log 2 ( a ) .
Set II
a 0 1 3 log 2 ( a ) , a 1 2 log 2 ( a ) , a 2 2 log 2 ( a ) , λ 1 log 2 ( a ) + 1 .
Consequently, the exact solutions of the fractional nonlinear HS equation are constructed in the following
E I ( Ξ , τ ) = 2 log 2 ( a ) 1 ± a τ ϱ ϱ ϱ log 2 ( a ) + Ξ 1 1 ± a τ ϱ ϱ ϱ log 2 ( a ) + Ξ 2 ,
E II ( Ξ , τ ) = 1 3 log 2 ( a ) 6 1 ± a τ ϱ ϱ log 2 ( a ) + ϱ + Ξ 1 1 ± a τ ϱ ϱ log 2 ( a ) + ϱ + Ξ 2 1 .

2.2. Solutions’ Accuracy

Checking the accuracy of the obtained exact solutions of the HSI equation along with ESE and MKud methods with respect to Equations (12) and (32) for h 1 = 1 , h 3 = 1 , h 2 = 0 & a = 3 , gets the following semi–analytical solutions;
E 0 | E I , 3 ( Ξ ) = 2 3 ,
E 0 | E I ( Ξ ) = log 2 ( 3 ) 2 ,
E 1 | E I , 3 ( Ξ ) = 2 Ξ 2 ,
E 1 | E I ( Ξ ) = 1 8 Ξ 2 log 4 ( 3 ) 1 4 Ξ 2 log 2 ( 3 ) + Ξ 2 log 2 ( 3 ) 4 1 log 2 ( 3 ) Ξ 2 log 4 ( 3 ) 4 1 log 2 ( 3 ) ,
E 2 | E I , 3 ( Ξ ) = 4 Ξ 4 3 ,
E 2 | E I ( Ξ ) = 1 48 Ξ 4 log 6 ( 3 ) + 1 96 Ξ 4 log 4 ( 3 ) + Ξ 4 log 6 ( 3 ) 96 1 log 2 ( 3 ) Ξ 4 log 4 ( 3 ) 96 1 log 2 ( 3 ) ,
E 3 | E I , 3 ( Ξ ) = 2 Ξ 4 3 26 Ξ 6 45 ,
E 3 | E I ( Ξ ) = Ξ 6 log 8 ( 3 ) 1152 Ξ 6 log 6 ( 3 ) 1440 Ξ 6 log 8 ( 3 ) 1440 1 log 2 ( 3 ) + Ξ 6 log 6 ( 3 ) 1440 1 log 2 ( 3 ) + 1 32 Ξ 4 log 6 ( 3 ) .
Consequently, the semi–analytical solutions are given by
E appro | E I , 3 ( Ξ ) = 26 Ξ 6 45 + 2 Ξ 4 2 Ξ 2 + 2 3 ,
E appro | E I ( Ξ ) = Ξ 6 log 8 ( 3 ) 1152 Ξ 6 log 6 ( 3 ) 1440 Ξ 6 log 8 ( 3 ) 1440 1 log 2 ( 3 ) + Ξ 6 log 6 ( 3 ) 1440 1 log 2 ( 3 ) + 5 96 Ξ 4 log 6 ( 3 ) + 1 96 Ξ 4 log 4 ( 3 ) + Ξ 4 log 6 ( 3 ) 96 1 log 2 ( 3 ) Ξ 4 log 4 ( 3 ) 96 1 log 2 ( 3 ) 1 8 Ξ 2 log 4 ( 3 ) 1 4 Ξ 2 log 2 ( 3 ) + Ξ 2 log 2 ( 3 ) 4 1 log 2 ( 3 ) Ξ 2 log 4 ( 3 ) 4 1 log 2 ( 3 ) + log 2 ( 3 ) 2 .
Calculating the exact, semi–analytical solutions based on Equations (42) and (43) gets the following value in Table 1 and Table 2:

3. Results’ Explanation

This paper has constructed some novel solutions of the fractional HSI equation by implementing ESE and MKud methods. These solutions have been represented through some different forms (Figure 1, Figure 2, Figure 3 and Figure 4) in three–dimension, density and spherical plot three–dimensional to illustrate more novel properties of the considered model. Comparing our results with that obtained in [35] which has applied the Hirota bilinear method and symbolic computation on the integer–order of the same model, explains our results’ novelty where all our solutions are entirely different from their obtained solutions. Additionally, employing the AD method explains our solutions’ accuracy, where the analytical and semi-analytical solutions are almost matching. This matching has been cleared along with Table 1 and Table 2 and Figure 5 and Figure 6. Still, it also shows the superiority of the MKud method’s solution over the ESE method, as shown in Figure 7.

4. Conclusions

This article has successfully implemented two recent analytical schemes (ESE and MKud techniques), and many novel solutions have been obtained for the considered model. The conformable fractional derivative has been employed to convert the fractional system to a system with an integer. The exact solutions have been demonstrated through 3D, density, spherical plot 3D sketches. Moreover, the accuracy of the obtained solutions has been illustrated by calculating the absolute value of error between the exact and semi-analytical methods accepted by the AD method. The novelty of the obtained results in this article has been explained by comparing our results with the previously published research paper.

Author Contributions

Conceptualization, C.Y.; methodology, C.Y.; formal analysis, M.M.A.K.; investigation, M.M.A.K.; writing—original draft, C.Y.; writing—review and editing, C.Y.; visualization, M.M.A.K.; supervision, D.L.; data curation, C.Y.; resources, D.L. All authors have read and agreed to the published version of the manuscript.

Funding

This paper has been funded by the Research Innovation Program for College Graduates of Jiangsu Province (Grant No.KYCX19 1609).

Data Availability Statement

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

Acknowledgments

The author would like to thank the journal stuff (Editor & Reviewers).

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Abdel-Aty, A.H.; Khater, M.M.; Dutta, H.; Bouslimi, J.; Omri, M. Computational solutions of the HIV-1 infection of CD4+ T-cells fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy. Chaos Solitons Fractals 2020, 139, 110092. [Google Scholar] [CrossRef]
  2. Abdel-Aty, A.H.; Khater, M.M.; Attia, R.A.; Abdel-Aty, M.; Eleuch, H. On the new explicit solutions of the fractional nonlinear space-time nuclear model. Fractals 2020, 28, 2040035. [Google Scholar] [CrossRef]
  3. Jena, R.; Chakraverty, S.; Yavuz, M. Two-hybrid techniques coupled with an integral transform for Caputo time-fractional Navier-Stokes Equations. Prog. Fract. Differ. Appl. 2020, 6, 201–213. [Google Scholar]
  4. Abdel-Aty, A.H.; Khater, M.M.; Baleanu, D.; Khalil, E.; Bouslimi, J.; Omri, M. Abundant distinct types of solutions for the nervous biological fractional FitzHugh–Nagumo equation via three different sorts of schemes. Adv. Differ. Equ. 2020, 2020, 1–17. [Google Scholar] [CrossRef]
  5. Khater, M.M.; Attia, R.A.; Park, C.; Lu, D. On the numerical investigation of the interaction in plasma between (high & low) frequency of (Langmuir & ion-acoustic) waves. Results Phys. 2020, 18, 103317. [Google Scholar]
  6. Khater, M.M.; Nofal, T.A.; Abu-Zinadah, H.; Lotayif, M.S.; Lu, D. Novel computational and accurate numerical solutions of the modified Benjamin–Bona–Mahony (BBM) equation arising in the optical illusions field. Alex. Eng. J. 2020, 60, 1797–1806. [Google Scholar] [CrossRef]
  7. Yavuz, M.; Yokus, A. Analytical and numerical approaches to nerve impulse model of fractional-order. Numer. Methods Partial. Differ. Equ. 2020, 36, 1348–1368. [Google Scholar] [CrossRef]
  8. Khater, M.M.; Attia, R.A.; Mahmoud, E.E.; Abdel-Aty, A.H.; Abualnaja, K.M.; Mohamed, A.B.; Eleuch, H. On the interaction between (low & high) frequency of (ion–acoustic & Langmuir) waves in plasma via some recent computational schemes. Results Phys. 2020, 19, 103684. [Google Scholar]
  9. Khater, M.M.; Attia, R.A.; Baleanu, D. Abundant new solutions of the transmission of nerve impulses of an excitable system. Eur. Phys. J. Plus 2020, 135, 1–12. [Google Scholar] [CrossRef]
  10. Khater, M.M.; Attia, R.A.; Lu, D. Computational and numerical simulations for the nonlinear fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation. Phys. Scr. 2020, 95, 055213. [Google Scholar] [CrossRef]
  11. Yavuz, M.; Sulaiman, T.A.; Usta, F.; Bulut, H. Analysis and numerical computations of the fractional regularized long-wave equation with damping term. Math. Methods Appl. Sci. 2020, 44, 7538–7555. [Google Scholar] [CrossRef]
  12. Khater, M.M.; Mohamed, M.S.; Park, C.; Attia, R.A. Effective computational schemes for a mathematical model of relativistic electrons arising in the laser thermonuclear fusion. Results Phys. 2020, 19, 103701. [Google Scholar] [CrossRef]
  13. Abdel-Aty, A.H.; Khater, M.M.; Baleanu, D.; Abo-Dahab, S.; Bouslimi, J.; Omri, M. Oblique explicit wave solutions of the fractional biological population (BP) and equal width (EW) models. Adv. Differ. Equ. 2020, 2020, 1–17. [Google Scholar] [CrossRef]
  14. Yavuz, M.; Sulaiman, T.A.; Yusuf, A.; Abdeljawad, T. The Schrödinger-KdV equation of fractional order with Mittag-Leffler nonsingular kernel. Alex. Eng. J. 2021, 60, 2715–2724. [Google Scholar] [CrossRef]
  15. Khater, M.M.; Lu, D.; Hamed, Y. Computational simulation for the (1+1)-dimensional Ito equation arising quantum mechanics and nonlinear optics. Results Phys. 2020, 19, 103572. [Google Scholar] [CrossRef]
  16. Rezazadeh, H.; Souleymanou, A.; Korkmaz, A.; Khater, M.M.; Mukam, S.P.; Kuetche, V.K. New exact solitary waves solutions to the fractional Fokas-Lenells equation via Atangana-Baleanu derivative operator. Int. J. Mod. Phys. B 2020, 34, 2050309. [Google Scholar] [CrossRef]
  17. Khater, M.; Chu, Y.M.; Attia, R.A.; Inc, M.; Lu, D. On the Analytical and Numerical Solutions in the Quantum Magnetoplasmas: The Atangana Conformable Derivative (2+1)-ZK Equation with Power-Law Nonlinearity. Adv. Math. Phys. 2020, 2020, 1–10. [Google Scholar] [CrossRef]
  18. Attia, R.A.; Alfalqi, S.; Alzaidi, J.; Khater, M.; Lu, D. Computational and Numerical Solutions for-Dimensional Integrable Schwarz–Korteweg–de Vries Equation with Miura Transform. Complexity 2020, 2020, 2394030. [Google Scholar] [CrossRef]
  19. Qin, H.; Khater, M.; Attia, R.A.; Lu, D. Approximate Simulations for the Non-linear Long-Short Wave Interaction System. Front. Phys. 2020, 7, 230. [Google Scholar] [CrossRef]
  20. Gao, W.; Rezazadeh, H.; Pinar, Z.; Baskonus, H.M.; Sarwar, S.; Yel, G. Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique. Opt. Quantum Electron. 2020, 52, 1–13. [Google Scholar] [CrossRef]
  21. Korkmaz, A.; Hepson, O.E.; Hosseini, K.; Rezazadeh, H.; Eslami, M. Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class. J. King Saud-Univ.-Sci. 2020, 32, 567–574. [Google Scholar] [CrossRef]
  22. Jajarmi, A.; Yusuf, A.; Baleanu, D.; Inc, M. A new fractional HRSV model and its optimal control: A non-singular operator approach. Phys. Stat. Mech. Its Appl. 2020, 547, 123860. [Google Scholar] [CrossRef]
  23. Inc, M.; Korpinar, Z.; Almohsen, B.; Chu, Y.M. Some numerical solutions of local fractional tricomi equation in fractal transonic flow. Alex. Eng. J. 2020, 60, 1147–1153. [Google Scholar] [CrossRef]
  24. Wang, K.J.; Wang, G.D. Variational principle and approximate solution for the fractal generalized Benjamin-Bona-Mahony-Burgers Equation in Fluid Mechanics. Fractals 2020. [Google Scholar] [CrossRef]
  25. Yavuz, M.; Sene, N. Approximate solutions of the model describing fluid flow using generalized ρ-laplace transform method and heat balance integral method. Axioms 2020, 9, 123. [Google Scholar] [CrossRef]
  26. Yavuz, M.; Abdeljawad, T. Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel. Adv. Differ. Equ. 2020, 2020, 1–18. [Google Scholar] [CrossRef]
  27. Ali, A.T.; Khater, M.M.; Attia, R.A.; Abdel-Aty, A.H.; Lu, D. Abundant numerical and analytical solutions of the generalized formula of Hirota-Satsuma coupled KdV system. Chaos Solitons Fractals 2020, 131, 109473. [Google Scholar] [CrossRef]
  28. Zhao, Z.; He, L. M-lump and hybrid solutions of a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation. Appl. Math. Lett. 2021, 111, 106612. [Google Scholar] [CrossRef]
  29. Zhang, L.D.; Tian, S.F.; Peng, W.Q.; Zhang, T.T.; Yan, X.J. The dynamics of lump, lumpoff and rogue wave solutions of (2+1)-dimensional Hirota-Satsuma-Ito equations. East Asian J. Appl. Math 2020, 10, 243–255. [Google Scholar] [CrossRef]
  30. Tala-Tebue, E.; Rezazadeh, H.; Djoufack, Z.I.; Eslam, M.; Kenfack-Jiotsa, A.; Bekir, A. Optical solutions of cold bosonic atoms in a zig-zag optical lattice. Opt. Quantum Electron. 2021, 53, 1–13. [Google Scholar] [CrossRef]
  31. Chu, Y.; Shallal, M.A.; Mirhosseini-Alizamini, S.M.; Rezazadeh, H.; Javeed, S.; Baleanu, D. Application of Modified Extended Tanh Technique for Solving Complex Ginzburg–Landau Equation Considering Kerr Law Nonlinearity. CMC-Comput. Mater. Contin. 2021, 66, 1369–1378. [Google Scholar] [CrossRef]
  32. da Silva, T.F.; Casarotti, S.N.; de Oliveira, G.L.V.; Penna, A.L.B. The impact of probiotics, prebiotics, and synbiotics on the biochemical, clinical, and immunological markers, as well as on the gut microbiota of obese hosts. Crit. Rev. Food Sci. Nutr. 2021, 61, 337–355. [Google Scholar] [CrossRef] [PubMed]
  33. Osman, M.; Korkmaz, A.; Rezazadeh, H.; Mirzazadeh, M.; Eslami, M.; Zhou, Q. The unified method for conformable time fractional Schroö dinger equation with perturbation terms. Chin. J. Phys. 2018, 56, 2500–2506. [Google Scholar] [CrossRef]
  34. Liu, J.G.; Eslami, M.; Rezazadeh, H.; Mirzazadeh, M. Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation. Nonlinear Dyn. 2019, 95, 1027–1033. [Google Scholar] [CrossRef]
  35. Shen, W.; Ma, Z.; Fei, J.; Zhu, Q. Novel characteristics of lump and lump–soliton interaction solutions to the (2+1)-dimensional Alice–Bob Hirota–Satsuma–Ito equation. Mod. Phys. Lett. B 2020, 34, 2050419. [Google Scholar] [CrossRef]
Figure 1. Bright solitary wave solution of Equation (12) in (a) three–dimension, (b) density and (c) spherical plot three–dimensional for h 1 = 4 , h 3 = 32 , ϑ = 5 , ϱ = 1 .
Figure 1. Bright solitary wave solution of Equation (12) in (a) three–dimension, (b) density and (c) spherical plot three–dimensional for h 1 = 4 , h 3 = 32 , ϑ = 5 , ϱ = 1 .
Fluids 06 00235 g001
Figure 2. Solitary wave solution of Equation (16) in (a) three–dimension, (b) density and (c) spherical plot three–dimensional for h 1 = 1 , h 3 = 9 , ϑ = 1 , ϱ = 1 .
Figure 2. Solitary wave solution of Equation (16) in (a) three–dimension, (b) density and (c) spherical plot three–dimensional for h 1 = 1 , h 3 = 9 , ϑ = 1 , ϱ = 1 .
Fluids 06 00235 g002
Figure 3. Solitary wave solution of Equation (32) in (a) three–dimension, (b) density and (c) spherical plot three–dimensional for a = 3 , ϱ = 0.5 .
Figure 3. Solitary wave solution of Equation (32) in (a) three–dimension, (b) density and (c) spherical plot three–dimensional for a = 3 , ϱ = 0.5 .
Fluids 06 00235 g003
Figure 4. Solitary wave solution of Equation (33) in (a) three–dimension, (b) density and (c) spherical plot three–dimensional for a = 2 , ϱ = 0.5 .
Figure 4. Solitary wave solution of Equation (33) in (a) three–dimension, (b) density and (c) spherical plot three–dimensional for a = 2 , ϱ = 0.5 .
Fluids 06 00235 g004
Figure 5. Matching between exact and semi–analytical solutions based on Table 1.
Figure 5. Matching between exact and semi–analytical solutions based on Table 1.
Fluids 06 00235 g005
Figure 6. Matching between exact and semi–analytical solutions based on Table 2.
Figure 6. Matching between exact and semi–analytical solutions based on Table 2.
Fluids 06 00235 g006
Figure 7. MKud method’s superiority.
Figure 7. MKud method’s superiority.
Fluids 06 00235 g007
Table 1. Accuracy of the ESE method’s solutions through AD method.
Table 1. Accuracy of the ESE method’s solutions through AD method.
Value of Ξ E I , 3 E appro | E I , 3 ( Ξ ) Absolute ErrorValue of Ξ E I , 3 E appro | E I , 3 ( Ξ ) Absolute Error
00.6666666670.66666666700.260.5373342310.5404277020.003093471
0.010.666466680.6664666876.66684 × 10 9 0.270.5276705590.5312716440.003601085
0.020.665866880.6658669871.06678 × 10 7 0.280.5177123010.5218813610.00416906
0.030.6648677460.6648682865.40129 × 10 7 0.290.5074665620.5122686110.004802049
0.040.6634700770.6634717841.70739 × 10 6 0.30.496940590.5024454670.005504876
0.050.6616749880.6616791584.16943 × 10 6 0.310.4861417680.4924243070.006282539
0.060.6594839110.659492568.64823 × 10 6 0.320.4750775980.4822178030.007140204
0.070.6568985910.6569146191.60274 × 10 5 0.330.4637556960.4718389050.008083209
0.080.6539210830.6539484352.73526 × 10 5 0.340.4521837780.4613008330.009117055
0.090.6505537470.650597584.38328 × 10 5 0.350.4403696530.4506170580.010247405
0.10.6467992480.6468660896.68405 × 10 5 0.360.4283212080.439801290.011480082
0.110.642660550.6427584639.79132 × 10 5 0.370.4160464020.4288674670.012821065
0.120.6381409070.6382796610.0001387540.380.4035532530.4178297350.014276482
0.130.6332438650.6334350980.0001912330.390.3908498290.4067024350.015852606
0.140.6279732490.6282306360.0002573880.40.3779442390.3955000890.01755585
0.150.622333160.6226725850.0003394250.410.364844620.3842373820.019392762
0.160.6163279710.6167676930.0004397220.420.3515591310.3729291460.021370015
0.170.6099623130.6105231410.0005608270.430.338095940.3615903440.023494403
0.180.6032410750.6039465350.000705460.440.3244632180.350236050.025772832
0.190.5961693910.5970459050.0008765140.450.3106691250.3388814350.02821231
0.20.5887526330.5898296890.0010770560.460.2967218090.3275417480.03081994
0.210.5809964030.5823067330.001310330.470.2826293880.3162322960.033602908
0.220.5729065250.5744862780.0015797540.480.2683999470.3049684230.036568476
0.230.5644890350.5663779550.001888920.490.2540415310.2937654990.039723967
0.240.5557501710.5579917720.0022416010.50.2395621330.2826388890.043076756
0.250.5466963640.5493381080.0026417430.510.2249696870.2716039430.046634256
Table 2. Accuracy of the MKud method’s solutions through AD method.
Table 2. Accuracy of the MKud method’s solutions through AD method.
Value of Ξ E I E appro | E I ( Ξ ) Absolute ErrorValue of Ξ E I E appro | E I ( Ξ ) Absolute Error
00.6034740.60347400.260.5913310.5915830.000252
0.010.6034560.6034565.49 × 10 10 0.270.5903920.5906860.000294
0.020.6034020.6034028.79 × 10 9 0.280.5894210.5897610.00034
0.030.6033110.6033114.45 × 10 8 0.290.5884160.5888070.000391
0.040.6031830.6031831.41 × 10 7 0.30.5873790.5878270.000448
0.050.6030190.603023.43 × 10 7 0.310.5863080.586820.000511
0.060.6028190.602827.12 × 10 7 0.320.5852060.5857870.000581
0.070.6025830.6025841.32 × 10 6 0.330.5840710.5847280.000657
0.080.6023110.6023132.25 × 10 6 0.340.5829050.5836460.000741
0.090.6020020.6020063.61 × 10 6 0.350.5817070.5825390.000832
0.10.6016570.6016635.5 × 10 6 0.360.5804770.581410.000932
0.110.6012770.6012858.05 × 10 6 0.370.5792170.5802580.001041
0.120.600860.6008711.14 × 10 5 0.380.5779260.5790840.001159
0.130.6004080.6004231.57 × 10 5 0.390.5766040.577890.001286
0.140.599920.5999412.11 × 10 5 0.40.5752530.5766770.001424
0.150.5993960.5994242.79 × 10 5 0.410.5738710.5754440.001573
0.160.5988370.5988733.61 × 10 5 0.420.572460.5741930.001733
0.170.5982430.5982884.6 × 10 5 0.430.571020.5729250.001905
0.250.5922350.5924510.0002160.510.5584850.5622750.00379
0.180.5976130.5976715.78 × 10 5 0.440.5695510.5716410.00209
0.190.5969480.597027.18 × 10 5 0.450.5680530.5703410.002288
0.20.5962490.5963378.82 × 10 5 0.460.5665270.5690270.0025
0.210.5955150.5956220.0001070.470.5649730.5676990.002726
0.220.5947460.5948760.0001290.480.5633910.5663590.002968
0.230.5939430.5940980.0001540.490.5617830.5650080.003225
0.240.5931060.593290.0001830.50.5601470.5636460.003499
0.250.5922350.5924510.0002160.510.5584850.5622750.00379
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

Yue, C.; Lu, D.; Khater, M.M.A. Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation. Fluids 2021, 6, 235. https://doi.org/10.3390/fluids6070235

AMA Style

Yue C, Lu D, Khater MMA. Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation. Fluids. 2021; 6(7):235. https://doi.org/10.3390/fluids6070235

Chicago/Turabian Style

Yue, Chen, Dianchen Lu, and Mostafa M. A. Khater. 2021. "Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation" Fluids 6, no. 7: 235. https://doi.org/10.3390/fluids6070235

Article Metrics

Back to TopTop