Nonlinear wave train in an inhomogeneous medium with the fractional theory in a plane self-focusing

Muhammad Imran Asjad; Waqas Ali Faridi; Adil Jhangeer; Maryam Aleem; Abdullahi Yusuf; Ali S. Alshomrani; Dumitru Baleanu; Muhammad Imran Asjad; Waqas Ali Faridi; Adil Jhangeer; Maryam Aleem; Abdullahi Yusuf; Ali S. Alshomrani; Dumitru Baleanu

doi:10.3934/math.2022462

AIMS Mathematics

2022, Volume 7, Issue 5: 8290-8313. doi: 10.3934/math.2022462

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Nonlinear wave train in an inhomogeneous medium with the fractional theory in a plane self-focusing

1.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
2.
Department of Mathematics, Namal Institute, Talagang Road, Mianwali 42250, Pakistan
3.
Department of Computer Engineering, Biruni University, Istanbul, Turkey
4.
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
5.
Department of Mathematics, King Abdul Aziz University, Jeddah, Saudi Arabia
6.
Department of Mathematics, Cankaya University, Ankara, Turkey
7.
Institute of Space Sciences, Magurele, Bucharest, Romania
8.
Department of Medical Research, China Medical University, Taichung, Taiwan

Received: 27 December 2021 Revised: 05 February 2022 Accepted: 13 February 2022 Published: 25 February 2022
MSC : 34K25, 34K40, 37B25, 58K25

The aim of study is to investigate the Hirota equation which has a significant role in applied sciences, like maritime, coastal engineering, ocean, and the main source of the environmental action due to energy transportation on floating anatomical structures. The classical Hirota model has transformed into a fractional Hirota governing equation by using the space-time fractional Riemann-Liouville, time fractional Atangana-Baleanu and space-time fractional $ \beta $ differential operators. The most generalized new extended direct algebraic technique is applied to obtain the solitonic patterns. The utilized scheme provided a generalized class of analytical solutions, which is presented by the trigonometric, rational, exponential and hyperbolic functions. The analytical solutions which cover almost all types of soliton are obtained with Riemann-Liouville, Atangana-Baleanu and $ \beta $ fractional operator. The influence of the fractional-order parameter on the acquired solitary wave solutions is graphically studied. The two and three-dimensional graphical comparison between Riemann-Liouville, Atangana-Baleanu and $ \beta $-fractional derivatives for the solutions of the Hirota equation is displayed by considering suitable involved parametric values with the aid of Mathematica.
- multi-wave non-linear Hirota equation,
- fractional derivatives,
- travelling wave transformation,
- new extended direct algebraic method,
- soliton solutions
Citation: Muhammad Imran Asjad, Waqas Ali Faridi, Adil Jhangeer, Maryam Aleem, Abdullahi Yusuf, Ali S. Alshomrani, Dumitru Baleanu. Nonlinear wave train in an inhomogeneous medium with the fractional theory in a plane self-focusing[J]. AIMS Mathematics, 2022, 7(5): 8290-8313. doi: 10.3934/math.2022462

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Abstract

The aim of study is to investigate the Hirota equation which has a significant role in applied sciences, like maritime, coastal engineering, ocean, and the main source of the environmental action due to energy transportation on floating anatomical structures. The classical Hirota model has transformed into a fractional Hirota governing equation by using the space-time fractional Riemann-Liouville, time fractional Atangana-Baleanu and space-time fractional $ \beta $ differential operators. The most generalized new extended direct algebraic technique is applied to obtain the solitonic patterns. The utilized scheme provided a generalized class of analytical solutions, which is presented by the trigonometric, rational, exponential and hyperbolic functions. The analytical solutions which cover almost all types of soliton are obtained with Riemann-Liouville, Atangana-Baleanu and $ \beta $ fractional operator. The influence of the fractional-order parameter on the acquired solitary wave solutions is graphically studied. The two and three-dimensional graphical comparison between Riemann-Liouville, Atangana-Baleanu and $ \beta $-fractional derivatives for the solutions of the Hirota equation is displayed by considering suitable involved parametric values with the aid of Mathematica.

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