Symmetry analysis and conservation laws of time fractional Airy type and other KdV type equations

Miguel Vivas-Cortez; Yasir Masood; Absar Ul Haq; Imran Abbas Baloch; Abdul Hamid Kara; F. D. Zaman; Miguel Vivas-Cortez; Yasir Masood; Absar Ul Haq; Imran Abbas Baloch; Abdul Hamid Kara; F. D. Zaman

doi:10.3934/math.20231514

AIMS Mathematics

2023, Volume 8, Issue 12: 29569-29576. doi: 10.3934/math.20231514

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Symmetry analysis and conservation laws of time fractional Airy type and other KdV type equations

1.
Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, Pontifical Catholic University of Ecuador, Sede Quito, Ecuador
2.
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
3.
Department of Natural Sciences and Humanities, University of Engineering and Technology (Narowal Campus), Lahore 54000, Pakistan
4.
Higher Education Department, Govt. Graduate College for Boys Gulberg Lahore, Punjab, Pakistan
5.
School of Mathematics, University of Witwatersrand, Johannesburg, Wits 2050, South Africa

Received: 12 July 2023 Revised: 13 September 2023 Accepted: 24 September 2023 Published: 01 November 2023
MSC : 05C35, 05C50

We study the invariance properties of the fractional time version of the nonlinear class of equations $ u_{t}^{\alpha}-g(u)\; u_{x}-f(u)\; u_{xxx} = 0 $, where $ 0 < \alpha < 1 $ using some recently developed symmetry-based techniques. The equations reduce to ordinary fractional Airy type, Korteweg-de Vries (KdV) and modified KdV equations through the change of variables provided by the symmetries. Furthermore, we utilize the symmetries to construct conservation laws for the fractional partial differential equations.
- Airy equation,
- KdV equation,
- modified KdV equation,
- transformation groups,
- Lie symmetries,
- conservation laws
Citation: Miguel Vivas-Cortez, Yasir Masood, Absar Ul Haq, Imran Abbas Baloch, Abdul Hamid Kara, F. D. Zaman. Symmetry analysis and conservation laws of time fractional Airy type and other KdV type equations[J]. AIMS Mathematics, 2023, 8(12): 29569-29576. doi: 10.3934/math.20231514

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Abstract

We study the invariance properties of the fractional time version of the nonlinear class of equations $ u_{t}^{\alpha}-g(u)\; u_{x}-f(u)\; u_{xxx} = 0 $, where $ 0 < \alpha < 1 $ using some recently developed symmetry-based techniques. The equations reduce to ordinary fractional Airy type, Korteweg-de Vries (KdV) and modified KdV equations through the change of variables provided by the symmetries. Furthermore, we utilize the symmetries to construct conservation laws for the fractional partial differential equations.

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