Investigation of time fractional nonlinear KdV-Burgers equation under fractional operators with nonsingular kernels

Asif Khan; Tayyaba Akram; Arshad Khan; Shabir Ahmad; Kamsing Nonlaopon; Asif Khan; Tayyaba Akram; Arshad Khan; Shabir Ahmad; Kamsing Nonlaopon

doi:10.3934/math.2023063

AIMS Mathematics

2023, Volume 8, Issue 1: 1251-1268. doi: 10.3934/math.2023063

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Research article

Investigation of time fractional nonlinear KdV-Burgers equation under fractional operators with nonsingular kernels

1.
Department of Mathematics, University of Malakand, Chakdara, Dir Lower, Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematics, COMSATS Institute of Information Technology, Lahore, 54000, Pakistan
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 14 July 2022 Revised: 30 September 2022 Accepted: 10 October 2022 Published: 19 October 2022
MSC : 35R11

In this manuscript, the Korteweg-de Vries-Burgers (KdV-Burgers) partial differential equation (PDE) is investigated under nonlocal operators with the Mittag-Leffler kernel and the exponential decay kernel. For both fractional operators, the existence of the solution of the KdV-Burgers PDE is demonstrated through fixed point theorems of $ \alpha $-type $ \digamma $ contraction. The modified double Laplace transform is utilized to compute a series solution that leads to the exact values when fractional order equals unity. The effectiveness and reliability of the suggested approach are verified and confirmed by comparing the series outcomes to the exact values. Moreover, the series solution is demonstrated through graphs for a few fractional orders. Lastly, a comparison between the results of the two fractional operators is studied through numerical data and diagrams. The results show how consistently accurate the method is and how broadly applicable it is to fractional nonlinear evolution equations.
- double Laplace transform,
- KdV equation,
- Burgers equation,
- fractional operators
Citation: Asif Khan, Tayyaba Akram, Arshad Khan, Shabir Ahmad, Kamsing Nonlaopon. Investigation of time fractional nonlinear KdV-Burgers equation under fractional operators with nonsingular kernels[J]. AIMS Mathematics, 2023, 8(1): 1251-1268. doi: 10.3934/math.2023063

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Abstract

In this manuscript, the Korteweg-de Vries-Burgers (KdV-Burgers) partial differential equation (PDE) is investigated under nonlocal operators with the Mittag-Leffler kernel and the exponential decay kernel. For both fractional operators, the existence of the solution of the KdV-Burgers PDE is demonstrated through fixed point theorems of $ \alpha $-type $ \digamma $ contraction. The modified double Laplace transform is utilized to compute a series solution that leads to the exact values when fractional order equals unity. The effectiveness and reliability of the suggested approach are verified and confirmed by comparing the series outcomes to the exact values. Moreover, the series solution is demonstrated through graphs for a few fractional orders. Lastly, a comparison between the results of the two fractional operators is studied through numerical data and diagrams. The results show how consistently accurate the method is and how broadly applicable it is to fractional nonlinear evolution equations.

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