Nonlinear analysis of a nonlinear modified KdV equation under Atangana Baleanu Caputo derivative

Gulalai; Shabir Ahmad; Fathalla Ali Rihan; Aman Ullah; Qasem M. Al-Mdallal; Ali Akgül; Gulalai; Shabir Ahmad; Fathalla Ali Rihan; Aman Ullah; Qasem M. Al-Mdallal; Ali Akgül

doi:10.3934/math.2022439

AIMS Mathematics

2022, Volume 7, Issue 5: 7847-7865. doi: 10.3934/math.2022439

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Nonlinear analysis of a nonlinear modified KdV equation under Atangana Baleanu Caputo derivative

1.
Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, Abu Dhabi, UAE
3.
Art and Science Faculty, Department of Mathematics, Siirt University, TR-56100 Siirt, Turkey

Received: 13 September 2021 Revised: 27 October 2021 Accepted: 28 October 2021 Published: 17 February 2022
MSC : 35R11

The focus of the current manuscript is to provide a theoretical and computational analysis of the new nonlinear time-fractional (2+1)-dimensional modified KdV equation involving the Atangana-Baleanu Caputo ($ \mathcal{ABC} $) derivative. A systematic and convergent technique known as the Laplace Adomian decomposition method (LADM) is applied to extract a semi-analytical solution for the considered equation. The notion of fixed point theory is used for the derivation of the results related to the existence of at least one and unique solution of the mKdV equation involving under $ \mathcal{ABC} $-derivative. The theorems of fixed point theory are also used to derive results regarding to the convergence and Picard's X-stability of the proposed computational method. A proper investigation is conducted through graphical representation of the achieved solution to determine that the $ \mathcal{ABC} $ operator produces better dynamics of the obtained analytic soliton solution. Finally, 2D and 3D graphs are used to compare the exact solution and approximate solution. Also, a comparison between the exact solution, solution under Caputo-Fabrizio, and solution under the $ \mathcal{ABC} $ operator of the proposed equation is provided through graphs, which reflect that $ \mathcal{ABC} $-operator produces better dynamics of the proposed equation than the Caputo-Fabrizio one.
- Atangana-Baleanu fractional operator,
- fixed point theory,
- Laplace Adomian decomposition
Citation: Gulalai, Shabir Ahmad, Fathalla Ali Rihan, Aman Ullah, Qasem M. Al-Mdallal, Ali Akgül. Nonlinear analysis of a nonlinear modified KdV equation under Atangana Baleanu Caputo derivative[J]. AIMS Mathematics, 2022, 7(5): 7847-7865. doi: 10.3934/math.2022439

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Abstract

The focus of the current manuscript is to provide a theoretical and computational analysis of the new nonlinear time-fractional (2+1)-dimensional modified KdV equation involving the Atangana-Baleanu Caputo ($ \mathcal{ABC} $) derivative. A systematic and convergent technique known as the Laplace Adomian decomposition method (LADM) is applied to extract a semi-analytical solution for the considered equation. The notion of fixed point theory is used for the derivation of the results related to the existence of at least one and unique solution of the mKdV equation involving under $ \mathcal{ABC} $-derivative. The theorems of fixed point theory are also used to derive results regarding to the convergence and Picard's X-stability of the proposed computational method. A proper investigation is conducted through graphical representation of the achieved solution to determine that the $ \mathcal{ABC} $ operator produces better dynamics of the obtained analytic soliton solution. Finally, 2D and 3D graphs are used to compare the exact solution and approximate solution. Also, a comparison between the exact solution, solution under Caputo-Fabrizio, and solution under the $ \mathcal{ABC} $ operator of the proposed equation is provided through graphs, which reflect that $ \mathcal{ABC} $-operator produces better dynamics of the proposed equation than the Caputo-Fabrizio one.

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