A novel method to study time fractional coupled systems of shallow water equations arising in ocean engineering

K. Pavani; K. Raghavendar; K. Pavani; K. Raghavendar

doi:10.3934/math.2024029

AIMS Mathematics

2024, Volume 9, Issue 1: 542-564. doi: 10.3934/math.2024029

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A novel method to study time fractional coupled systems of shallow water equations arising in ocean engineering

K. Pavani ,
K. Raghavendar ^,

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India

Received: 13 July 2023 Revised: 13 September 2023 Accepted: 19 September 2023 Published: 01 December 2023
MSC : 26A33, 35A22, 35R11

This study investigates solutions for the time-fractional coupled system of the shallow-water equations. The shallow-water equations are employed for the purpose of elucidating the dynamics of water motion in oceanic or sea environments. Also, the aforementioned system characterizes a thin fluid layer that maintains a hydrostatic equilibrium while exhibiting uniform density. Shallow water flows have a vertical dimension that is considerably smaller in magnitude than the typical horizontal dimension. In the current work, we employ an innovative and effective technique, known as the natural transform decomposition method, to obtain the solutions for these fractional systems. The present methodology entails the utilization of both singular and non-singular kernels for the purpose of handling fractional derivatives. The Banach fixed point theorem is employed to demonstrate the uniqueness and convergence of the obtained solution. The outcomes obtained from the application of the suggested methodology are compared to the exact solution and the results of other numerical methods found in the literature, including the modified homotopy analysis transform method, the residual power series method and the new iterative method. The results obtained from the proposed methodology are presented through the use of tabular and graphical simulations. The current framework effectively captures the behavior exhibited by different fractional orders. The findings illustrate the efficacy of the proposed method.
- shallow-water equation,
- Caputo-Fabrizio derivative,
- Atangana-Baleanu-Caputo derivative,
- Caputo derivative
Citation: K. Pavani, K. Raghavendar. A novel method to study time fractional coupled systems of shallow water equations arising in ocean engineering[J]. AIMS Mathematics, 2024, 9(1): 542-564. doi: 10.3934/math.2024029

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Abstract

This study investigates solutions for the time-fractional coupled system of the shallow-water equations. The shallow-water equations are employed for the purpose of elucidating the dynamics of water motion in oceanic or sea environments. Also, the aforementioned system characterizes a thin fluid layer that maintains a hydrostatic equilibrium while exhibiting uniform density. Shallow water flows have a vertical dimension that is considerably smaller in magnitude than the typical horizontal dimension. In the current work, we employ an innovative and effective technique, known as the natural transform decomposition method, to obtain the solutions for these fractional systems. The present methodology entails the utilization of both singular and non-singular kernels for the purpose of handling fractional derivatives. The Banach fixed point theorem is employed to demonstrate the uniqueness and convergence of the obtained solution. The outcomes obtained from the application of the suggested methodology are compared to the exact solution and the results of other numerical methods found in the literature, including the modified homotopy analysis transform method, the residual power series method and the new iterative method. The results obtained from the proposed methodology are presented through the use of tabular and graphical simulations. The current framework effectively captures the behavior exhibited by different fractional orders. The findings illustrate the efficacy of the proposed method.

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