Wave solution for time fractional geophysical KdV equation in uncertain environment

Mrutyunjaya Sahoo; Dhabaleswar Mohapatra; S. Chakraverty; Mrutyunjaya Sahoo; Dhabaleswar Mohapatra; S. Chakraverty

doi:10.3934/mmc.2025005

Mathematical Modelling and Control

2025, Volume 5, Issue 1: 61-72. doi: 10.3934/mmc.2025005

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Wave solution for time fractional geophysical KdV equation in uncertain environment

1.
Department of Mathematics, National Institute of Technology Rourkela, Odisha, India
2.
Department of Mathematics, Institute of Technical Education and Research, Siksha O Anusandhan Deemed to be University, Bhubaneswar, Odisha, India

Received: 30 March 2024 Revised: 03 October 2024 Accepted: 29 October 2024 Published: 13 March 2025

This work aims to develop an approximate analytical solution for the geophysical Korteweg-de Vries (GeoKdV) equation with time-fractional derivatives defined in the Caputo sense. This equation is relevant to shallow water wave (SWW) propagation, which may have uses in mathematical physics and engineering sciences. In real-world scenarios, factors such as environmental or climate changes or the dynamics of air and water waves introduce uncertainty or ambiguity into parameters like the Coriolis effect and initial or boundary conditions. Unlike previous studies that solely focused on either integer-order or fractional-order models, this research introduces fractional dynamics with fuzzy uncertainty. To deal with such uncertainty, this work aims to find the approximate fuzzy solution to the said physical problem by applying a double parametric approach with the help of an effective method called the fractional reduced differential transform method (FRDTM). This approach has been shown to be highly effective, thereby efficiently addressing both fractional calculus and fuzzy initial conditions. Furthermore, to validate the obtained solution, we conduct a comparison between the special case of the current fuzzy solution, thereby considering the fractional order of the index with the existing precise (crisp) solutions of the integer order governing equation. The solutions are presented in both fuzzy and precise formats, and graphical representations are provided to enhance the understanding of their physical significance across various parameter values.
- fractional theory,
- geophysical KdV equation,
- fractional reduced differential transform method,
- fuzzy number
Citation: Mrutyunjaya Sahoo, Dhabaleswar Mohapatra, S. Chakraverty. Wave solution for time fractional geophysical KdV equation in uncertain environment[J]. Mathematical Modelling and Control, 2025, 5(1): 61-72. doi: 10.3934/mmc.2025005

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Abstract

This work aims to develop an approximate analytical solution for the geophysical Korteweg-de Vries (GeoKdV) equation with time-fractional derivatives defined in the Caputo sense. This equation is relevant to shallow water wave (SWW) propagation, which may have uses in mathematical physics and engineering sciences. In real-world scenarios, factors such as environmental or climate changes or the dynamics of air and water waves introduce uncertainty or ambiguity into parameters like the Coriolis effect and initial or boundary conditions. Unlike previous studies that solely focused on either integer-order or fractional-order models, this research introduces fractional dynamics with fuzzy uncertainty. To deal with such uncertainty, this work aims to find the approximate fuzzy solution to the said physical problem by applying a double parametric approach with the help of an effective method called the fractional reduced differential transform method (FRDTM). This approach has been shown to be highly effective, thereby efficiently addressing both fractional calculus and fuzzy initial conditions. Furthermore, to validate the obtained solution, we conduct a comparison between the special case of the current fuzzy solution, thereby considering the fractional order of the index with the existing precise (crisp) solutions of the integer order governing equation. The solutions are presented in both fuzzy and precise formats, and graphical representations are provided to enhance the understanding of their physical significance across various parameter values.

References

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