The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method

Aslı Alkan; Halil Anaç; Aslı Alkan; Halil Anaç

doi:10.3934/math.20241237

AIMS Mathematics

2024, Volume 9, Issue 9: 25333-25359. doi: 10.3934/math.20241237

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The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method

Aslı Alkan ¹,
Halil Anaç ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, Fırat University, Elazığ 23119, Turkey
2.
Department of Computer Technologies, Torul Vocational School, Gumushane University, Gumushane 29802, Turkey

Received: 25 January 2024 Revised: 07 May 2024 Accepted: 20 May 2024 Published: 30 August 2024
MSC : 35C05, 35R11, 65R10

The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.
Citation: Aslı Alkan, Halil Anaç. The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method[J]. AIMS Mathematics, 2024, 9(9): 25333-25359. doi: 10.3934/math.20241237

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Abstract

The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.

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