Analytical insight into fractional Fornberg-Whitham equations using novel transform methods

Safyan Mukhtar; Wedad Albalawi; Faisal Haroon; Samir A. El-Tantawy; Safyan Mukhtar; Wedad Albalawi; Faisal Haroon; Samir A. El-Tantawy

doi:10.3934/math.2025375

AIMS Mathematics

2025, Volume 10, Issue 4: 8165-8190. doi: 10.3934/math.2025375

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Analytical insight into fractional Fornberg-Whitham equations using novel transform methods

1.
Department of Basic Sciences, General Administration of Preparatory Year, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia; fharoon@kfu.edu.sa
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia; wsalbalawi@pnu.edu.sa
4.
Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt; samireltantawy@yahoo.com, tantawy@sci.psu.edu.eg
5.
Department of Physics, Faculty of Science, Al-Baha University, Al-Baha 1988, Saudi Arabia

Received: 10 February 2025 Revised: 27 March 2025 Accepted: 29 March 2025 Published: 09 April 2025
MSC : 34G20, 35A20, 35A22, 35R11

This work addressed one of the most essential evolutionary equations that was widely used in describing various nonlinear wave propagation and dispersive phenomena in various scientific and engineering applications, which was called the nonlinear fractional Fornberg-Whitham (FFW). Due to the importance of this equation, we examined it by employing two highly effective techniques: the residual power series method (RPSM) and the new iterative approach (NIM), both distinguished by their efficacy in solving more complicated nonlinear fractional evolutionary equations. Moreover, we integrated the Elzaki transform with both approaches to create Elzaki RPSM (ERPSM) and Elzaki NIM (ENIM) to ease the calculations. The ERPSM effectively combined the power series approach with residual error analysis to generate highly accurate series solutions, while ENIM provided alternative frameworks for handling nonlinearities and achieving rapid convergence. Comparative studies of the obtained solutions highlighted these methods' efficiency, accuracy, and reliability in solving fractional-order differential equations. The results underscored the potential of these analytical techniques for modeling and solving complex fractional wave equations, contributing to the advancement of mathematical physics and computational fluid dynamics.
- Elzaki transform,
- residual power series transform method,
- new iterative transform method,
- fractional Fornberg-Whitham equations
Citation: Safyan Mukhtar, Wedad Albalawi, Faisal Haroon, Samir A. El-Tantawy. Analytical insight into fractional Fornberg-Whitham equations using novel transform methods[J]. AIMS Mathematics, 2025, 10(4): 8165-8190. doi: 10.3934/math.2025375

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Abstract

This work addressed one of the most essential evolutionary equations that was widely used in describing various nonlinear wave propagation and dispersive phenomena in various scientific and engineering applications, which was called the nonlinear fractional Fornberg-Whitham (FFW). Due to the importance of this equation, we examined it by employing two highly effective techniques: the residual power series method (RPSM) and the new iterative approach (NIM), both distinguished by their efficacy in solving more complicated nonlinear fractional evolutionary equations. Moreover, we integrated the Elzaki transform with both approaches to create Elzaki RPSM (ERPSM) and Elzaki NIM (ENIM) to ease the calculations. The ERPSM effectively combined the power series approach with residual error analysis to generate highly accurate series solutions, while ENIM provided alternative frameworks for handling nonlinearities and achieving rapid convergence. Comparative studies of the obtained solutions highlighted these methods' efficiency, accuracy, and reliability in solving fractional-order differential equations. The results underscored the potential of these analytical techniques for modeling and solving complex fractional wave equations, contributing to the advancement of mathematical physics and computational fluid dynamics.

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