Advanced analytical techniques for fractional Schrödinger and Korteweg-de Vries equations

Qasem M. Tawhari; Qasem M. Tawhari

doi:10.3934/math.2025530

AIMS Mathematics

2025, Volume 10, Issue 5: 11708-11731. doi: 10.3934/math.2025530

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Advanced analytical techniques for fractional Schrödinger and Korteweg-de Vries equations

Qasem M. Tawhari ^,

Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Saudi Arabia

Received: 10 February 2025 Revised: 28 April 2025 Accepted: 09 May 2025 Published: 21 May 2025
MSC : 34G20, 35A20, 35A22, 35R11

This paper investigated the Schrödinger and Korteweg-de Vries equations within the framework of fractional-order differential equations, utilizing the variational iteration transform method and the q-homotopy analysis transform method. These equations, crucial for modeling wave propagation and nonlinear dispersive systems, were analyzed using the Caputo fractional derivative to explore the influence of non-integer orders on their dynamics. The findings contributed to a deeper understanding of how fractional-order parameters affected the behavior of nonlinear wave models and oscillations, underscoring the growing importance of fractional calculus in mathematical physics and engineering. Both methods presented in this study effectively converted fractional-order problems into iterative schemes that were straightforward to solve, leading to quicker convergence of the analytical solutions. A comparative analysis evaluated the accuracy, computational efficiency, and convergence properties of the variational iteration transform method (VITM) and the q-homotopy analysis transform method q-HATM. The results, supported by numerical simulations and various graphical representations, validated the practicality and effectiveness of these methods for solving complex fractional differential equations. This study not only enhanced our comprehension of fractional wave dynamics but also strengthened the body of knowledge in both analytical and computational methods in mathematical physics and engineering.
Citation: Qasem M. Tawhari. Advanced analytical techniques for fractional Schrödinger and Korteweg-de Vries equations[J]. AIMS Mathematics, 2025, 10(5): 11708-11731. doi: 10.3934/math.2025530

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Abstract

This paper investigated the Schrödinger and Korteweg-de Vries equations within the framework of fractional-order differential equations, utilizing the variational iteration transform method and the q-homotopy analysis transform method. These equations, crucial for modeling wave propagation and nonlinear dispersive systems, were analyzed using the Caputo fractional derivative to explore the influence of non-integer orders on their dynamics. The findings contributed to a deeper understanding of how fractional-order parameters affected the behavior of nonlinear wave models and oscillations, underscoring the growing importance of fractional calculus in mathematical physics and engineering. Both methods presented in this study effectively converted fractional-order problems into iterative schemes that were straightforward to solve, leading to quicker convergence of the analytical solutions. A comparative analysis evaluated the accuracy, computational efficiency, and convergence properties of the variational iteration transform method (VITM) and the q-homotopy analysis transform method q-HATM. The results, supported by numerical simulations and various graphical representations, validated the practicality and effectiveness of these methods for solving complex fractional differential equations. This study not only enhanced our comprehension of fractional wave dynamics but also strengthened the body of knowledge in both analytical and computational methods in mathematical physics and engineering.

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