Document Type : Research Paper
Authors
1
1. Nanoelectronics Integrated Systems Center, Nile University, Giza, 12588, Egypt. 2. Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt.
2
1. Department of Mathematics, College of Sciences and Humanities, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia. 2. Department of Mathematics and Computer Sciences, Faculty of Science, Port-Said University, Egypt.
3
Department of Mathematics and Computer Sciences, Faculty of Science, Port-Said University, Egypt.
4
1. Department of Mathematics and Computer Sciences, Faculty of Science, Port-Said University, Egypt. 2. Higher Institute for Engineering and Technology, Manzala, Egypt.
Abstract
This study investigates the fractional-order Jaulent–Miodek system, which models nonlinear wave propagation in dispersive media with memory effects. A Petrov–Galerkin finite element method (PG-FEM) based on the Caputo fractional derivative is employed to capture long-range temporal correlations and nonlocal interactions. The proposed scheme achieves high accuracy and computational efficiency in resolving compacton solutions, soliton interactions, and nonlinear wave structures, overcoming limitations of traditional finite difference and spectral methods. Numerical simulations show that reducing the fractional order \((\alpha < 1)\) enhances dispersion and delays attenuation, consistent with behaviors observed in viscoelastic media, porous fluid dynamics, nonlinear optics, and biomechanical processes. Moreover, the results confirm that fractional-order models provide a more faithful representation of power-law wave decay and anomalous transport phenomena compared with classical integer-order formulations. These findings underscore the importance of fractional calculus in advancing the mathematical modeling of nonlinear wave systems and point to promising directions for adaptive numerical schemes and future experimental validation.
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