Efficient Laplace-based decomposition and iteration techniques for the fractional GHS coupled KdV system with Atangana-Baleanu derivative

Mofida Zaki; M. Abdelgaber; Hoda F. Ahmed; W. A. Hashem; Mofida Zaki; M. Abdelgaber; Hoda F. Ahmed; W. A. Hashem

doi:10.3934/math.20251232

AIMS Mathematics

2025, Volume 10, Issue 11: 28034-28058. doi: 10.3934/math.20251232

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Efficient Laplace-based decomposition and iteration techniques for the fractional GHS coupled KdV system with Atangana-Baleanu derivative

1.
Department of Mathematics, College of Science and Arts, Najran University, Najran 66445, Saudi Arabia
2.
Mathematics Department, Faculty of Science, Minia University, Minia, Egypt
3.
Faculty of Computer Science, Nahda University, Beni Suef, Egypt

Received: 23 September 2025 Revised: 29 October 2025 Accepted: 03 November 2025 Published: 28 November 2025
MSC : 26A33, 65D12, 91G20

This study explored efficient solution strategies for the fractional generalized Hirota-Satsuma (GHS) coupled KdV system, which serves as a fundamental model for describing nonlinear wave interactions across various branches of physics and engineering, including nonlinear dispersive-wave phenomena in plasma and optical fibers. In particular, attention was focused on the GHS-KdV system governed by the Atangana-Baleanu Caputo fractional derivative (AB-CFD), which effectively captures the distinctive memory and hereditary features of complex materials and wave propagation phenomena. We presented two advanced Laplace-based methods: the Laplace Adomian decomposition method (LADM) and Laplace variational iteration method (LVIM) that effectively capture the system's nonlinear dynamics and memory effects without requiring linearization or perturbation. Numerical validation confirmed the methods' accuracy through excellent agreement with exact solutions, demonstrating systematic convergence as additional terms were included and complete recovery of classical solutions when fractional orders approached integer values. The mathematical consistency of both approaches is verified through comprehensive error analysis. These results establish LADM and LVIM as efficient computational tools for nonlinear fractional PDEs, particularly valuable for modeling wave propagation in complex media and other systems exhibiting hereditary properties. The methods offer significant advantages in handling the nonlocal characteristics inherent to fractional systems while maintaining computational efficiency.
Citation: Mofida Zaki, M. Abdelgaber, Hoda F. Ahmed, W. A. Hashem. Efficient Laplace-based decomposition and iteration techniques for the fractional GHS coupled KdV system with Atangana-Baleanu derivative[J]. AIMS Mathematics, 2025, 10(11): 28034-28058. doi: 10.3934/math.20251232

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Abstract

This study explored efficient solution strategies for the fractional generalized Hirota-Satsuma (GHS) coupled KdV system, which serves as a fundamental model for describing nonlinear wave interactions across various branches of physics and engineering, including nonlinear dispersive-wave phenomena in plasma and optical fibers. In particular, attention was focused on the GHS-KdV system governed by the Atangana-Baleanu Caputo fractional derivative (AB-CFD), which effectively captures the distinctive memory and hereditary features of complex materials and wave propagation phenomena. We presented two advanced Laplace-based methods: the Laplace Adomian decomposition method (LADM) and Laplace variational iteration method (LVIM) that effectively capture the system's nonlinear dynamics and memory effects without requiring linearization or perturbation. Numerical validation confirmed the methods' accuracy through excellent agreement with exact solutions, demonstrating systematic convergence as additional terms were included and complete recovery of classical solutions when fractional orders approached integer values. The mathematical consistency of both approaches is verified through comprehensive error analysis. These results establish LADM and LVIM as efficient computational tools for nonlinear fractional PDEs, particularly valuable for modeling wave propagation in complex media and other systems exhibiting hereditary properties. The methods offer significant advantages in handling the nonlocal characteristics inherent to fractional systems while maintaining computational efficiency.

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