Generalized fractional modeling of the Jaulent-Miodek system using the ϕ-Caputo operator with semi-analytical solutions

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In this paper, we consider the fractional-order Jaulent-Miodek (JM) model within the framework of ϕ-Caputo fractional derivative operator which generalizes classical model by taking into accounts nonlocal and memory-related effects. In this paper, we apply two efficient semi-analytical methods: the Residual Power Series (RPS) method and the New Iterative Method (NIM), to obtain approximate analytical solutions of fractional JM system. The effectiveness, convergence and accuracy of the two methods are studied and compared with the exact integer-order result. It is shown that the ϕ-Caputo fractional model can well describe the transition from diffusive character to dispersive characteristics. The amplitude, width and propagation of the soliton structures are all affected significantly by the fractional-order parameter τ. The comparisons between RPS, NIM and the exact solution demonstrate that they are in good agreement to each other with small absolute errors, which further verify the computational stability and accuracy of the proposed methods. The fractional model based on the new definition of ϕ-Caputo derivative is more generalized and complex, compared to earlier formulations with nonlinear dynamical systems. The findings reveal that the RPS and NIM are effective methods for solving nonlinear fractional partial differential equations (FPDEs), and provide new ideas in investigating soliton dynamics of nonlocality and memory dependent media.

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