Novel approximate solutions of the fractional form of graphene sheet thermophoretic motion model.

In this paper, we investigate the dynamics of travelling waves and wrinkle formation in graphene sheets using the thermophoretic motion equation and generalized Mittag-Leffler function (GMLF) with Caputo sense. The practical importance of understanding wrinkles in graphene to crucial for understanding the wave propagation in nanomaterials, as well as nanomechanical resonators and energy storage devices, and thermal management systems, where wrinkle dynamics can affect heat transfer. The limitations of prior models are that integer models may not fully capture the complex, memory-dependent, viscoelastic, and anomalous diffusion behaviors in graphene sheets. This limitation leads us to present this work, where a fractional-order model provides a more effective framework for accurately describing the intricate dynamics of wrinkle formation and diffusion. To bridge this gap, we introduce a new fractional form of the wrinkle graphene sheet model and transform the thermophoretic motion equation into a fractional form using Caputo fractional derivative (CFD) and GMLF. We obtain novel wrinkle-like soliton solutions for the proposed model. Our outcome is in good agreement with the exact solution of the original equation when the fractional order is equal to 1. We compare the exact and approximate solutions and present the results through 2D and 3D figures. We discuss the impact of system parameters on the dynamics of the obtained solutions. The findings reveal that the fractional order and model parameters significantly impact the solution dynamics. This implies that the fractional order and system parameters can serve as controllers for the dynamics of graphene wrinkles. The outcomes reveal the advantages and efficacy of the MGMLFM, which include that it does not require any transformation, perturbation, or linearization, unlike other techniques in the published papers. It is easily computable components, implemented directly on the problems, and produces approximate solutions of high accuracy with a small absolute error.