Abstract
This study investigates the analytical solutions of a nonlinear evolution model governing the dynamics of graphene sheets, a material renowned for its exceptional electronic properties and versatile applications in nanotechnology. Three advanced analytical approaches-the Khater II (Khat II) method, the Khater III (Khat III) method, and the Generalized Rational (GRat) approach-are employed to derive exact solutions for this model with high precision. The accuracy and reliability of these solutions are validated by comparing them to numerical results obtained via He's Variational Iteration (HVI) method, which serves as a benchmark for numerical verification. The analysis reveals a remarkable agreement between the analytical and numerical solutions, highlighting the robustness and effectiveness of the proposed methodologies. Furthermore, this study provides new insights into the nonlinear dynamics and physical properties of graphene sheets, while also identifying connections to other prominent nonlinear evolution equations. The innovative use of these analytical techniques offers practical frameworks for addressing complex nonlinear models in mathematical physics, thus advancing solution methodologies for such equations. This research contributes significantly to applied mathematics, material science, and nanotechnology by delivering accurate solutions and enhancing our understanding of graphene's nonlinear behavior. Finally, the findings have far-reaching implications, offering potential applications in designing advanced materials with tailored properties to support technological advancements, thereby pushing the boundaries of nanotechnology and materials engineering.