Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method

Abstract

In this study, we introduce an efficient analysis of a new equation, termed the time-fractional qq-deformed tanh-Gordon equation (TGE), which is the fractional form of the qq-deformed TGE that was recently introduced by Ali and Alharbi. This equation represents a significant advancement in the field of mathematical physics, which is due to its applications in many fields including superconductivity and fiber optics. It has many applications in condensed matter physics and in modeling physical systems that exhibit violated symmetries. We investigate the qq-deformed TGE in fractional form using Caputo fractional derivative to capture non-local and memory effects, which means they can take into account the entire history of a function rather than just its current value. Notably, this equation has not been previously solved in fractional form, making our approach pioneering in its analysis. We solve this equation utilizing the modified double Laplace transform method, which is considered a semi-analytical technique that combines the double Laplace transform with Adomian polynomials to enable us to extract nonlinear terms. This method renowned for its efficacy in handling fractional differential equations; this is evident from the results obtained in the tables by comparing the analytical solution with the approximate solution we obtained, as well as by calculating the absolute error between them. We examine the existence and the uniqueness of the solution utilizing Schaefer’s fixed-point theorem. Different graphs in 2D and 3D are presented to clarify the effect of different parameters on the behavior of the solution, specially the fractional operator and the deformation parameter qq.

Keywords

• fractional calculus
• q-deformed tanh-gordon equation
• double laplace transform method
• adomian polynomials
• schaefer’s fixed-point theorem
• existence and uniqueness analysis