Fractional q-Deformed Model: Analysis via Singular and Non-Singular Kernels

In this research, we study the solution of the time-fractional q-deformed Sinh-Gordon equation. This equation can be used to model physical systems with broken symmetries, incorporating the effects of amplification or dissipation. We apply the modified double Laplace transform method to solve this equation. We consider two different fractional derivative definitions, the Caputo and Caputo-Fabrizio definitions, to analyze their impact on the solution. The detailed analysis of the double Laplace transform method demonstrates its effectiveness in solving fractional differential equations. By comparing the results obtained using the Caputo and Caputo-Fabrizio definitions, we highlight the influence of the fractional derivative definition on the solutions. Our findings provide intriguing insights into the behavior of solutions to the q-deformed Sinh-Gordon equation when different fractional definitions are employed. Additionally, we examine the convergence, stability, existence, and uniqueness of the solution. Various 2D and 3D graphs are presented to illustrate the effect of different parameters on the solution behavior. Overall, this research offers a comprehensive investigation into the time-fractional q-deformed Sinh-Gordon equation, providing valuable insights into the role of fractional derivatives in modelling complex physical systems.