Symmetry Reductions and Exact Solutions for Caputo-Type Fractional Differential Equations and Nonlinear Fractional Wave-Diffusion Systems

Abstract

This study explores a class of (2+1)-dimensional nonlinear fractional wave-diffusion equations of the Caputo type by examining their associated Lie symmetry groups. We first derive an explicit formula for the prolonged group action on the fractional order Caputo-type derivatives with a general $\alpha$-th order. This work builds upon previous research by Gazizov et al. \cite{Gazizov} involving the Riemann-Liouville  fractional derivative. Utilizing the optimal system method, we determine the symmetries of the studied class of nonlinear fractional wave-diffusion equations and present corresponding reductions. Furthermore, we obtain a set of exact invariant and solitary solutions for the fractional Burger's equation. Our findings highlight the effectiveness of Lie symmetry methods in analyzing nonlinear fractional wave-diffusion equations, providing valuable insights into the mathematical structure underlying these equations. The invariances discovered may find potential applications across various disciplines, including physics and engineering.