Invariant analysis of the time-fractional (2+1)-dimensional dissipative long-wave system

In this paper, we delve into the study of Lie symmetries and conservation laws for fractional partial differential equations featuring mixed derivatives of fractional and integer orders. Firstly, we explore the Lie symmetries permitted by the (2+1)-dimensional time-fractional dissipative long-wave system. Subsequently, leveraging these symmetries, we reduce the original (2+1)-dimensional partial differential equations to (1+1)-dimensional counterparts. We then proceed to develop explicit power series solutions along with several exact solutions. Lastly, employing the concept of nonlinear self-adjointness in the fractional context, we establish the conservation properties of the system, resulting in the derivation of novel non-trivial conservation law.