Lindblad equation solution

The analogy is based on writing the Lindblad equation in a Liouville form $dot{rho} = cal{L} rho$ with the Liouville superoperator $cal{L}$ being the generator of the (semigroup) evolution for the density matrix $rho$. The solution is then formally $rho(t) = e^{cal{L} t} rho(0)$.

We always solve the Lindblad form linear differential equations by numerical methods, such as fourth order Runge-Kutta method. If you want the steady state solution in analytic method, you can read this paper: PHYSICAL REVIEW A 92, 022116 (2015). I don't know if I solved your question.