I have studied the fractional Langevin equation ${^C} D^{\beta}_t( {^C} D^{\alpha}_t + \lambda)u=f(t, u)$ $\alpha. \beta \in (0, 1]$ subject to the conditions $u(0)=a$, ${^C} D^{\alpha}_t u(0) = b$, where ${^C} D^{\alpha}_t, {^C} D^{\beta}_t$ are Caputo fractional derivatives. However, I cannot find any papers that mentioned the physical meaning of the condition ${^C} D^{\alpha}_t u(0) = b$.

In your opinion, what is the physical meaning of the initial condition ${^C} D^{\alpha}_t u(0) = b$?

Dear Prof. Jocelyn Sabatier I would like to receive your opinion on the above question. Thank you so much.