On the Cauchy problem for the Langevin-type fractional equation

Abstract

In this article, the Cauchy problem for the Langevin-type time-fractional equation $D_t^{\beta }(D_t^{\alpha }u(t))+D_t^{\beta }(Au(t))=f(t),(0<t\le T)$ is studied. Here $\alpha ,\beta \in (0,1)$, $D_t^{\alpha },D_t^{\beta }$ is the Caputo derivative and $A$ is an unbounded self-adjoint operator in a separable Hilbert space. Under certain conditions, we establish the existence and uniqueness of the solution and provide an explicit representation of it using eigenfunction expansions.

10 pages