On the Cauchy problem for the Langevin-type fractional equation

TL;DR

This paper investigates the well-posedness of a fractional Langevin-type equation with Caputo derivatives, establishing existence, uniqueness, and explicit solution representations under specific conditions.

Contribution

It introduces a novel analysis of the Cauchy problem for a fractional Langevin-type equation with explicit solution formulas.

Findings

Existence and uniqueness of solutions are proven.

Explicit eigenfunction expansion representations are derived.

Conditions for the well-posedness of the fractional equation are identified.

Abstract

In this article, the Cauchy problem for the Langevin-type time-fractional equation $D_t^\beta(D_t^\alpha u(t))+D_t^\beta(Au(t))=f(t),(0<t\leq T)$ 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.