On the Cauchy problem for the Langevin-type fractional equation

TL;DR

This work analyzes the Cauchy problem for a Langevin-type fractional equation $D_t^beta(D_t^alpha u(t))+D_t^beta(Au(t))=f(t)$ in a separable Hilbert space, with $0<alpha,beta<1$ and an unbounded self-adjoint operator $A$. The authors employ an eigenfunction expansion along the eigenbasis of $A$, reducing the problem to scalar fractional equations and deriving an explicit solution in terms of Mittag-Leffler functions $E_{alpha,mu}$. They prove existence and uniqueness under milder regularity assumptions ($varphi in D(A)$, $psi in H$, $f(t) in C([0,T];D(A^varepsilon))$) and present a spectral representation $u(t)=\sum_k T_k(t) v_k$ with $T_k(t)$ given by a closed-form expression involving $E_{alpha,1}$, $E_{alpha,alpha+1}$, and a convolution with $E_{alpha,alpha+beta}$. Additional regularity results show $u\in AC([0,T];H)$ and continuity properties of $D_t^beta(Au(t))$ and $D_t^alpha u(t)$, establishing a rigorous well-posedness framework for this two-order fractional Langevin-type equation.

Abstract

In this article, the Cauchy problem for the Langevin-type time-fractional equation $D_t^β(D_t^αu(t))+D_t^β(Au(t))=f(t),(0<t\leq T)$ is studied. Here $α,β\in(0,1)$, $D_t^α, D_t^β$ 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.

Theorems & Definitions (10)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof