Forward and Inverse Problems for a Langevin-Type Fractional Equation Involving Non-Local Time Condition

TL;DR

This paper investigates direct and inverse problems for a Langevin-type fractional differential equation with a non-local time condition, establishing existence, uniqueness, and stability of solutions.

Contribution

It introduces new criteria for uniqueness and analyzes the well-posedness of fractional Langevin equations with non-local conditions.

Findings

Existence and uniqueness of solutions are proven.

Stability inequalities are derived.

Criteria for solution uniqueness are provided.

Abstract

The paper addresses the formulation and analysis of direct and inverse problems for a Langevin-type fractional differential equation under a non-local condition imposed on the time variable. An additional condition for solving the inverse problem has the form $u(t_0)=\omega$. The existence and uniqueness of a solution to the problems are established, and stability inequalities are derived. Criteria ensuring the uniqueness of the solution are identified.