A class of Langevin time-delay differential equations with general fractional orders and their applications to vibration theory

Abstract Langevin differential equations with fractional orders play a significant role due to their applications in vibration theory, viscoelasticity and electrical circuits. In this paper, we mainly study the explicit analytical representation of solutions to a class of Langevin time-delay differential equations with general fractional orders, for both homogeneous and inhomogeneous cases. First, we propose a new representation of the solution via a recently defined delayed Mittag-Leffler type function with double infinite series to homogeneous Langevin differential equation with a constant delay using the Laplace transform technique. Second, we obtain exact formulas of the solutions of the inhomogeneous Langevin type delay differential equation via the fractional analogue of the variation constants formula and apply them to vibration theory. Moreover, we prove the existence and uniqueness problem of solutions of nonlinear fractional Langevin equations with constant delay using Banach’s fixed point theorem in terms of a weighted norm with respect to exponential functions. Furthermore, the concept of stability analysis in the mean of solutions to Langevin time-delay differential equations based on the fixed point approach is proposed. Finally, an example is given to demonstrate the effectiveness of the proposed results.