Multi-order fractional differential equations have been studied due to their applications in modeling, and solved using various mathematical methods. We present explicit analytical solutions for several families of Langevin differential equations with general fractional orders, both homogeneous and inhomogeneous cases. The results can be written, in general and special cases, by means of a recently defined bivariate Mittag-Leffler function and the associated operators of fractional calculus. The novelty of this work is to apply an appropriate norm on the proof of existence and uniqueness theorem, and discuss the application of Langevin differential equation with fractional orders in several interesting cases to the electrical circuit. Moreover, we investigate Ulam–Hyers stability of Caputo type fractional Langevin differential equation. At the end, we provide an example to verify our main results.

由于它们在建模中的应用，已经研究了多阶分数阶微分方程，并使用各种数学方法对其进行了求解。我们提供了几个具有一般分数阶的Langevin微分方程族的显式解析解，包括齐次和非齐次情况。在一般情况和特殊情况下，都可以通过最近定义的二元Mittag-Leffler函数和相关的分数演算符来写结果。这项工作的新颖之处在于在存在性和唯一性定理的证明中应用适当的范数，并讨论在几种有趣的情况下分数阶Langevin微分方程在电路中的应用。此外，我们研究了Caputo型分数阶Langevin微分方程的Ulam-Hyers稳定性。在最后，