Langevin differential equations with general fractional orders and their applications to electric circuit theory

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. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

This study presents explicit analytical solutions for families of Langevin differential equations with general fractional orders, incorporating homogeneous and inhomogeneous cases. Novelty lies in applying appropriate norms in the proof of existence and uniqueness theorem, and discussing the application of fractional order Langevin differential equations in electrical circuits. Furthermore, Ulam-Hyers stability of Caputo type fractional Langevin differential equations is investigated, with an example provided to validate the main results.