Fuzzy Langevin fractional delay differential equations under granular derivative

Analytical studies of the class of the fuzzy Langevin fractional delay differential equations (FLFDDEs) are frequently complex and challenging. It is necessary to construct an effective technique for the solution of FLFDDEs. This article presents an explicit analytical representation of the solution to the class of FLFDDEs with the general fractional orders under granular differentiability. The closed-form solution to the FLFDDEs is extracted for both the homogeneous and non-homogeneous cases using the Laplace transform technique and presented in terms of the delayed Mittag-Leffler type function with double infinite series. Moreover, the existence and uniqueness of the solutions of the FLFDDEs are investigated using the generalized contraction principle. An illustrative example is provided to support the proposed technique. To add to the originality of the presented work, the FLFDDEs with constant delay are solved by applying vibration theory and visualizing their graphs to support the theoretical results.