Fractional Langevin equations play an important role in describing a wide range of physical processes. For instance, they have been used to describe single-file predominance and the behavior of unshackled particles propelled by internal sounds. This article investigates fractional Langevin equations incorporating recent extensive fractional operators of different orders. Nonperiodic and nonlocal integral boundary conditions are assumed for the model. The Hyres-Ulam stability, existence, and uniqueness of the solution are defined and analyzed for the suggested equations. Also, we utilize Banach contraction principle and Krasnoselskii fixed point theorem to accomplish our results. Moreover, it will be apparent that the findings of this study include various previously obtained results as exceptional cases.

This research explored stochastic soliton and periodic wave (SPW) solutions for the geophysical Korteweg-de Vries (KdV) equation with variable coefficients, incorporating the effects of Earth's rotation, fluid stratification, and topographical variations. The classical KdV equation, widely used to model nonlinear wave propagation, was extended to describe geophysical wave dynamics in atmospheric and oceanic systems. Exact solutions for both the deterministic and Wick-type stochastic (W-TS) forms of the geophysical KdV equation were obtained using white noise (WN) theory, the Hermite transform (HT), and the exp-function method. By employing the HT, the stochastic equation was transformed into a deterministic counterpart, facilitating the derivation of novel SPW solutions expressed as rational functions involving exponential terms. The inverse HT is then applied to retrieve stochastic SPW solutions under Gaussian WN conditions. Numerical analysis highlights the influence of Brownian motion (B-M) on the formation and behavior of SPWs in geophysical settings. Additionally, numerical simulations illustrate how random fluctuations affect wave stability and evolution, offering deeper insights into nonlinear wave interactions in oceanic and atmospheric environments.

This research paper investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution lies in the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which is pivotal for deriving the main results. The paper systematically proved weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results not only contribute to the understanding of Milne-type inequalities but also offer insights that pave the way for potential future research in the considered topics. Furthermore, it is evident that the results obtained encompass numerous findings that were previously presented in various studies as special cases.