Using a novel norm that is comfy for Fractional and singular differential equations the existence and uniqueness of IVP for new type nonlinear Langevin equations involving three Fractional orders are discussed. This norm is a tool to measure how far a numerical solution is from the exact one. New results are based on the contraction mapping principle. Lemma 2.2 has a prominent role in proving the main theorem. The Fractional derivatives are described in Caputo sense. Two examples are presented to illustrate the theory.

A combined dynamical and statistical model for fission was employed in our calculation. There is no doubt that a Langevin description plus a Monte Carlo treatment of the evaporation processes provide the most adequate dynamical description. In this paper, we would consider a strongly shaped dependent friction force and we use the numerical method rather than the analytical one. The objective of this article is to calculate the time dependent fission widths of the224Th nucleus. The fission widths were calculated with both chaos-weighted wall friction (CWWF) and wall friction (WF) dissipations. The calculations are repeated for 100000 trajectories. The result was compared to the others' work. We use nuclear elongation coordinate with time and it is necessary to repeat the small steps many times to improve the accuracy.

The Jump-Diffusion equation is a generalization of the Langevin equation; it has been usually applied to reconstruct discontinuous stochastic processes. In this article, by using this equation, we investigate the electrocardiogram of the electric activity of the heart beat, for three groups of subjects with normal, atrial fibrillation and ventricular arrhythmia. At first, we demonstrate that the time series of electrocardiogram is a discontinuous process that can be modeled by the jump-diffusion equation. Then, by calculating the Kramers-Moyal coefficients related to this equation, we show that there is a significant difference between the heart dynamics of the normal subjects and the ones with heart failure exists. Finally, we introduce a measure that may be used for the diagnosis of heart failures.

This paper presents an investigation of the existence and the unique feature of solutions for nonlinear Langevin equations involving two Fractional orders with anti-periodic boundary conditions (APBCs). As a result of employing some fixed point theorems like  Schauder and contraction mapping principles, the existence and uniqueness of solutions are examined. On top of this,   the stability within the scope of Ulam–Hyers of solutions to this problem is also considered. The distinctive features of the present study are its similar variant and the existence of derivatives of Caputo and Riemann in the problem structure. Finally, to illustrate the result of the study, an example is presented.

In this research article, we study the existence, uniqueness and Ulam-Hyers stability of solutions in connection to the generalized Caputo Langevin Fractional differential equations in Banach Space. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the technique of measure of noncompactness combined with the Monch fixed point theorem, the implementation Banach contraction principle fixed point theorem.  Moreover,  the Ulam--Hyers stability is discussed by utilizing the Urs's. Lastly, we deliver an example to check the efficiency and accuracy of the proposed methods.

In this paper, we , nd an integral representation for the fundamental solution of the Fractional Ostrovsky equation in terms of the Airy and Bessel-Wright functions. The equation is studied in the sense of the Weyl Fractional derivative and the solu-tion is presented as the Airy transforms of Wright functions. Using the asymptotic expansion of Wright function the asymptotic behavior of solution is also discussed.

In this article, we study a new nonlinear Langevin equation of two Fractional orders with Fractional boundary value conditions which is a generalization of previous Langevin equations. Based on Banach and Schauder fixed point theorems, the existence and uniqueness of solutions of this equation are investigated. Moreover, our hypotheses are simpler than similar works.

In this research paper, we present an exact matrix form analytical solution of the multi-dimensional generalized Langevin equation with quadratic potentials. Our investigation provides detailed expressions for the two-dimensional probability distribution and extends the understanding of the dynamics governed by harmonic potentials. By utilizing the inverse Laplace transformation, we offer a precise method to solve these equations, corroborated by specific examples. This study contributes to the fundamental understanding of stochastic processes in multi-dimensional systems with harmonic potentials and clarifies the limitations of our approach. While the findings are specific to quadratic potentials, they provide a robust framework for exploring related phenomena within this context.