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.

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.

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.

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.

A model is presented to study the random growth of the number of tumor cells. It contains deterministic growth and therapy terms, as well as a random term. The model is formulated as a Langevin equation and its corresponding Fokker–, Planck equation is studied. Three forms for the time-dependence of the therapy are used and the results are compared to each other. Specifically, the ratio of the probability that the number of tumor cells be large to the probability that the number of tumor cells be small is investigated. The large time behavior of this ratio is considered as a figure of merit. Better therapies correspond to smaller values for this figure of merit. The behavior of this figure of merit in terms of various parameters of the therapy is investigated. It is seen that decreasing the amplitude or the period, decreases this figure of merit, hence improves the therapy.

Using Langevin dynamics and the dissipative nature of the fission process, we have studied dynamical variations of nucleus from the formation of the compound nucleus to separation stage of two fission fragments. During this dissipative process, particles such as neutron, proton, alpha particle and gamma ray emit from the compound system. In the present work, the number of emitted particles using Langevin equation are calculated, dynamically. Obtained results for emitted alpha particles, protons and neutrons are compared with the experimental data. Also, we have studied the influence of the dissipative coefficient on these quantities. These results show that the dissipative coefficient affects the results, as well as the good agreement between the experimental data and theoretical results can be reproduced by using the dynamical approach.

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.

Physical systems are mostly modeled by certain differential equations. However, in most cases random effects are omitted and therefore, the solutions are not in agreement with the experimental results. In the present work, we have investigated the effect of white noise perturbation on two physical models. At first, we have reviewed the random variables and processes and then, we have solved the Langevin equation, which is a general form of a random equation with a random white noise perturbation. We have also proposed a stochastic model for both the electric current and radiation of radioactive materials. This is done by considering the white noise perturbation sentence in the corresponding ordinary differential equations. By solving these random equations, we have obtained the mean value function, the variance function, and the random process function, as well. Finally, the results have been simulated and the corresponding diagrams presented. In order to simulate the desired random movement, the Monte Carlo method in Microsoft Excel environment was used.