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This paper aims to investigate the existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions. With regard to this nonlinear boundary value problem, three popular fixed point theorems, namely, Krasnoselskii’s theorem, Leray–Schauder’s theorem and Banach contraction principle, are employed to theoretically prove and guarantee three novel theorems. The main outcomes of this work are verified and confirmed via several numerical examples.

In order to accomplish our purpose, three fixed point theorems are applied to the problem under consideration according to some conditions that have been established to this end. These theorems are Krasnoselskii's theorem, Leray Schauder's theorem and Banach contraction principle.

In accordance to the applied fixed point theorems on our main problem, three corresponding theoretical results are stated, proved, and then verified via several numerical examples.

The existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions are studied. To the best of the authors’ knowledge, this work is original and has not been published elsewhere.

The purpose of this paper is to determine confidence intervals for the stochastic solutions in RLCG cells with a potential source influenced by coloured noise.

The deterministic model of the basic RLCG cell leads to an ordinary differential equation. In this paper, a stochastic model is formulated and the corresponding stochastic differential equation is analysed using the Itô stochastic calculus.

Equations for the first and the second moment of the stochastic solution of the coloured noise-affected RLCG cell are obtained, and the corresponding confidence intervals are determined. The moment equations lead to ordinary differential equations, which are solved numerically by an implicit Euler scheme, which turns out to be very effective. For comparison, the confidence intervals are computed statistically by an implementation of the Euler scheme using stochastic differential equations.

The theoretical results are illustrated by examples. Numerical simulations in the examples are carried out using Matlab. A possible generalization for transmission line models is indicated.

The Itô-type stochastic differential equation describing the coloured noise RLCG cell is formulated, and equations for the respective moments are derived. Owing to this original approach, the confidence intervals can be found more effectively by solving a system of ordinary differential equations rather than by using statistical methods.

The problem of ethnic conflicts and unrest is at the forefront in a diverse world today. This paper aims to identify ways of resolving social conflicts and establishing a balanced way of taking into account the diverse interests of a multinational society.

Modeling of dynamical ethnic processes in the Republic of Kazakhstan was based on the Parson's sociological scheme and the Nash equilibrium theory. The model consists of differential equations describing the development of four ethnic subsystems: political system, economic system, societal community and Institute of the Assembly of the People of Kazakhstan. This model allows investigating how the interests of various ethnic groups change over time and identifying the states of equilibrium in which the interests of all groups are satisfied.

The results of computer simulation showed that one of the solutions to the problem of social stability is establishing social equilibrium. For this, the Institute of the Assembly of the People of Kazakhstan (APK Institute) must take changes that occur within ethnic groups into account. The proposed model can reveal states of equilibrium with respect to positive and negative dynamic processes that exist between different ethnic groups.

The proposed model can be used to predict changes in social behavior and find balance between ethnic subsystems in the research on ethnic processes in multinational countries to early detect conflicts of interest and crisis situations. Future studies will benefit from expanding the range of subsystems that can affect alterations in the ethnic community.

Recently, Stein et al. (2016) studied theoretical properties and parameter estimation of continuous time processes derived as solutions of a generalized Langevin equation (GLE). In this paper, the authors extend the model to a wider class of memory kernels and then propose a bond and bond option valuation model based on the extension of the generalized Langevin process of Stein et al. (2016).

Bond and bond option pricing based on the proposed interest rate models presents new difficulties as the standard partial differential equation method of stochastic calculus for bond pricing cannot be used directly. The authors obtain bond and bond option prices by finding the closed form expression of the conditional characteristic function of the integrated short rate process driven by a general Lévy noise.

The authors obtain zero-coupon default-free bond and bond option prices for short rate models driven by a variety of Lévy processes, which include Vasicek model and the short rate model obtained by solving a second-order Langevin stochastic differential equation (SDE) as special cases.

Bond and bond option pricing plays an important role in capital markets and risk management. In this paper, the authors derive closed form expressions for bond and bond option prices for a wider class of interest rate models including second-order SDE models. Closed form expressions may be especially instrumental in facilitating parameter estimation in these models.

The aim of the present paper is to investigate the behavior of collective motion of living biological organisms in the two-dimensional (2D) plane by adopting a new approach based on the use of Langevin dynamics. Langevin dynamics is a powerful tool to study these systems because they present a stochastic process due to collisions between their constituents.

In this paper, the dynamical properties and scaling behavior of self-propelled particles were studied numerically by using Langevin dynamics. These dynamics have been affected by the use of only the alignment zone of radius R.

The results indicated that the system’s velocity increases with time and reaches to finite value at the equilibrium phase.

This result is more consistent with that of Vicsek’s model. However, the system’s velocity decreases exponentially with the applied noise without taking the zero value for the highest noise value.

As well as, the crossover time of the growth kinetic system decreases exponentially with noise.

Scaling behavior has been checked for this system and the corresponding results prove that behavior scales with the same law of the one in Vicsek’s model but with different scaling exponents.

The phase transition observed in Vicsek’s model cannot be reproduced by the Langevin dynamics model, which describes more about the dynamical properties of self-propelled systems.

In this study, the authors introduce a solvability of special type of Langevin differential equations (LDEs) in virtue of geometric function theory. The analytic solutions of the LDEs are considered by utilizing the Caratheodory functions joining the subordination concept. A class of Caratheodory functions involving special functions gives the upper bound solution.

The methodology is based on the geometric function theory.

The authors present a new analytic function for a class of complex LDEs.

The authors introduced a new class of complex differential equation, presented a new technique to indicate the analytic solution and used some special functions.

The paper aims to illustrate the two kinds of analysis approach for which finite element method (FEM) can be successfully employed: the Poisson-Nernst-Planck (PNP) model and the Langevin-Lorentz-Poisson (LLP) one.

The approach of this work is to try making a survey of the use of the FEM in the modelling of charge transport/ion flow across membrane channels, in particular for the PNP analysis and for a particle based model such as LLP model.

In this paper, the two kinds of analysis approach for which FEM can be successfully employed, the PNP model and the LLP one, have been shown. In both cases the FEM is extremely useful to carry out these analysis and the simulation results obtained are in good agreement with experimental results.

The value of this paper is to demonstrate the FEM is extremely useful to carry out analysis and results which are in good agreement with experimental ones.

The increase of the accuracy of a mathematical model of hysteresis by the choice of the optimum saturation curve for a given material.

Hysteresis loops of typical soft magnetic materials are approximated with the help of the Taka´cs magnetization model using different saturation curves. The quality of approximations is determined by the deviation of computed magnetic induction amplitudes, iron losses, apparent remanences and coercivities from the measured values.

By the proper choice of saturation curve, the relative inaccuracy of approximations can be reduced with reference to the original model based on tangent hyperbolic function.

The accuracy of approximations worsens close to saturation because of the excessive rise of magnetization due to the linear term of the model. This effect should be minimized by the application of complex saturation curves using greater number of parameters.

Owing to the convenient analytical form and increased accuracy, the model equations can be used in simpler practical evaluations of hysteresis effects and for teaching purposes.

Presented form of model equations enables approximation of hysteresis loops and the evaluation of main characteristics of magnetic materials on the basis of any saturation curve.

The conventional treatment of thermal noise is based on Nyquist’s theorem. This theorem has only been derived for linear, reciprocal (we define “reciprocal networks” as networks that are built of reciprocal network elements) networks. In this paper a description of thermal noise in reciprocal non‐linear RLC networks is presented. This description is derived from first principles, i.e. from a direct application of non‐equilibrium thermodynamics (irreversible thermodynamics) to electrical networks. As an example, the class of “complete” non‐linear networks is considered. Using the idea of equivalent n‐ports, the theory’s extension to certain classes of transistor circuits should be possible.

A transport equation for the one‐point velocity probability density function (pdf) of turbulence is derived, modelled and solved. The new pdf equation is obtained by two modeling steps. In the first step, a dynamic equation for the fluid elements is proposed in terms of the fluctuating part of Navier‐Stokes equation. A transition probability density function (tpdf) is extracted from the modelled dynamic equation. Then the pdf equation of Fokker‐Planck type is obtained from the tpdf. In the second step, the Fokker‐Planck type pdf equation is modified by Lundgren’s formal pdf equation to ensure it can properly describe the turbulence intrinsic mechanism. With the new pdf equation, the turbulent plane Couette flow is solved by the direct finite difference method coupled with dimensionality reduction and QUICKER scheme. A simple boundary treatment is proposed such that the near‐wall solution is tractable and then no refined grid is required. The calculated mean velocity, friction coefficient, and turbulence structure are in good agreement with available experimental data. In the region departed from the center of flow field, the contours of isojoint pdf of V₁ and V₂ is very similar to that of experimental result of channel flow. These agreements show the validity of the new pdf model and the availability of the boundary treatment and QUICKER scheme for solving the turbulent plane Couette flow.