An Improved Bingham Model and the Parameter Identification of Coal (Rock) Containing Water Based on the Fractional Calculus Theory

The rheological properties of coal (rock) containing water cannot be characterized by the traditional Bingham model. This problem was addressed in this study through theoretical analysis and experimental research. Based on fractional calculus theory, a fractional calculus soft element was introduced into the traditional Bingham model. An improved Bingham model creep equation and a relaxation equation were obtained through theoretical derivations. Triaxial creep experiments of coal (rock) with different moisture contents were conducted. The parameters of the improved Bingham model were obtained by the least-squares method. Conclusions are as follows: (1) in the improved Bingham model, the stage of nonlinear accelerated creep could be characterized by the creep curves of the soft element; (2) with the increasing moisture content of the coal (rock), the transient strain and the slope of the steady creep stage increased and the total creep time showed a decreasing trend; and (3) the parameters of the creep model were obtained by nonlinear fitting of experimental data, and the fitted curve could better describe the whole creep process. The rationality of the improved creep model was verified. It can provide a theoretical basis for the study and engineering analysis of coal (rock).

Dynamical Effective Field Model for Interacting Ferrofluids: I. Derivations for homogeneous, inhomogeneous, and polydisperse cases

Quite recently I have proposed a nonperturbative dynamical effective field model (DEFM) to quantitatively describe the dynamics of interacting ferrofluids. Its predictions compare very well with the results from Brownian dynamics simulations. In this paper I put the DEFM on firm theoretical ground by deriving it within the framework of dynamical density functional theory (DDFT), taking into account nonadiabatic effects. The DEFM is generalized to inhomogeneous finite-size samples for which the macroscopic and mesoscopic scale separation is nontrivial due to the presence of long-range dipole-dipole interactions. The demagnetizing field naturally emerges from microscopic considerations and is consistently accounted for. The resulting mesoscopic dynamics only involves macroscopically local quantities such as local magnetization and Maxwell field. Nevertheless, the local demagnetizing field essentially couples to magnetization at distant macroscopic locations. Thus, a two-scale parallel algorithm, involving information transfer between different macroscopic locations, can be applied to fully solve the dynamics in an inhomogeneous sample. I also derive the DEFM for polydisperse ferrofluids, in which different species can be strongly coupled to each other dynamically. I discuss the underlying assumptions in obtaining a thermodynamically consistent polydisperse magnetization relaxation equation, which is of the same generic form as that for monodisperse ferrofluids. The theoretical advances presented in this paper are important for both qualitative understanding and quantitative modeling of the dynamics of ferrofluids and other dipolar systems.

Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory

A new class of fractional differential equations with exponential memory was recently defined in the space A C δ n [ a , b ] $A{C}_{\delta }^{n}\left[a,b\right]$ . In order to use the famous predictor–corrector method, a new quasi-linear interpolation with a non-equidistant partition is suggested in this study. New Euler and Adams–Moulton methods are proposed for the fractional integral equation. Error estimates of the generalized fractional integral and numerical solutions are provided. The predictor–corrector method for the new fractional differential equation is developed and numerical solutions of fractional nonlinear relaxation equation are given. It can be concluded that the non-equidistant partition is needed for non-standard fractional differential equations.

Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative

This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ-Caputo fractional derivative. The existence of solution is carried out with the help of Schauder’s fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results.

The Limited Validity of the Conformable Euler Finite Difference Method and an Alternate Definition of the Conformable Fractional Derivative to Justify Modification of the Method

A method recently advanced as the conformable Euler method (CEM) for the finite difference discretization of fractional initial value problem Dtαyt = ft;yt, yt0 = y0, a≤t≤b, and used to describe hyperchaos in a financial market model, is shown to be valid only for α=1. The property of the conformable fractional derivative (CFD) used to show this limitation of the method is used, together with the integer definition of the derivative, to derive a modified conformable Euler method for the initial value problem considered. A method of constructing generalized derivatives from the solution of the non-integer relaxation equation is used to motivate an alternate definition of the CFD and justify alternative generalizations of the Euler method to the CFD. The conformable relaxation equation is used in numerical experiments to assess the performance of the CEM in comparison to that of the alternative methods.

Censored stable subordinators and fractional derivatives

Based on the popular Caputo fractional derivative of order β in (0, 1), we define the censored fractional derivative on the positive half-line ℝ+. This derivative proves to be the Feller generator of the censored (or resurrected) decreasing β-stable process in ℝ+. We provide a series representation for the inverse of this censored fractional derivative. We are then able to prove that this censored process hits the boundary in a finite time τ ∞, whose expectation is proportional to that of the first passage time of the β-stable subordinator. We also show that the censored relaxation equation is solved by the Laplace transform of τ ∞. This relaxation solution proves to be a completely monotone series, with algebraic decay one order faster than its Caputo counterpart, leading, surprisingly, to a new regime of fractional relaxation models. Lastly, we discuss how this work identifies a new sub-diffusion model.

Using regional scaling for temperature forecasts with the Stochastic Seasonal to Interannual Prediction System (StocSIPS).

Over time scales between 10 days and 10–20 years – the macroweather regime – atmospheric fields, including the temperature, respect statistical scale symmetries, such as power-law correlations, that imply the existence of a huge memory in the system that can be exploited for long-term forecasts. The Stochastic Seasonal to Interannual Prediction System (StocSIPS) is a stochastic model that exploits these symmetries to perform long-term forecasts. It models the temperature as the high-frequency limit of the (fractional) energy balance equation (fractional Gaussian noise) which governs radiative equilibrium processes when the relevant equilibrium relaxation processes are power law, rather than exponential. They are obtained when the order of the relaxation equation is fractional rather than integer and they are solved as past value problems rather than initial value problems. StocSIPS was first developed for monthly and seasonal forecast of globally averaged temperature. In this paper, we extend it to the prediction of the spatially resolved temperature field by treating each grid point as an independent time series. Compared to traditional global circulation models (GCMs), StocSIPS has the advantage of forcing predictions to converge to the real-world climate. It extracts the internal variability (weather noise) directly from past data and does not suffer from model drift. Here we apply StocSIPS to obtain monthly and seasonal predictions of the surface temperature and show some preliminary comparison with multi-model ensemble (MME) GCM results. For one month lead time, our simple stochastic model shows similar values of the skill scores than the much more complex deterministic models.

Correlations versus causality in Stochastic Long-range Forecasting as a Past Value Problem

<p>Over time scales between 10 days and 10-20 years – the macroweather regime – atmospheric fields, including the temperature, respect statistical scale symmetries, such as power-law correlations, that imply the existence of a huge memory in the system that can be exploited for long-term forecasts. The Stochastic Seasonal to Interannual Prediction System (StocSIPS) is a stochastic model that exploits these symmetries to perform long-term forecasts. It models the temperature as the high-frequency limit of the (fractional) energy balance equation (fractional Gaussian noise) which governs radiative equilibrium processes when the relevant equilibrium relaxation processes are power law, rather than exponential. They are obtained when the order of the relaxation equation is fractional rather than integer and they are solved as past value problems rather than initial value problems.</p><p>Long-range weather prediction is conventionally an initial value problem that uses the current state of the atmosphere to produce ensemble forecasts. In contrast, StocSIPS predictions for long-memory processes are “past value” problems that use historical data to provide conditional forecasts. Cross-correlations can be used to define teleconnection patterns, and for identifying possible dynamical interactions, but they do not necessarily imply any causation. Using the precise notion of Granger causality, we show that for long-range stochastic temperature forecasts, the cross-correlations are only relevant at the level of the innovations – not temperatures. Extended here to the multivariate case, (m-StocSIPS) produces realistic space-time temperature simulations. Although it has no Granger causality, we are able to reproduce emergent properties including realistic teleconnection networks and El Niño events and indices.</p>

Modeling the dynamic properties of iii-nitrides in strong electric fields

This paper proposes a method of modeling the dynamic properties of multi-valley semiconductors. The model is applied to the relevant materials GaN, AlN, and InN, which are now known by the general name of III-nitrides. The method is distinguished by economical use of computational resources without significant loss of accuracy and the possibility of application for both dynamic time-dependent tasks and the fields variable in space. The proposed approach is based on solving a system of differential equations, which are known as relaxation ones, and derived from the Boltzmann kinetic equation in the approximation of relaxation time by the function of distribution over k-space. Unlike the conventional system of equations for the concentration of carriers, their pulse and energy, we have used, instead of the energy relaxation equation, an equation of electronic temperature as a measure of the energy of the chaotic motion only. Relaxation times are defined not as integral values from the static characteristics of the material but the averaging of quantum-mechanic speeds for certain types of scattering is used. Averaging was carried out according to the Maxwellian distribution function in the approximation of electronic temperature, as a result of which various mechanisms of dispersion of carriers are taken into consideration through specific relaxation times. The system of equations includes equations in partial derivatives from time and coordinates, which makes it possible to investigate the pulse properties of the examined materials. In particular, the dynamic effect of the "overshoot" in drift velocity and a spatial "ballistic transport" of carriers. The use of Fourier transforms of pulse dependence of the drift carrier velocity to calculate maximum conductivity frequencies is considered. It has been shown that the limit frequencies are hundreds of gigahertz and, for aluminum nitride, exceed a thousand gigahertz

Fractional Stress Relaxation Model of Rock Freeze-Thaw Damage

Freeze-thaw cycle is a type of fatigue loading, and rock stress relaxation under freeze-thaw cycles takes into account the influence of the freeze-thaw cycle damage and deterioration. Rock stress relaxation under freeze-thaw cycles is one of the paramount issues in tunnel and slope stability research. To accurately describe the mechanical behaviour of stress relaxation of rocks under freeze-thaw, the software element is constructed based on the theory of fractional calculus to replace the ideal viscous element in the traditional element model. The freeze-thaw damage degradation of viscosity coefficient is considered. A new three-element model is established to better reflect the nonlinear stress relaxation behavior of rocks under freeze-thaw. The freeze-thaw and stress relaxation of rock are simulated by ABAQUS, the relevant model parameters are determined, and the stress relaxation equation of rock under freeze-thaw cycle is obtained based on numerical simulation results. The research shows that the test results are consistent with the calculated results, indicating that the constitutive equation can better describe the stress relaxation characteristics of rocks under freeze-thaw and provide theoretical basis for surrounding rock support in cold region.