Nonlinear Diffusion Coefficient Research Articles

Exact higher-order moments for linear non-homogeneous stochastic differential equation

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

Fractal relaxation model with a nonlinear diffusion coefficient for fitting anomalous diffusion data in magnetic resonance imaging

In this paper a relaxation model of signal attenuation in diffusion-weighted magnetic resonance imaging (dMRI) with varying diffusion coefficient in terms of fractal derivative is proposed, in which the diffusion coefficient is a power law of the effective diffusion time. The relaxation model provides measures of diffusion constant, fractal dimension of diffusive trajectory of water molecule and the time power-law behavior of the diffusion coefficient. The proposed model was used to describe the magnetic resonance attenuation signal of the bullfrog sciatic nerve, and the corresponding spectral entropy was calculated to detect the environmental complexity in bullfrog sciatic nerve for water molecular diffusion. The results showed that the fractal derivative relaxation model (the VDC model) can accurately depict the diffusion pattern of water molecules in complex heterogeneous biological media at large b values. The VDC model provides an alternative theoretical reference for biological tissue detection based on time-dependent diffusion of water molecules.

Remaining useful life prediction of mechanical system based on improved adaptive fractional Lévy stable motion with statistical dependence measurement error

Prediction of remaining useful life (RUL) is a critical component of prognostics and health management (PHM) strategies for mechanical systems. Although some degradation models based on stochastic processes have played a pivotal role in the field of RUL prediction, the diffusion coefficients associated with these models are often fixed values that remain independent of the drift coefficient. Additionally, the measurement error is typically assumed to follow a Gaussian distribution that is independently and identically distributed with respect to the level of degradation. To address these constraints, an RUL prediction framework is proposed based on improved adaptive fractional Lévy stable motion (IAFLSM) with statistical dependence measurement error. The established IAFLSM model is capable of effectively characterizing the individual differences and time-varying uncertainties inherent in equipment degradation processes, with the nonlinear drift and diffusion coefficients exhibiting positive correlation in their variability. Moreover, the state space model is employed to realize the synchronous adaptive dynamic update of the drift coefficient and diffusion coefficient, thus accommodating the mechanical equipment degradation trajectory. Furthermore, a statistical dependence measurement error based on fractional Lévy stable motion is constructed, and the corresponding scale parameter incorporates a dynamic update mechanism with statistical correlation of degradation incremental behavior. The hidden variables related to performance degradation model are estimated through parameter estimation method and characteristic function. Expanding upon the proposed framework for RUL prediction, the quantification of the uncertainty intrinsic in the forecasting results is accomplished by employing the stability theorem of stable distribution and Monte Carlo technique. The RUL prediction framework is validated through the use of authentic truck rear axle full-life data and benchmark rolling bearing data. The comparative analysis results demonstrate the effectiveness and superiority of the proposed methodology.

Numerical Solution of a Subdiffusion Equation with Variable Order Time Fractional Derivative and Nonlinear Diffusion Coefficient

Numerical Solution of a Subdiffusion Equation with Variable Order Time Fractional Derivative and Nonlinear Diffusion Coefficient

Influence of a nonlinear degenerate diffusion on an advection-diffusion equation in a diffuse interface framework

This work is motivated by the modelling a liquid-vapour flows with phase transition describing the evolution of the coolant within an heat exchanger (e.g. the core of a Pressurized Water Reactor). We investigate an advection-diffusion equation with a degenerate and nonlinear diffusion coefficient. The degeneracy corresponds to a liquid-vapor mixture in the original model whereas the diffusion coefficient is non-degenerate in the pure phase cases. We focus on the influence of the diffusion coefficient on a simple 1D configuration for which some analytical computations can be done, leading to a surprising behavior of the phase transition with respect to the diffusion.

Propagation of neuronal signals in intact and injured peripheral nerves: The role of nonlinear diffusion of transmembrane potential

We present results of a computational study of the influence of nonlinear diffusion on propagation of electrical excitation generated in normal and injured peripheral nerves (PN) using a one-dimensional Fitzhugh-Nagumo (FN) model. Changes in action potential duration (APD) and repolarization intervals (RI) were observed based upon modification of the FN model. Namely, the model has been altered by adding an additional power function type nonlinear diffusion term which accounted for substantial intra-extracellular charge balance changes developing during action potential formation in small peripheral nerves. It was found that changes in magnitude of nonlinear diffusion coefficient result in oscillations of APD and RI in both normal and injured PNs with partially reduced conductivity. It was also found that these oscillations in injured PNs had markedly higher amplitudes dependent on the PN's length as well as on the width of its injured part. Results of our study may be useful in improving of monitoring of peripheral nerve growth and adjusting propagation of excitation to enhance impaired connectivity in injured peripheral nerves.

Scattering of energetic electrons through nonlinear cyclotron resonance with coherent whistler-mode hiss emissions

Recent observations have revealed that plasmaspheric hiss consists of many discrete waves called “hiss elements.” However, the interaction of energetic electrons (10 keV to several MeV) with the plasmaspheric hiss has only been simulated by the quasilinear (QL) diffusion theory, which does not take the fine wave structure into account. The QL theory cannot address nonlinear particle motions determined by the inhomogeneity factor, which influences the scattering of electrons in pitch angle and energy. This study aims to identify differences between the nonlinear wave–particle interaction and QL theory for plasmaspheric hiss emissions. We conduct test particle simulations to demonstrate the nonlinear interactions between hiss waves and electrons. The nonlinear theory is used to model hiss elements consisting of discrete frequencies and continuous phases. Unlike the other theories, the frequency and amplitude variations in time of the hiss packet are taken into account. Frequencies of the packets are determined to satisfy the separability criterion; when the criterion is met, resonance overlapping is absent, and the electrons can generate each wave element independently. The realistic simulation model of hiss waves reproduces the scattering of electrons by both first- and second-order resonances. We also evaluate the efficiency of electron scattering by calculating nonlinear diffusion coefficients. The diffusion coefficient of equatorial pitch angle is of the same order of magnitude as those calculated by the QL diffusion theory, while we find the effective acceleration of resonant electrons by successive nonlinear trapping, which is not evaluated by the QL theory.

An iterative scheme for solving a coupled Darcy–convection–diffusion model

We present an iterative scheme for the numerical analysis of a class of coupled Darcy-convection-diffusion problems modelling flow and heat transfer in porous media. The governing equations consist of the Darcy equations for the flow coupled to a convection-diffusion equation for heat transfer with nonlinear viscosity and diffusion coefficient depending on the temperature. Existence and uniqueness of the weak solution for the considered problem are first analyzed using a fixed-point method and then convergence study of two iterative schemes for the fixed-point algorithm is presented. Two numerical examples are selected to validate the theoretical estimates and to demonstrate the performance of the proposed algorithm. The obtained results support our theoretical expectations for a good numerical convergence with the developed estimates.

Frequency Transform Method for Transient Analysis of Nuclear Reactors in Monte Carlo

The Frequency Transform method is used for the first time to efficiently model a multiple-second transient problem with Monte Carlo (MC). This is achieved by coupling MC with a time-dependent coarse mesh finite difference (TD-CMFD) diffusion solver. TD-CMFD presents a large advantage over commonly used point kinetics equations since it preserves spatial resolution during the transient and provides equivalence with the high-order method through nonlinear diffusion coefficients. As TD-CMFD computes time-dependent and spatially dependent neutronics information, it also computes frequencies that describe the rate of change of neutron and delayed precursor concentrations. These frequencies are used in MC shape function calculations as an approximation for the time derivatives. As the simulation proceeds, MC calculations update the multigroup cross sections, currents, and diffusion coefficients that are needed in TD-CMFD, and in turn, TD-CMFD updates the frequencies. Our results show the success of the Frequency Transform method in prescribed transient problems on the C5G7 geometry and on a fuel pin geometry. The Frequency Transform method showed significant improvement compared to the Adiabatic approximation, which does not use any frequency information in the MC calculation. The improvements in spatial resolution are shown to be a direct result of frequencies.

Nonlinear evolution of Buneman instability in inhomogeneous plasmas using Adomian decomposition method

Nonlinear development of the Buneman instability in current-driven plasmas is investigated using Adomian decomposition method (ADM). Considering the continuity and momentum transfer equations for electrons and ions and using Poisson’s equation, a nonlinear diffusion equation with positive and negative nonlinear diffusion coefficients is obtained. Assuming the inhomogeneity of the collision frequency, the nonlinear equation is solved using ADM. Results show that by increasing the time and electron velocity, the amplitude of the electron density increases. Furthermore, it is shown that the collision frequency can decrease the amplitude of the electron density which leads to stabilizing the plasma system. Finally, it is found that ADM is a convergent method in the Buneman instability with real parameters.

A modified Kačanov iteration scheme with application to quasilinear diffusion models

The classical Kačanov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we introduce a modified Kačanov method, which allows for (adaptive) damping, and, thereby, to derive a new convergence analysis under more general assumptions and for a wider range of applications. For instance, in the specific context of quasilinear diffusion models, our new approach does no longer require a standard monotonicity condition on the nonlinear diffusion coefficient to hold. Moreover, we propose two different adaptive strategies for the practical selection of the damping parameters involved.

Error Analysis of an Unfitted HDG Method for a Class of Non-linear Elliptic Problems

We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter.

A small modification of interpolation method for diffusion coefficients at boundary interfaces

In this paper, a small modification of interpolation method for diffusion coefficients at boundary interfaces is presented. A one dimension steady source-free diffusion equation with variable diffusion coefficient is taken as an example to verify the effectiveness of the modification. The existing interpolation method used in literatures for computing diffusion coefficients at boundary interfaces is analyzed, the existing method seems reasonable in physics but not in mathematics, and it is accurate only if diffusion coefficients are constant or weakly nonlinear. The conclusion that diffusion coefficients at boundary interfaces should be specified by interior nodes near the boundary, is thus derived, such as λb should be specified by λP. The modification is evaluated by solving several diffusion problems with non-linear diffusion coefficients. Numerical tests show that the modification proposed in this paper can obviously improve the accuracy of non-linear diffusion coefficients problems numerical solution. More reasonable solutions can be obtained through the modified method, Especially for the problem IX the average error is reduced from 24.94 to 9.94e−1.

Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient

A new approach to the study of multidimensional singularly perturbed problems of nonlinear heat conduction is proposed, based on the further development and use of asymptotic analysis methods. We study the question of the existence of classical Lyapunov stable stationary solutions with boundary and internal transition layers (stationary thermal structures) of the nonlinear heat transfer equation. We suggest the efficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed formal asymptotics, we use the principle of comparison.We consider the application of the results of asymptotic analysis to solving inverse problem of reconstructing the temperature dependence of the thermal conductivity coefficient from a known position of the internal layer of thermal structure.

The logarithmic mean method for Nonlinear Conductivities at Cell Faces

In this work, logarithmic mean method, which is based on the assumption that the diffusion coefficients profile is linear between nodes, is developed to calculate diffusion coefficients at cell interfaces. The new method is evaluated by solving one dimension steady diffusion equation and convection diffusion equation with strongly non-linear diffusion coefficients. Results indicate that the logarithmic mean method can achieve high precision as the Kirchhoff integral mean method even if the number of nodes is small, while the harmonic mean method needs more nodes to achieve the same accuracy. Furthermore, the logarithmic mean method only needs the diffusion coefficients of nodes, the disadvantage of Kirchhoff method which needs concrete expression of diffusion coefficients and numerous integral operation thus can be overcome.

Singular limit for reactive transport through a thin heterogeneous layer including a nonlinear diffusion coefficient

<p style='text-indent:20px;'>Reactive transport processes in porous media including thin heterogeneous layers play an important role in many applications. In this paper, we investigate a reaction-diffusion problem with nonlinear diffusion in a domain consisting of two bulk domains which are separated by a thin layer with a periodic heterogeneous structure. The thickness of the layer, as well as the periodicity within the layer are of order <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is much smaller than the size of the bulk domains. For the singular limit <inline-formula><tex-math id="M3">\begin{document}$ \epsilon \to 0 $\end{document}</tex-math></inline-formula>, when the thin layer reduces to an interface, we rigorously derive a macroscopic model with effective interface conditions between the two bulk domains. Due to the oscillations within the layer, we have the combine dimension reduction techniques with methods from the homogenization theory. To cope with these difficulties, we make use of the two-scale convergence in thin heterogeneous layers. However, in our case the diffusion in the thin layer is low and depends nonlinearly on the concentration itself. The low diffusion leads to a two-scale limit depending on a macroscopic and a microscopic variable. Hence, weak compactness results based on standard <i>a priori</i> estimates are not enough to pass to the limit <inline-formula><tex-math id="M4">\begin{document}$ \epsilon \to 0 $\end{document}</tex-math></inline-formula> in the nonlinear terms. Therefore, we derive strong two-scale compactness results based on a variational principle. Further, we establish uniqueness for the microscopic and the macroscopic model.</p>

Interaction between vegetation patches and gaps: A self-organized response to water scarcity

The dynamics of ecological systems are often described by integrodifferential equations that incorporate nonlocal interactions associated with facilitative, competitive interactions between plants, and seed dispersion. In the weak-gradient limit, these models can be reduced to a simple partial-differential equation in the form of a nonvariational Swift–Hohenberg equation. In this contribution, we perform this reduction for any type of kernels provided that their Taylor series converge. Some parameters such as linear and nonlinear diffusion coefficients are affected by the spatial form of the kernel. In particular, Gaussian and exponential kernels are used to evaluate all coefficients of the reduced model. This weak gradient approximation is greatly useful for the investigation of periodic and localized vegetation patches, and gaps. Based on this simple model, we investigate the interaction between two-well separated patches and gaps. In the case of patches, the interaction is always repulsive. As a consequence, bounded states of patches are excluded. However, when two gaps are close to one another, they start to interact through their oscillatory tails. The interaction alternates between attractive and repulsive depending on the distance separating them. This allows for the stabilization of bounded gaps and clusters of them. The analytical formula of the interaction potential is derived for both patches and gaps interactions and checked by numerical investigation of the model equation.This volume is dedicated to Professor Ehud Meron on the occasion of his sixtieth birthday. We take this opportunity to express our warmest and most sincere wishes to him.

Strong convergence of the split-step backward Euler method for stochastic delay differential equations with a nonlinear diffusion coefficient

This paper is concerned with the strong convergence of numerical methods for stochastic differential equations with a variable delay. Based on the concepts of stochastic C-stability and B-consistency, we first derive a fundamental strong convergence theorem for general one-step methods applied to nonlinear autonomous stochastic delay differential equations under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. These conditions admit that the diffusion coefficient is highly nonlinear. Then, we construct the split-step backward Euler method. It is verified that the method is stochastically C-stable and B-consistent of order 1/2 so that the method is strongly convergent of order 1/2 for the considered problems. Finally, the theoretical results are confirmed by numerical experiments.

Topology-Dependent Brownian Gyromotion of a Single Skyrmion.

Noninteracting particles exhibiting Brownian motion have been observed in many occasions of sciences, such as molecules suspended in liquids, optically trapped microbeads, and spin textures in magnetic materials. In particular, a detailed examination of Brownian motion of spin textures is important for designing thermally stable spintronic devices, which motivates the present study. In this Letter, through using temporally and spatially resolved polar magneto-optic Kerr effect microscopy, we have experimentally observed the thermal fluctuation-induced random walk of a single isolated Néel-type magnetic skyrmion in an interfacially asymmetric Ta/CoFeB/TaO_{x} multilayer. An intriguing topology-dependent Brownian gyromotion behavior of skyrmions has been identified. The onset of Brownian gyromotion of a single skyrmion induced by thermal effects, including a nonlinear temperature-dependent diffusion coefficient and topology-dependent gyromotion are further formulated based on the stochastic Thiele equation. The experimental and numerical demonstration of topology-dependent Brownian gyromotion of skyrmions can be useful for understanding the nonequilibrium magnetization dynamics and implementing spintronic devices.

The geometric structure on a degradation model with application to optimal design under a cost constraint

Information geometry has been attracting considerable attention in different scientific fields including information theory, neural networks, machine learning, and statistical physics. In reliability and survival analysis, methods of information geometry are employed to discuss the geometry on a reliability model. Most of the existing work of information geometry in reliability analysis focused on time-to-failure data. For highly reliable devices, it is difficult to obtain failure data in a reasonable period of time, and hence degradation measurements are used to extrapolate the failure time. In this paper, we investigate the geometry on a statistical manifold induced by the Wiener degradation process with nonlinear drift and diffusion coefficients, where the drift parameter is assumed to be a random variable to incorporate the unit-to-unit characteristics. The Fisher information metric, Amari–Chentsov tensor and α-connection on the manifold for the degradation model are discussed. As an application in the design of engineering experiments, the information metric is employed to develop the optimal design of degradation experiments under a cost constraint. Monte Carlo simulation and a numerical study are used to illustrate the methodologies developed in this paper.