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The dynamics of a chiral active particle moving on two-dimensional random landscapes (with Gaussian distribution and Gaussian correlation) is investigated. We demonstrate that the ergodic uncertain diffusion and the non-ergodic subdiffusion occur in the system. The ergodic uncertain diffusion arises from the periodic motion of the chiral active particle along a circular trajectory. In contrast, the non-ergodic subdiffusion is primarily caused by the competitive effect between local high heterogeneity due to random landscapes (or chirally induced rotational motion) and the self-propulsion of the active particle. Our work provides a theoretical basis for understanding the dynamic behavior of a chiral active particle in complex environments. [ABSTRACT FROM AUTHOR]

Let R + N = [ 0 , ∞) N . We here make new contributions concerning a class of random fields (X t) t ∈ R + N which are known as multiparameter Lévy processes. Related multiparameter semigroups of operators and their generators are represented as pseudo-differential operators. We also provide a Phillips formula concerning the composition of (X t) t ∈ R + N by means of subordinator fields. We finally define the composition of (X t) t ∈ R + N by means of the so-called inverse random fields, which gives rise to interesting long-range dependence properties. As a byproduct of our analysis, we present a model of anomalous diffusion in an anisotropic medium which extends the one treated in Beghin et al. (Stoch Proc Appl 130:6364–6387, 2020), by improving some of its shortcomings. [ABSTRACT FROM AUTHOR]

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the problem valid at any spatial dimension. The aim of this paper is to extend this general analysis to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solution of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provide a clear picture of the role played by the time-fractional derivatives in this kind of random motions. They display an anomalous behavior and are useful to describe several complex systems arising in statistical physics and biology. In particular, we focus on the one-dimensional random flight, called telegraph process, studying the time-fractional version of the classical telegraph equation and providing a suitable interpretation of its stochastic solutions. [ABSTRACT FROM AUTHOR]

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations. [ABSTRACT FROM AUTHOR]

The spatiotemporal evolution of earthquakes induced by fluid injections into the subsurface can be erratic owing to the complexity of the physical process. To effectively mitigate the associated hazard and to draft appropriate regulatory strategies, a detailed understanding of how induced seismicity may evolve is needed. In this work, we build on the well-established continuous-time random walk (CTRW) theory to develop a purely stochastic framework that can delineate the essential characteristics of this process. We use data from the 2003 and 2012 hydraulic stimulations in the Cooper Basin geothermal field that induced thousands of microearthquakes to test and demonstrate the applicability of the model. Induced seismicity in the Cooper Basin shows all the characteristics of subdiffusion, as indicated by the fractional order power-law growth of the mean square displacement with time and broad waiting-time distributions with algebraic tails. We further use an appropriate master equation and the time-fractional diffusion equation to map the spatiotemporal evolution of seismicity. The results show good agreement between the model and the data regarding the peak earthquake concentration close to the two injection wells and the stretched exponential relaxation of seismicity with distance, suggesting that the CTRW model can be efficiently incorporated into induced seismicity forecasting. [ABSTRACT FROM AUTHOR]

The issues of parametric identification of fractional differential models describing the processes of anomalous diffusion/heat conductivity are considered. The emphasis is on the option with a spatially localized initial condition; this corresponds to the experimental approach to determining diffusion characteristics. Methods are proposed for solving the identification problem that do not require repeated solution of the direct problem. Testing the methods has been carried out in a quasireal experiment mode. [ABSTRACT FROM AUTHOR]

We investigate the fully nonlinear model for convection in a Darcy porous material where the diffusion is of anomalous type as recently proposed by Barletta. The fully nonlinear model is analysed but we allow for variable gravity or penetrative convection effects which result in spatially dependent coefficients. This spatial dependence usually requires numerical solution even in the linearized case. In this work, we demonstrate that regardless of the size of the Rayleigh number, the perturbation solution will decay exponentially in time for the superdiffusion case. In addition, we establish a similar result for convection in a bidisperse porous medium where both macro- and microporosity effects are present. Moreover, we demonstrate a similar result for thermosolutal convection. [ABSTRACT FROM AUTHOR]

This research employs advanced computational methodologies to analyze solitary wave solutions associated with the fractional nonlinear Chaffee–Infante ( C I ) equation, extending classical diffusion models with broad applications in materials science, fluid dynamics, and signal processing. The study makes notable contributions to the modeling of anomalous diffusion in porous materials, the comprehension of nonlinear dynamics, and the analysis of wave behavior incorporating memory and non-local effects. The research enhances our understanding of practical applications and provides valuable insights into complex wave dynamics within fluid dynamics, nonlinear optics, and plasma physics. The computational techniques utilized in this investigation, specifically the extended unified ( E U ) and trigonometric–quantic–B-spline (TQBS ) approaches, demonstrate superior effectiveness in comparison to existing methods, promising heightened accuracy and efficiency in solving fractional partial differential equations. The numerical scheme, Trigonometric–Quantic–B–spline, in tandem with analytical solutions, serves to validate the model and adapt it to real-world complexities. This integrated approach not only quantifies accuracy but also supports parameter estimation, ensuring the model's applicability in diverse engineering and scientific scenarios. Graphical representations of both analytical and numerical solutions are presented, offering a visual elucidation of the model's characteristics and validating solution accuracy. The incorporation of the Trigonometric–Quantic–B–spline scheme provides a robust foundation for further exploration and applications in varied scientific and engineering domains. [ABSTRACT FROM AUTHOR]

The article is devoted to the studies of the existence of solutions of an integro-differential equation in the case of the double scale anomalous diffusion with the sum of the two negative Laplacians raised to two distinct fractional powers in R d , d = 4 , 5 . The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for the non-Fredholm elliptic operators in unbounded domains are used. [ABSTRACT FROM AUTHOR]

Two-stage kinetics of moisture sorption at a temperature of 23°C and humidity of 68% in glass-fiber and basalt-fiber composites after exposure to cold climate conditions (Yakutsk, Russia) for 2 and 4 years is presented. The relaxation model of anomalous moisture diffusion at the first stage is shown to be adequate and an assessment of the increase in moisture content at the second stage is provided. [ABSTRACT FROM AUTHOR]

In this work, eddy diffusivity is derived from the energy spectra for the stable and convective regimes in the planetary boundary layer. The energy spectra are obtained from a spectral model for the inertial subrange that considers the anomalous behavior of turbulence. This spectrum is expressed as a function of the Hausdorff fractal dimension. The diffusivity eddies are employed in a classical Eulerian dispersion model, where the derivatives are of integer order and in fractional dispersion model, where the derivatives are of fractional order. The eddy diffusivity proposed considers the anomalous nature of geophysical turbulent flow. The results obtained with the fractional and classic dispersion models using the eddy diffusivity proposed is satisfactory when compared with the experimental data of the Prairie Grass and Hanford experiments in a stable regime, and the Copenhagen experiment in a convective regime. [ABSTRACT FROM AUTHOR]

We describe a compact finite-difference discretization and Gaussian-radial basis function for the two-dimensional local fractional elliptic PDEs that describe anomalous diffusion-convection of groundwater contamination. Precisely estimating pollutant concentration over a long period helps protect water reservoirs. The local fractional partial differential equations and their discretization described here are the generalization of the integer order elliptic partial differential equations and their high-order scheme. The high-order discretization of fractal gradient and anomalous diffusion on a non-uniformly spaced nine-point single-cell grid network gives the result in small computing time. The new scheme is supported by a detailed convergence analysis describing the monotone property and a strongly connected Jacobian (iteration) matrix graph. The computational illustration of various anomalous diffusion-convection models demonstrates the proposed methodology's effectiveness. [ABSTRACT FROM AUTHOR]

We consider the persistence probability of a certain fractional Gaussian process M H that appears in the Mandelbrot-van Ness representation of fractional Brownian motion. This process is self-similar and smooth. We show that the persistence exponent of M H exists, is positive and continuous in the Hurst parameter H. Further, the asymptotic behaviour of the persistence exponent for H ↓ 0 and H ↑ 1 , respectively, is studied. Finally, for H → 1 / 2 , the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that M 1 / 2 vanishes. [ABSTRACT FROM AUTHOR]

It is known that at least ten equivalent definitions of the fractional Laplacian exist in an unbounded domain. Here we derive a further equivalent definition that is based on the Mellin transform and it can be used when the fractional Laplacian is applied to radial functions. The main finding is tested in the case of the space-fractional diffusion equation. The one-dimensional case is also considered, such that the Mellin transform of the Riesz (namely the symmetric Riesz–Feller) fractional derivative is established. This one-dimensional result corrects an existing formula in literature. Further results for the Riesz fractional derivative are obtained when it is applied to symmetric functions, in particular its relation with the Caputo and the Riemann–Liouville fractional derivatives. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical solution of fractional differential equations with interval parameters in terms of derivatives describing anomalous diffusion processes. Computational algorithms for solving initial–boundary value problems as well as the corresponding inverse problems for equations containing interval fractional derivatives with respect to time and space are presented. The algorithms are based on the previously developed and theoretically substantiated adaptive interpolation algorithm tested on a number of applied problems for modeling dynamical systems with interval parameters; this makes it possible to explicitly obtain parametric sets of states of dynamical systems. The efficiency and workability of the proposed algorithms are demonstrated in several problems. [ABSTRACT FROM AUTHOR]

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings. [ABSTRACT FROM AUTHOR]

The temporal and spatiotemporal linear stability analyses of viscoelastic, subdiffusive, plane Poiseuille flow obeying the Fractional Upper Convected Maxwell (FUCM) equation in the limit of low to moderate Reynolds number (Re) and Weissenberg number (We) is reported to identify the regions of topological transition of the advancing flow interface. In particular, we demonstrate how the exponent of the power-law scaling ( t α , with 0 < α ≤ 1 ) in viscoelastic microscale models (Mason and Weitz in Phys Rev Lett 74:1250–1253, 1995) is related to the fractional order of the time derivative, α , of the corresponding non-linear stress constitutive equation in the continuum. The stability studies are limited to two exponents: monomer diffusion in Rouse chain melts, , and in Zimm chain solutions,. The temporal stability analysis indicates that with decreasing order of the fractional derivative (a) the most unstable mode decreases with decreasing values of α , (b) the peak of the most unstable mode shifts to lower values of Re, and (c) the peak of the most unstable mode, for the Rouse model precipitates towards the limit R e → 0 . The Briggs idea of analytic continuation is deployed to classify regions of temporal stability, absolute and convective instabilities, and evanescent modes. The spatiotemporal phase diagram indicates an abnormal region of temporal stability at high fluid inertia, revealing the presence of a non-homogeneous environment with hindered flow, thus highlighting the potential of the model to effectively capture certain experimentally observed, flow-instability transition in subdiffusive flows. [ABSTRACT FROM AUTHOR]

Expansive clay is of multi-fissures and deterioration with the changing moisture. To ensure the safety of earth constructions founded on expansive clay in the semiarid area, the effects of intensive water migration have received much more attention, together with global warming and climate change. Herein, water migration was analyzed from the perspective of the anomalous diffusion equation, and the time-fractional vapor–liquid migration equation in one-dimensional unsaturated clay was discussed based on Caputo and conformable fractional derivatives. The model’s validity was also verified by the measured data obtained from the migration experiments under various temperatures and water gradients in a closed system. The sensitivity of the fractional order was further analyzed. Results show that the fractional-order water migration model can describe the migration process of water in unsaturated clay well. Its error was about 30% of the integer-order model. The presented model may help study the law of water migration in unsaturated expansive clay. [ABSTRACT FROM AUTHOR]

A recurrent neural network is developed for segmenting between anomalous and normal diffusion in single-particle trajectories. Accurate segmentation infers a distinct change point that is used to approximate an Einstein linear regime in the mean-squared displacement curve via the transition density function, a unique physical descriptor for short-lived and delayed transiency. Through several artificial and simulated scenarios, we demonstrate the compelling accuracy of our model for dissecting linear and nonlinear behaviour. The inherent practicality of our model lies in its ability to substantiate the self-diffusion coefficient through offline trajectory segmentation, which is opposed to the common 'best-guess' linear fitting standard. Additionally, we show that the transition density function has fundamental implications and correspondence to underlying mechanisms that influence transition. In particular, we show that the known proportionality between salt concentration and diffusion of water also influences delayed anomalous behaviour. Article Highlights: Recurrent neural network is developed to identify diffusive change points in single-particle trajectories RNN predictions are accurate in a variety of synthetic and experimental scenarios simulated by molecular dynamics Particle trajectory transitions follow a distribution that can be affected by salinity in the context of CO2 storage [ABSTRACT FROM AUTHOR]

This paper introduces three types of dynamical indicators that capture the effect of uncertainty on the time evolution of dynamical systems. Two indicators are derived from the definition of finite-time Lyapunov exponents, while a third indicator directly exploits the property of the polynomial expansion of the dynamics with respect to the uncertain quantities. The paper presents the derivation of the indicators and a number of numerical experiments that illustrates the use of these indicators to depict a cartography of the phase space under parametric uncertainty and to identify robust initial conditions and regions of practical stability in the restricted three-body problem. [ABSTRACT FROM AUTHOR]

Nowadays, several applications in engineering and science are considering fractional partial differential equations. However, this type of equation presents new challenges to obtaining analytical solutions, since most existing techniques have been developed for integer order differential equations. In this sense, this work aims to investigate the potential of fractional derivatives in the mathematical modeling of the dispersion of atmospheric pollutants by obtaining a semi-analytical solution of the time-dependent fractional, two-dimensional advection–diffusion equation. To reach this goal, the GILTT (Generalized Integral Laplace Transform Technique) and conformal derivative methods were combined, taking fractional parameters in the transient and longitudinal advective terms. This procedure allows the anomalous behavior in the dispersion process to be considered, resulting in a new methodology called α-GILTT. A statistical comparison between the traditional Copenhagen experiment dataset (moderately unstable) with the simulations from the model showed little influence on the fractional parameters under lower fractionality conditions. However, the sensitivity tests with the fractional parameters allow us to conclude that they effectively influence the dispersion of pollutants in the atmosphere, suggesting dependence on atmospheric stability. [ABSTRACT FROM AUTHOR]

We consider a forager diffusing via a fractional heat equation and we introduce several efficiency functionals whose optimality is discussed in relation to the Lévy exponent of the evolution equation. Several biological scenarios, such as a target close to the forager, a sparse environment, a target located away from the forager and two targets are specifically taken into account. The optimal strategies of each of these configurations are here analyzed explicitly also with the aid of some special functions of classical flavor and the results are confronted with the existing paradigms of the Lévy foraging hypothesis. Interestingly, one discovers bifurcation phenomena in which a sudden switch occurs between an optimal (but somehow unreliable) Lévy foraging pattern of inverse square law type and a less ideal (but somehow more secure) classical Brownian motion strategy. Additionally, optimal foraging strategies can be detected in the vicinity of the Brownian one even in cases in which the Brownian one is pessimizing an efficiency functional. [ABSTRACT FROM AUTHOR]

This study aims to investigate the potential of fractional derivatives in the mathematical modelling of the dispersion of air pollutants. For this purpose, an analytical solution of the fractional two-dimensional advection–diffusion equation by combining generalized integral Laplace transform technique (GILTT) and conformable derivatives methods was proposed. Although the use of the conformable derivatives loses the non-local character contained in the fractional derivatives, fractional parameters remain in the solution. Thus, this procedure allows considering the anomalous behaviour in the diffusion process, resulting in a new methodology here called the α -GILTT method. The concentrations calculated with the model were compared with ground-level crosswind-integrated concentrations data from the Copenhagen and Prairie Grass experiments. The statistical indices showed the best results for the moderately unstable Copenhagen experiment under conditions of low fractionality (values close to unity). However, for the strongly convective Prairie Grass experiment, the results showed greater dependence on the fractional parameters (integer-order: NMSE = 0.90, COR = 0.81, FAT2 = 0.63; non-integer order: NMSE = 0.56, COR = 0.89, FAT2 = 0.84). The results suggest that fractional parameters are dependent on atmospheric stability and open a new direction to improve the knowledge of the atmospheric pollutant dispersion processes. [ABSTRACT FROM AUTHOR]

The aim of this work is to devise and analyse an accurate numerical scheme to solve Erdélyi–Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion, and grey Brownian motion. To obtain a convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erdélyi–Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singularity in the time term appearing in the main equation, the proposed method converges slower than first order. Finally, we provide a numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions. [ABSTRACT FROM AUTHOR]

The diffusion kinetics of a molecular probe—rhodamine B—in ternary aqueous solutions containing poly(vinyl alcohol), glycerol, and surfactants was investigated using fluorescence correlation spectroscopy and dynamic light scattering. We show that the diffusion characteristics of rhodamine B in such complex systems is determined by a synergistic effect of molecular crowding and intermolecular interactions between chemical species. The presence of glycerol has no noticeable impact on rhodamine B diffusion at low concentration, but significantly slows down the diffusion of rhodamine B above 3.9% (w/v) due to a dominating steric inhibition effect. Furthermore, introducing surfactants (cationic/nonionic/anionic) to the system results in a decreased diffusion coefficient of the molecular probe. In solutions containing nonionic surfactant, this can be explained by an increased crowding effect. For ternary poly(vinyl alcohol) solutions containing cationic or anionic surfactant, surfactant—polymer and surfactant—rhodamine B interactions alongside the crowding effect of the molecules slow down the overall diffusivity of rhodamine B. The results advance our insight of molecular migration in a broad range of industrial complex formulations that incorporate multiple compounds, and highlight the importance of selecting the appropriate additives and surfactants in formulated products. [ABSTRACT FROM AUTHOR]

We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Marichev-Saigo-Maeda time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional Lévy motion for suitable pseudo-differential operators in space. [ABSTRACT FROM AUTHOR]

In a porous fractal medium, the transport dynamics is sometimes anomalous as well as the crossover between numerous transport regimes occurs. In this paper, we experimentally investigate the mass transfer of the diffusing agents of various classes in the composite porous particle with fractal geometry. It is shown that transport mechanisms differ at short and long times. At the beginning, pure advection is observed, whereas the longtime transport follows a convective mechanism. Moreover, the longtime transport experiences either Fickian or non-Fickian kinetics depending on the diffusing agent. The non-Fickian transport is justified for the diffusing agent with higher adsorption energy. Therefore, we speculate that non-Fickian transport arises due to the strong irreversible adsorption sticking of the diffusing molecules on the surface of the porous particle. For the distinguishing of the transport regimes, an approach admitting the transformations of the experimental data and the relevant analytic solutions in either semi-logarithmic or logarithmic coordinates is developed. The time-fractional advection–diffusion equation is used on a phenomenological basis to describe the experimental data exhibiting non-Fickian kinetics. The obtained anomalous diffusion exponent corresponds to the superdiffusive transport. [ABSTRACT FROM AUTHOR]

Fractional telegraph equation is introduced to promote a sub-diffusive transport model of the neutrons inside a nuclear reactor. Here, the time derivatives of the classical telegraph equation have been replaced by the Caputo fractional derivative. In this paper, the fractional solutions are obtained through a technique that relies heavily on Fourier and Laplace transformations and is given in terms of the M-Wright function. The neutron flux density is clarified for different values of the time-fractional order. [ABSTRACT FROM AUTHOR]

A conducting biodegradable solid polymer nanocomposite electrolyte (conductivity of the order ~ 10−3 S/cm) comprising of the blend of chitosan (CS) and carboxymethyl cellulose (CMC), plasticized with glycerol, lithium perchlorate (LiClO4) as dopant salt, and carbon nanotube (CNT) as nanofiller, has been prepared. The ion dynamics and charge carrier relaxation properties in CNT-incorporated solid biopolymer nanocomposite electrolytes are studied. The scaling of the conductivity and permittivity spectra is performed to study the effect of dopant composition on the relaxation mechanism. The anomalous intra-diffusive behavior of the charge carriers inside the blended polymer matrix has been investigated based on the fractional diffusion model. FTIR spectroscopy results suggest that the secondary amide functional groups of chitosan dominate the conduction mechanism. Transference number analysis confirms the predominant ionic conduction in the CNT-incorporated CS/CMC blended solid nanocomposite electrolytes. A very low leakage current of ~ 6 µA has been obtained with the CNT-incorporated solid nanocomposite electrolyte which makes it attractive for energy storage applications. A symmetric hybrid solid-state supercapacitor has been fabricated with active carbon and nanocrystallites of manganese cobalt ferrite (MnCoFeO4) as electrode materials and CNT-incorporated CS/CMC blended biopolymers as solid-state nanocomposite electrolyte, offering a fairly good effective specific capacitance of ~ 94 Fg−1. [ABSTRACT FROM AUTHOR]

A problem of porous media cleaning by a pulsating external flow is studied numerically. We consider a closed domain of porous media, which was initially clogged by an impurity. The purification of such a domain is performed by an external vertical pulsating flow of a pure fluid. The transport of the impurity with immobilization process is modeled into framework of the nonlinear MIM model with saturation of the immobile phase. The clogging of media is taken into account which, in turn, leads to a decrease of the pore volume due to the immobilized impurity on the pores. The one dimensional solution as in the form of vertical seepage was obtained. It is shown that the most effective cleaning is achieved for the maximal possible value of the external flow intensity and can be controlled by the variation of the pulsation frequency and amplitude. However at the high intensities of the external flow the one dimensional solution becomes unstable. The stability problem was solved within the quasistatic approach. The critical curves in the system parameters are obtained and discussed. [ABSTRACT FROM AUTHOR]

Quantifying ecological memory could be done at several levels from the rate of physiological changes in an ecosystem all the way down to responses at the genetic level. One way of unlocking the information encoded in a collective environmental memory is to examine the recorded time-series data generated by different components of an ecosystem. In this paper, we probe into the case of the Great Barrier Reef (GBR) which is threatened by elevated sea surface temperatures (SST) and ocean acidification attributed to rising atmospheric CO2 levels. Specifically, we investigate the interrelated dynamics between the degradation of the GBR, SST, and rising atmospheric CO2 levels, by considering three datasets: (a) the mean percentage hard coral cover of the GBR from the archives of the Australian Institute of Marine Science; (b) SST close to the GBR from the National Oceanic and Atmospheric Administration; and (c) the Keeling curve for atmospheric CO2 levels measured by the Mauna Loa Observatory. We show that fluctuating observables in these datasets have the same memory behavior described by a non-Markovian stochastic process. All three datasets show a good match between empirical and analytical mean square deviation. An explicit analytical form for the corresponding probability density function is obtained which obeys a modified diffusion equation with a time dependent diffusion coefficient. This study provides a new perspective on the similarities of and interaction between the GBR's declining hard coral cover, SST, and rising atmospheric CO2 levels by putting all three systems into one unified framework indexed by a memory parameter μ and a characteristic frequency ν . The short-time dynamics of CO2 levels and SST fall in the superdiffusive regime, while the GBR exhibits hyperballistic fluctuation in percent coral cover with the highest values for μ and ν . [ABSTRACT FROM AUTHOR]

Anomalous diffusion behavior can be observed in many single-particle (contained in crowded environments) tracking experimental data. Numerous models can be used to describe such data. In this paper, we focus on two common processes: fractional Brownian motion (fBm) and scaled Brownian motion (sBm). We proposed novel methods for sBm anomalous diffusion parameter estimation based on the autocovariance function (ACVF). Such a function, for centered Gaussian processes, allows its unique identification. The first estimation method is based solely on theoretical calculations, and the other one additionally utilizes neural networks (NN) to achieve a more robust and well-performing estimator. Both fBm and sBm methods were compared between the theoretical estimators and the ones utilizing artificial NN. For the NN-based approaches, we used such architectures as multilayer perceptron (MLP) and long short-term memory (LSTM). Furthermore, the analysis of the additive noise influence on the estimators' quality was conducted for NN models with and without the regularization method. [ABSTRACT FROM AUTHOR]

Earthquake diffusion and the migration behaviour of seismic clusters are commonly studied to provide insight on the spatiotemporal evolution of seismicity and the interplaying driving mechanisms. Using a high-resolution relocated catalogue, we study the variations of the earthquake diffusion rates in the Western Gulf of Corinth during 2013–2014, a period with abundant local seismicity, including intense microseismic background, seismic swarms and mainshock-aftershock sequences. We treat earthquake occurrence as a point process in time and space and estimate the diffusion rates of the main seismic sequences and the background seismicity in terms of normalized spatial histograms and the evolutions of the mean squared distance of seismicity with time. The statistical analysis of the studied seismic sequences reveals that the mean squared distance of the hypocentres increases slowly with time, at a much lower rate than for a normal diffusion process. Such findings confirm previous results on weak earthquake diffusion, analogous to subdiffusion, in regional and clustered seismicity. In addition, seismic swarms associated with pore fluid pressure diffusion present considerably higher diffusion exponents compared to mainshock-aftershock-type sequences that are consistent with primary or secondary stress triggering effects and stress corrosion. The observed variations of the earthquake diffusion rates indicate the stochastic nature of the phenomenon and may provide novel constraints on the triggering mechanisms of clustered seismic activity in the Western Gulf of Corinth and in other seismically active regions. [ABSTRACT FROM AUTHOR]

A model is developed for moisture absorption by rebar made of basalt fiber-reinforced polymer composite (BFRP), taking account of Fick diffusion, structural relaxation, and the design features of the rebar in the initial stages of climatic aging. The model is experimentally confirmed. The analytical solution may be used for kinematic approximation of moisture absorption by other reinforced polymer composites. [ABSTRACT FROM AUTHOR]

In this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space-dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in u ∈ L ∞ ((0 , T) , H 0 1 (Ω)) to the problem if the initial data belongs to H 0 1 (Ω) . We show that the solution belongs to C ([ 0 , T ] , H 0 1 (Ω) ∗) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form d d t (k ∗ v) (t) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains. [ABSTRACT FROM AUTHOR]

This paper intends on obtaining the explicit solution of -dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the parabolic integro-differential equation with memory in which the kernel is , where is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox H-function and convolution theorem, explicit solution for anomalous diffusion equation is obtained. [ABSTRACT FROM AUTHOR]

We study the drift-diffusion equation with fractional dissipation (- Δ) θ / 2 arising from a model of semiconductors. First, we prove the existence and uniqueness of the small solution to the corresponding stationary problem in the whole space. Moreover, it is proved that the unique solution of non-stationary problem exists globally in time and decays exponentially, if initial data are suitably close to the stationary solution and the stationary solution is sufficiently small. [ABSTRACT FROM AUTHOR]

This work is devoted to examining the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator and a multiplicative noise. Our proof for uniqueness is based upon the analysis on double variables method and the existence is enabled by a parabolic approximation. [ABSTRACT FROM AUTHOR]

High magnetic fields (up to 17.6 T) in combination with large magnetic field gradients (up to 25 T/m) were successfully utilized in pulsed field gradient (PFG) NMR studies of gas and liquid diffusion in nanoporous materials. In this mini-review, we present selected examples of such studies demonstrating the ability of high field PFG NMR to gain unique insights and differentiate between various types of diffusion. These examples include identifying and explaining an anomalous relationship between molecular size and self-diffusivity of gases in a zeolitic imidazolate framework (ZIF), as well as revealing and explaining an influence of mixing different linkers in a ZIF on gas self-diffusion. Different types of normal and restricted self-diffusion were quantified in hybrid membranes formed by dispersing ZIF crystals in polymers. High field PFG NMR studies of such membranes allowed observing and explaining an influence of the ZIF crystal confinement in a polymer on intra-ZIF self-diffusion of gases. This technique also allowed measuring and understanding anomalous single-file diffusion (SFD) of mixed sorbates. Furthermore, the presented examples demonstrate a high potential of combining high field PFG NMR with single-crystal infrared microscopy (IRM) for obtaining greater physical insights into the studied diffusion processes. [ABSTRACT FROM AUTHOR]

A brief review of the emergence and development of the nonlocal approach to the problem of turbulent diffusion with a discussion of the physical reasons of the nonlocality is given. The main attention is paid to fractional differential operators. In concluding the paper, the author's original results on applications to the diffusion of cosmic rays in the interstellar galactic medium are presented. [ABSTRACT FROM AUTHOR]

Many processes in science and engineering are described by fractional systems which may in general be stiff and involve a nonsmooth source term. In this paper, we develop robust first, second, and third order accurate exponential time differencing schemes for solving such systems. Rather than imposing regularity requirements on the solution to account for the singularity caused by the fractional derivative, we only consider regularity requirements on the source term for preserving the optimal order of accuracy of the proposed schemes. Optimal convergence rates are proved for both smooth and nonsmooth source terms using uniform and graded meshes, respectively. For efficient implementation, high-order global Padé approximations together with their fractional decompositions are developed for Mittag–Leffler functions. We present numerical experiments involving a typical stiff system, a fractional two-compartment pharmacokinetics model, a two-term fractional Kelvin–Viogt model of viscoelasticity, and a large system obtained by spatial discretization of a sub-diffusion problem. Demonstrations of the efficiency of the rational approximation implementation technique and the newly constructed high-order schemes are provided. [ABSTRACT FROM AUTHOR]

On the basis of a well-based model (Model I) developed in a previous work (Liu and Valkó in SPE J 2019. https://doi.org/10.2118/197049-PA), in which a fractional production decline model is developed based on anomalous diffusion to structurally account for the heterogeneity related to the complex fracture network, we incorporate two more components, i.e., the tempered anomalous diffusion and a source term, in the present work to develop a generalized model, namely Model II. Because of the two new components, Model II is capable of physically considering both the scale lower bound of heterogeneity and the influx from the matrix into the conductive fracture system. It consequently enables the new model to describe the complete sequence of regimes for the production of the slightly compressible single-phase fluid from the fractured unconventional reservoirs. Then, in the case studies, the synthetic data used in the previous work are better fitted to the new type curves in all observed periods, which demonstrates the advantage of Model II over Model I regarding the late-time flow regimes. In addition, the good fit of other sets of synthetic data to the Model II type curves exhibits the claimed capabilities of the new model and indicates the production of this type could be well characterized by seven parameters, i.e., α , c , λ D , ω , σ , τ , and EUR . Finally, the characteristics of Model II are further investigated by analyzing the effects of several parameters on the production performance, based on which some insights about the practice of developing unconventional reservoirs using massive hydraulic fracturing are provided. [ABSTRACT FROM AUTHOR]

The third boundary-value problem in the half-strip for a partial differential equation with Bessel operator is studied. Existence and uniqueness theorems are proved. The representation of the solution is found in terms of the Laplace convolution of the exponential function and Mittag-Leffler type function with power multipliers. Uniqueness is proved for the class of bounded functions. [ABSTRACT FROM AUTHOR]

In this work, an analytical solution for the steady-state fractional advection-diffusion equation was investigated to simulate the dispersion of air pollutants in a finite media. The authors propose a method that uses classic integral transform technique (CITT) to solve the transformed problem with a fractional derivative, resulting in a more general solution. We compare the solutions with data from real experiment. Physical consequences are discussed with the connections to generalised diffusion equations. In the wake of these analysis, the results indicate that the present solutions are in good agreement with those obtained in the literature. This report demonstrates that fractional equations have come of age as a decisive tool to describe anomalous transport processes. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the reasons of anomalous transport observation in the experiments with microchannels. Usually, retardation of such transport is associated with the interaction of solute particles with channel walls. In the present paper, we have shown that the viscous interaction of the flow and solid wall can be the reason for the anomalous transport observation. It was illustrated on a specific example of a passive solute transport through the channel. The effect of diffusion on such transport was investigated numerically. The power law decline of concentration with respect to time was obtained in a wide range of parameters. [ABSTRACT FROM AUTHOR]

We consider an inverse source two-parameter sub-diffusion model subject to a non-local initial condition. The problem models several physical processes, among them are the microwave heating and light propagation in photoelectric cells. A bi-orthogonal pair of bases is employed to construct a series representation of the solution and a Volterra integral equation for the source term. We develop a stable numerical algorithm, based on discontinuous collocation method, for approximating the unknown time-dependent source term. Due to the singularity of the solution near t = 0 , a graded mesh is used to maintain optimal convergence rates, both theoretically and numerically. Numerical experiments are provided to illustrate the expected analytical order of convergence. [ABSTRACT FROM AUTHOR]

We study the dynamics of a particle moving in a square two-dimensional Lorentz lattice-gas. The underlying lattice-gas is occupied by two kinds of rotators, "right-rotator (R)" and "left-rotator (L)" and some of the sites are empty viz. vacancy "V".The density of R and L are the same and density of V is one of the key parameters of our model. The rotators deterministically rotate the direction of a particle's velocity to the right or left and vacancies leave it unchanged. We characterise the dynamics of particle motion for different densities of vacancies. Since the system is deterministic, the particle forms a closed trajectory asymptotically. The probability of the particle being in a closed or open trajectory at time t is a function of the density of vacancies. The motion of the particle is uniform throughout in a fully occupied lattice. However, it is divided in two distinct phases in partially vacant lattices: The first phase of the motion, which is the focus of this study, is characterised by anomalous diffusion and a power-law decay of the probability of being in an open trajectory. The second phase of the motion is characterised by subdiffusive motion and an exponential decay of the probability of being in an open trajectory. For lattices with a non-zero density of vacancies, the first phase of motion lasts for a longer period of time as the density of vacancies increases. [ABSTRACT FROM AUTHOR]