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The authors of the paper (Vitali et al 2022 J. Phys. A: Math. Theor. 55 224012) analyzed a simple CTRW model with a waiting time distribution defined as the weighted sum of two exponential distributions. They showed that their model meets many paradigmatic features that belong to the anomalous diffusion as it is observed in living systems. This comment point out the previous paper that considers a similar model and improves on the authors' result regarding the time dependence of the mean-square displacement. [ABSTRACT FROM AUTHOR]

This paper presents numerical methods for solving the one-dimensional fractional reaction-diffusion equation with the fractional Caputo derivative. The proposed methods are based on transformation of the fractional differential equation to an equivalent form of a integro-differential equation. The paper proposes an improvement of the existing implicit method, and a new explicit method. Stability and convergence tests of the methods were also conducted. [ABSTRACT FROM AUTHOR]

In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d ⩾ 1 , in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form (1 + | v | 2) − β 2 , in the range β ∈ ] d , d + 4 [. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity28 (2015) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ : = ∫ R d f d v , with a fractional Laplacian κ (− Δ) β − d + 2 6 and a positive diffusion coefficient κ. [ABSTRACT FROM AUTHOR]

This paper deals with a time-space fractional Schrödinger equation with homogeneous Dirichlet boundary conditions. A common strategy for discretizing time-fractional operators is finite difference schemes. In these methods, the time-step size should usually be chosen sufficiently small, and subsequently, too many iterations are required which may be time-consuming. To avoid this issue, we utilize the Laplace transform method in the present work to discretize time-fractional operators. By using the Laplace transform, the equation is converted to some time-independent problems. To solve these problems, matrix transformation and improved matrix transformation techniques are used to approximate the spatial derivative terms which are defined by the spectral fractional Laplacian operator. After solving these stationary equations, the numerical inversion of the Laplace transform is used to obtain the solution of the original equation. The combination of finite difference schemes and the Laplace transform creates an efficient and easy-to-implement method for time-space fractional Schrödinger equations. Finally, some numerical experiments are presented and show the applicability and accuracy of this approach. [ABSTRACT FROM AUTHOR]

Riverbed elevations play a crucial role in sediment transport and flow resistance, making it essential to understand and quantify their effects. This knowledge is vital for various fields, including river engineering and stream ecology. Previous observations have revealed that fluctuations in the bed surface can exhibit both multifractal and monofractal behaviors. Specifically, the probability distribution function (PDF) of elevation increments may transition from Laplace (two‐sided exponential) to Gaussian with increasing scales or consistently remain Gaussian, respectively. These differences at the finest timescale lead to distinct patterns of bedload particle exchange with the bed surface, thereby influencing particle resting times and streamwise transport. In this paper, we utilize the fractional Laplace motion (FLM) model to analyze riverbed elevation series, demonstrating its capability to capture both mono‐ and multi‐fractal behaviors. Our focus is on studying the resting time distribution of bedload particles during downstream transport, with the FLM model primarily parameterized based on the Laplace distribution of increments PDF at the finest timescale. Resting times are extracted from the bed elevation series by identifying pairs of adjacent deposition and entrainment events at the same elevation. We demonstrate that in cases of insufficient data series length, the FLM model robustly estimates the tail exponent of the resting time distribution. Notably, the tail of the exceedance probability distribution of resting times is much heavier for experimental measurements displaying Laplace increments PDF at the finest scale, compared to previous studies observing Gaussian PDF for bed elevation. Plain Language Summary: The evolution of riverbed elevations is difficult to describe due to its highly variable and strongly non‐linear nature. Understanding and quantifying the dynamics of riverbed elevations are important for river engineering and stream ecology, and serve as the basis for numerical models of predicting sediment transport as well as interpreting stratigraphy from the past records. Through laboratory experiments, we have observed that the form of elevation increment PDF can change from Laplace to Gaussian as the timescale increases. This phenomenon is successfully modeled in this paper for the first time by the fractional Laplace motion, which essentially generates bed elevation series for the evolution of bed surface height at a certain spatial location of the bed. This series contains information on how long a bedload particle can rest (resting time) in the riverbed before it can be re‐entrained to move downstream, the determination of which by other means (e.g., particle‐tracking measurements) is challenging. By extracting resting times embedded in this bed elevation series, we obtain statistics (i.e., the tail behavior of the resting time distribution) that are key for correctly modeling the transport of bedload particles, and more specifically, that can help us to understand the anomalous bedload diffusion process. Key Points: The Fractional Laplace motion (FLM) can be used to describe the evolution of the distribution of bed elevation increments over different timescalesCorrectly predicting bed elevation fluctuations at the finest timescale is critical for estimating resting times for bedload tracer transportFLM model provides a means of robustly estimating the tail exponent of the resting time distribution in case of insufficient data series length [ABSTRACT FROM AUTHOR]

First hitting times (FHTs) describe the time it takes a random "searcher" to find a "target" and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order s ε (0, 1) (describing a (2s)-stable Lévy flight whose squared displacement scales as t1/s in time t) and targets of radius \varepsilon \ll 1, we show that the MFHT is order one for s ε (1/2, 1) and diverges as log(1/\varepsilon) for s = 1/2 and \varepsilon 2s 1 for s ε (0, 1/2). We then use our asymptotic results to identify the value of s ε (0, 1] which minimizes the average MFHT and find that (a) this optimal value of s vanishes for sparse targets and (b) the value s = 1/2 (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations. [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]

Brownian motion (BM) is the archetypal model of regular diffusion. BM is a Gaussian and Markov process, whose increments are stationary, and whose non-overlapping increments are independent. Elevating from regular diffusion to anomalous diffusion, fractional Brownian motion (FBM) and scaled Brownian motion (SBM) are arguably the two most popular Gaussian anomalous-diffusion models. Each of these two models maintains some BM properties, abandons other, and displays certain anomalous behaviors. This paper explores a Gaussian anomalous-diffusion model— Power Brownian Motion (PBM)—that is attained by a coupled amplitudal and temporal 'tinkering' with BM. The PBM model combines 'the better of FBM and SBM'. Indeed, as FBM, PBM displays the anomalous behaviors of persistence and anti-persistence. And, as SBM, PBM is a Markov process that displays the anomalous behaviors of aging and anti-aging. On their own, neither FBM nor SBM can provide the 'features package' that PBM provides. The PBM 'features package' on the one hand, and its simple construction on the other hand, render PBM a compelling anomalous-diffusion model. [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]

Turbulence is a non-local phenomenon and has multiple-scales. Non-locality can be addressed either implicitly or explicitly. Implicitly, by subsequent resolution of all spatio-temporal scales. However, if directly solved for the temporal or spatially averaged fields, a closure problem arises on account of missing information between two points. To solve the closure problem in Reynolds-averaged Navier–Stokes equations (RANS), an eddy-viscosity hypotheses has been a popular modelling choice, where it follows either a linear or nonlinear stress–strain relationship. Here, a non-constant diffusivity is introduced. Such a non-constant diffusivity is also characteristic of non-Fickian diffusion equation addressing anomalous diffusion process. An alternative approach, is a fractional derivative-based diffusion equations. Thus, in the paper, we formulate a fractional stress–strain relationship using variable-order Caputo fractional derivative. This provides new opportunities for future modelling effort. We pedagogically study of our model construction, starting from one-sided model and followed by two-sided model applied to channel, couette and pipe flow. Non-locality at a point is the amalgamation of all the effects, thus we find the two-sided model is physically consistent. Further, our construction can also addresses viscous effects, which is a local process. Thus, our fractional model addresses the amalgamation of local and non-local process. We also show its validity at infinite Reynolds number limit. An expression for the fractional order is also found, thereby solving the closure problem for the considered cases. This study is further extended to tempered fractional calculus, where tempering ensures finite jump lengths, this is an important remark for unbounded flows. Within the context of this paper, we limit ourselves to wall bounded flow. Two tempered definitions are introduced with a smooth and sharp cutoff, by the exponential term and Heaviside function, respectively and we also define the horizon of non-local interactions. We further study the equivalence between the two definitions, as truncating the domain has computational advantages. For the above investigations, we carefully designed algorithms, notably, the pointwise version of fractional physics-informed neural network to find the fractional order as an inverse problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a revised fractional Brownian motion run with a nonlinear clock (fBm-nlc) model and utilize it to illustrate the microscopic mechanism analysis of the fractal derivative diffusion model with variable coefficient (VC-FDM). The power-law mean squared displacement (MSD) links the fBm-nlc model and the VC-FDM via the two-parameter power law clock and the Hurst exponent is 0.5. The MSD is verified by using the experimental points of the chloride ions diffusion in concrete. When compared to the linear Brownian motion, the results show that the power law MSD of the fBm-nlc is much better in fitting the experimental points of chloride ions in concrete. The fBm-nlc clearly interprets the VC-FDM and provides a microscopic strategy in characterizing different types of non-Fickian diffusion processes with more different nonlinear functions. [ABSTRACT FROM AUTHOR]

This paper focuses on tracing the connectivity of white matter fascicles in the brain. In particular, a generalized order algorithm based on mixture of non-central Wishart distribution model is proposed for this purpose. The proposed algorithm utilizes the generalization of integer order based approach with the mixture of non-central Wishart distribution model. Pseudo super anomalous behavior of water diffusion inside human brain is the prime motivation of the the present study. We have shown results on multiple synthetic simulations with fibers orientations in two and three directions in each voxel as well as experiments on real data. Synthetic simulations were performed with varying noise levels and diffusion weighting gradient i.e. |$b-$| values. The proposed model performed outstanding especially for distinguishing closely oriented fibers. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation's solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank–Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through the Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver–Wynn–Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained via the two methods. [ABSTRACT FROM AUTHOR]

This paper presents and explores a diffusion model that generalizes Brownian motion (BM). On the one hand, as BM: the model's mean square displacement grows linearly in time, and the model is Gaussian and selfsimilar (with Hurst exponent 1 2 ). On the other hand, in sharp contrast to BM: the model is not Markov, its increments are not stationary, and its non-overlapping increments are not independent. Moreover, the model exhibits a host of statistical properties that are dramatically different than those of BM: aging and anti-aging, positive and negative momenta, correlated velocities, persistence and anti-persistence, aging Wiener–Khinchin spectra, and more. Conventionally, researchers resort to anomalous-diffusion models—e.g. fractional BM and scaled BM (both with Hurst exponents different than 1 2 )—to attain such properties. This model establishes that such properties are attainable well within the realm of diffusion. As it is seemingly Brownian yet highly non-Brownian, the model is termed Weird BM. [ABSTRACT FROM AUTHOR]

Richards' equation is a classical differential equation describing water transport in unsaturated porous media, in which the moisture content and the soil matrix depend on the spatial derivative of hydraulic conductivity and hydraulic potential. This paper proposes a nonlocal model and the peridynamic formulation replace the temporal and spatial derivative terms. Peridynamic formulation utilizes a spatial integration to describe the path-dependency, so the fast diffusion process of water transport in unsaturated porous media can be captured, while the Caputo derivative accurately describes the sub-diffusion phenomenon caused by the fractal nature of heterogeneous media. A one-dimensional water transport problem with a constant permeability coefficient is first addressed. Convergence studies on the nonlocal parameters are carried out. The excellent agreement between the numerical and analytical solutions validates the proposed model for its accuracy and parameter stability. Subsequently, the wetting process in two porous building materials is simulated. The comparison of the numerical results with experimental observations further demonstrates the capability of the proposed model in describing water transport phenomena in unsaturated porous media. [ABSTRACT FROM AUTHOR]

This paper deals with homogeneous and non-homogeneous fractional diffusion difference equations. The fractional operators in space and time are defined in the sense of Gr¨unwald and Letnikov. Applying results on the existence of eigenvalues and corresponding eigenfunctions of the Sturm--Liouville problem, we show that solutions of fractional diffusion difference equations exist and are given by a finite series. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the well-posedness of weak solutions to the time-fractional Fokker–Planck equation. Its dynamics is governed by anomalous diffusion, and we consider the most general case of space-time dependent forces. Consequently, the fractional derivatives appear on the right-hand side of the equation, and they cannot be brought to the left-hand side, which would have been preferable from an analytical perspective. For showing the model's well-posedness, we derive an energy inequality by considering nonstandard and novel testing methods that involve a series of convolutions and integrations. We close the estimate by a Henry–Gronwall-type inequality. Lastly, we propose a numerical algorithm based on a nonuniform L1 scheme and present some simulation results for various forces. [ABSTRACT FROM AUTHOR]

Molecular communication in nature can incorporate a large number of nano-things in nanonetworks as well as demonstrate how nano-things communicate. This paper presents molecular communication where transmit nanomachines deliver information molecules to a receive nanomachine over an anomalous diffusion channel. By considering a random molecule concentration in a space-time fractional diffusion channel, an analytical expression is derived for the first passage time (FPT) of the molecules. Then, the bit error rate of the $\ell $ th nearest molecular communication with timing binary modulation is derived in terms of Fox’s $H$ -function. In the presence of interfering molecules, the mean and variance of the number of the arrived interfering molecules in a given time interval are presented. Using these statistics, a simple mitigation scheme for timing modulation is provided. The results in this paper provide the network performance on the error probability by averaging over a set of random distances between the communicating links as well as a set of random FPTs caused by the anomalous diffusion of molecules. This result will help in designing and developing molecular communication systems for various design purposes. [ABSTRACT FROM AUTHOR]

In this paper, we propose the W-, K-, τ-, U-, V-, and O-functions for the first time. The K series representations for the W-, τ-, U-, V-, and O-functions are discussed. The derivatives, and integral transforms and special cases for the obtained special functions are presented. The anomalous diffusion models via derivative operators associated with the τ- and L-functions are suggested. The obtained results are used to give the series representations for the I-, H-, and G-functions. [ABSTRACT FROM AUTHOR]

In this paper, we study traveling wavefronts in an anomalous diffusion predator–prey model with the modified Leslie–Gower and Holling-type II schemes. We perform a traveling wave analysis to show that the model has heteroclinic trajectories connecting two steady state solutions of the resulting system of fractional partial differential equations and corresponding to traveling wavefronts. This also includes numerical results to show the existence of traveling wavefronts. Furthermore, we obtain the numerical time-dependent solutions in order to show the evolution of wavefronts. We find that wavefronts exist that travel faster in the anomalous subdiffusive regime than in the normal diffusive one. Our results emphasize that the main properties of traveling waves and invasions are altered by anomalous subdiffusion in this model. [ABSTRACT FROM AUTHOR]

In this paper, the anomalous diffusion models are studied in the framework of the scaling-law calculus with the Mandelbrot scaling law. A analytical technology analogous to the Fourier transform is proposed to deal with the one-dimensional scaling-law diffusion equation. The scaling-law series formula via Kohlrausch–Williams–Watts function is efficient and accurate for finding exact solutions for the scaling-law PDEs arising in the Mandelbrot scaling-law phenomena. [ABSTRACT FROM AUTHOR]

Continuous-time random walks (CTRWs) are generic models of anomalous diffusion and fractional dynamics in statistical physics. They are typically defined in the way that their trajectories are discontinuous step functions. In this paper, we propose alternative definition of CTRWs with continuous trajectories. We also give the scaling limit theorem for sequence of such random walks. In general case this result requires the use of strong Skorohod 1 topology instead of Skorohod 1 topology, which is usually used in limit theorems for ordinary CTRW processes. We also give additional conditions under which convergence of sequence of considered random walks holds in the 1 topology. [ABSTRACT FROM AUTHOR]

In this paper we study the properties of a non-markovian version of the telegraph process, whose non-markovian character comes from a nonlocal in-time evolution equation that is satisfied by its probability density function. In the first part of the paper, using the theory of Volterra integral equations, we obtain an explicit formula for its moments, and we prove that the Carleman condition is satisfied. This shows that the distribution of the process is uniquely determined by its moments. We also obtain an explicit formula for the moment generating function. In the second part of the paper, we prove that the distribution of this process coincides with the distribution of a process of the form T (| W (t) |) where T (t) is the classical telegraph process, and | W (t) | is a random time whose distribution is related to a nonlocal in-time version of the wave equation. To this end, we construct the probability density function via subordination from the distribution of the classic telegraph process. Our results exhibit a strong interplay between this type of processes and subdiffusion theory. [ABSTRACT FROM AUTHOR]

Diffusion in narrow curved channels with dead-ends as in extracellular space in the biological tissues, e.g., brain, tumors, muscles, etc. is a geometrically induced complex diffusion and is relevant to different kinds of biological, physical, and chemical systems. In this paper, we study the effects of geometry and confinement on the diffusion process in an elliptical comb-like structure and analyze its statistical properties. The ellipse domain whose boundary has the polar equation ρ (θ) = b 1 − e 2 cos 2 θ with 0 < e < 1, θ ∈ [0,2 π ], and b as a constant, can be obtained through stretched radius r such that Υ = r ρ (θ) with r ∈ [0,1]. We suppose that, for fixed radius r = R, an elliptical motion takes place and is interspersed with a radial motion inward and outward of the ellipse. The probability distribution function (PDF) in the structure and the marginal PDF and mean square displacement (MSD) along the backbone are obtained numerically. The results show that a transient sub-diffusion behavior dominates the process for a time followed by a saturating state. The sub-diffusion regime and saturation threshold are affected by the length of the elliptical channel lateral branch and its curvature. [ABSTRACT FROM AUTHOR]

The application of fractional calculus in the field of kinetic theory begins with questions raised by Bernoulli, Clausius, and Maxwell about the motion of molecules in gases and liquids. Causality, locality, and determinism underly the early work, which led to the development of statistical mechanics by Boltzmann, Gibbs, Enskog, and Chapman. However, memory and nonlocality influence the future course of molecular interactions (e.g., persistence of velocity and inelastic collisions); hence, modifications to the thermodynamic equations of state, the non-equilibrium transport equations, and the dynamics of phase transitions are needed to explain experimental measurements. In these situations, the inclusion of space- and time-fractional derivatives within the context of the continuous time random walk (CTRW) model of diffusion encodes particle jumps and trapping. Thus, we anticipate using fractional calculus to extend the classical equations of diffusion. The solutions obtained illuminate the structure and dynamics of the materials (gases and liquids) at the molecular, mesoscopic, and macroscopic time/length scales. The development of these models requires building connections between kinetic theory, physical chemistry, and applied mathematics. In this paper, we focus on the kinetic theory of gases and liquids, with particular emphasis on descriptions of phase transitions, inter-phase mixing, and the transport of mass, momentum, and energy. As an example, we combine the pressure–temperature phase diagrams of simple molecules with the corresponding anomalous diffusion phase diagram of fractional calculus. The overlap suggests links between sub- and super-diffusion and molecular motion in the liquid and the vapor phases. [ABSTRACT FROM AUTHOR]

The unexpectable variations of the diagnosed disease symptoms are quite often observed during medical diagnosis. In stochastics, such behavior is called "grey swan" or "black swan" as synonyms of sudden, unpredictable change. Evolution of the disease's symptoms is usually described by means of Markov processes, where dependency on process history is neglected. The common expectation is that such processes are Gaussian. It is demonstrated here that medical observation can be described as a Markov process and is non-Gaussian. Presented non-Gaussian processes have "fat tail" probability density distribution (pdf). "Fat tail" permits a slight change of probability density distribution and triggers an unexpectable big variation of the diagnosed parameter. Such "fat tail" solution is delivered by the anomalous diffusion model applied here to describe disease evolution and to explain the possible presence of "swans" mentioned above. The proposed model has been obtained as solution of the Fractal Fokker–Planck equation (FFPE). The paper shows a comparison of the results of the theoretical model of anomalous diffusion with experimental results of clinical studies using bioimpedance measurements in cardiology. This allows us to consider the practical usefulness of the proposed solutions. [ABSTRACT FROM AUTHOR]

In March 2024 Construction and Building Materials published "Water distribution characteristics of capillary absorption in internally cured concrete with superabsorbent polymers", which claims to present a water distribution model characterising the spatiotemporal moisture content evolution in internally cured concrete with superabsorbent polymers during capillary absorption. This discussion, considered a post-publication critique, establishes that the paper's water distribution model conflicts with the paper's capillary absorption tests. While the model is based on the premise of the square-root-of-time behaviour of capillary absorption, the tests do not demonstrate that square-root-of-time behaviour. This critique in addition questions the paper's forthright application of the power-law diffusivity expression with exponent 4, given that its validity for the paper's materials has not been verified. It is shown that different but equivalent diffusivity expressions lead to different moisture diffusivities at lower moisture contents. • This post-publication review formulates concerns on a previously published paper. • The paper's water distribution model and capillary absorption tests are in conflict. • The paper's adoption of an unverified moisture diffusivity expression is doubtful. [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]

The anomalous diffusion in fractal isotropic/anisotropic porous media is characterized by the Hausdorff derivative diffusion model with the varying fractal orders representing the fractal structures in different directions. This paper presents a comprehensive understanding of the Hausdorff derivative diffusion model on the basis of the physical interpretation, the Hausdorff fractal distance and the fundamental solution. The concept of the Hausdorff fractal distance is introduced, which converges to the classical Euclidean distance with the varying orders tending to 1. The fundamental solution of the 3-D Hausdorff fractal derivative diffusion equation is proposed on the basis of the Hausdorff fractal distance. With the help of the properties of the Hausdorff derivative, the Huasdorff diffusion model is also found to be a kind of time-space dependent convection-diffusion equation underlying the anomalous diffusion behavior. [ABSTRACT FROM AUTHOR]

Stochastic processes (SPs) appear in a wide field, such as ecology, biology, chemistry, and computer science. In transport dynamics, deviations from Brownian motion leading to anomalous diffusion (AnDi) are found, including transport mechanisms, cellular organization, signaling, and more. For various reasons, identifying AnDi is still challenging; for example, (i) a system can have different physical processes running simultaneously, (ii) the analysis of the mean-squared displacements (MSDs) of the diffusing particles is used to distinguish between normal diffusion and AnDi. However, MSD calculations are not very informative because different models can yield curves with the same scaling exponent. Recently, proposals have suggested several new approaches. The majority of these are based on the machine learning (ML) revolution. This paper is based on ML algorithms known as the convolutional neural network to classify SPs. To do this, we generated the dataset from published paper codes for 12 SPs. We use a pre-trained model, the ResNet-50, to automatically classify the dataset. Accuracy of 99% has been achieved by running the ResNet-50 model on the dataset. We also show the comparison of the Resnet18 and GoogleNet models with the ResNet-50 model. The ResNet-50 model outperforms these models in terms of classification accuracy. [ABSTRACT FROM AUTHOR]

In the present paper, we propose two methods for tracking molecular species in water layers via two approaches of the diffusion equation with a power-law time-dependent diffusion coefficient. The first approach shows the species densities and the growth of different species via numerical simulation. At the same time, the second approach is built on the fractional diffusion equation with a time-dependent diffusion coefficient in the sense of regularised Caputo fractional derivative. As an illustration, we present here the species densities profiles and track the normal and anomalous growth of five molecular species OH, H2O2, HO2, NO3-, and NO2- via the calculation of the mean square displacement using the two methods. [ABSTRACT FROM AUTHOR]

In this paper, a semi-analytical mathematical model of pressure transient analysis (PTA) for multi-wing fractured vertical well (MWFV) in coalbed methane (CBM) reservoir is proposed, which considers the complexity of porous media by fractal geometry, the anomalous diffusion based on fractional calculus and the stress sensitivity represented by the exponential expression. Then through line source theory, dimensionless transformation, Pedrosa transformation, and other methods, the solution of the bottom hole pressure is obtained. At last, the PTA curve is presented by the Stehfest inversion method, and seven flowing regimes are identified. It can be observed that after introducing fractal geometry and fractional calculus, the PTA curve is quite different from the traditional curve in the slope of the derivative curve. The influences of related parameters are discussed, including mass fractal dimension, anomalous diffusion coefficient, number of hydraulic fractures, fracture angle, and stress sensitivity factor. Relevant results can provide better guidance to understand the CBM production performance in MWFV. [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]