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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]

In this paper, we introduce an innovative hybrid discrete‐continuum model that provides a comprehensive framework for understanding the intricate interactions between tumor cells, immune cells, and the effects of immunotherapy. This study distinguishes itself by addressing the limitations of traditional diffusion models, which often fail to capture the irregular and nonlocal movements of cells within the complex tumor microenvironment. To overcome these challenges, we employ fractional time derivatives and the fractional Laplacian operator, offering a more accurate representation of anomalous diffusion processes that are critical in cancer dynamics. Our research begins with a rigorous mathematical analysis, where we establish the global existence of a unique mild solution, laying a solid theoretical foundation for the model. A key innovation of our study is the introduction of the “invasion threshold,” a critical parameter inspired by the next‐generation operator used in epidemiological modeling. This threshold provides a powerful tool for determining the existence and stability of equilibrium points, offering deep insights into the conditions that either promote or inhibit tumor growth under immunotherapeutic interventions. By integrating a hybrid discrete‐continuum approach, we capture the individual behavior of each cell, allowing for a more detailed exploration of the dynamic interplay within the tumor microenvironment. This model not only advances our understanding of tumor‐immune interactions but also holds potential for informing more effective therapeutic strategies, making a significant contribution to the field of mathematical oncology. [ABSTRACT FROM AUTHOR]

In this work, a framework for deriving theoretical equations for mean squared displacement (MSD) and fractional Fokker–Planck is developed for any arbitrary rheological model. The obtained general results are then specified for different fractional rheological models. To test the novel equations extracted from our framework and bridge the gap between microrheology and fractional rheological models, microrheology of polystyrene in tetrahydrofuran solutions at several polymer concentrations is measured. By comparing the experimental and theoretical MSDs, we find the fractional rheological parameters and demonstrate for the first time that the polymer concentration regimes can be distinguished using the fractional exponent and relaxation time data because of the existence of a distinct behavior in each regime. We suggest simple approximations for the critical overlap concentration and the shear viscosity of viscoelastic liquidlike solutions. This work provides a more sensitive approach for distinguishing different polymer concentration regimes and measuring the critical overlap concentration and shear viscosity of polymeric solutions, which is useful when conventional rheological characterization methods are unreliable due to the volatility and low viscosity of the samples. [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]

We investigate the diffusion phenomenon of particles in the vicinity of a spherical colloidal particle where particles may be adsorbed/desorbed and react on the surface of the colloidal particle. The mathematical model comprises a generalized diffusion equation to govern bulk dynamics and kinetic equations which can describe non-Debye relaxations and is used for the colloid's surface. For the reaction processes, we also consider the presence of convolution kernels, which offer the flexibility of describing a single process or process with intermediate reactions before forming the final species. Our analysis focuses on analytical and numerical calculations to obtain the particles' behavior on the colloidal particle's surface and to determine how it affects the diffusion of particles around it. The solutions obtained show various behaviors that can be connected to anomalous diffusion phenomena and may be used to describe the ever-richer science of colloidal particles better. [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]

Alzheimer's disease (AD) is a progressive degenerative disorder that causes behavioral changes, cognitive decline, and memory loss. Currently, AD is incurable, and the few available medicines may, at best, improve symptoms or slow down AD progression. One main challenge in drug delivery to the brain is the presence of the blood–brain barrier (BBB), a semi-permeable layer around cerebral capillaries controlling the influx of blood-borne particles into the brain. In this paper, a mathematical model of drug transport to the brain is proposed that incorporates two mechanisms of BBB crossing: transcytosis and diffusion. To account for the structural damage and accumulation of harmful waste in the brain caused by AD, the diffusion is assumed to be anomalous and is modeled using spatial Riemann–Liouville fractional-order derivatives. The model's parameters are taken from published experimental observations of the delivery to mice brains of the orally administered AD drug donepezil hydrochloride. Numerical simulations suggest that drug delivery modalities should depend on the BBB fitness and anomalous diffusion and be tailored to AD severity. These results may inspire novel brain-targeted drug carriers for improved AD therapies. [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]

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]

This contribution considers the time‐fractional subdiffusion with a time‐dependent variable‐order fractional operator of order β(t)$$ \beta (t) $$. It is assumed that β(t)$$ \beta (t) $$ is a piecewise constant function with a finite number of jumps. A proof technique based on the Fourier method and results from constant‐order fractional subdiffusion equations has been designed. This novel approach results in the well‐posedness of the problem. [ABSTRACT FROM AUTHOR]

Fractional calculus is a method to predict processes mathematically. This study uses fractional order models to determine whether starch hydration is governed by Fickian or anomalous diffusion. Native and modified starches are compared and classified based on their diffusive characteristics and the type of diffusion observed. The study aims to adjust the equation of the analytical solution of the diffusion model to study the hydration of both native and modified starches. The fractional order diffusion model is generalized to compare the two models and identify whether anomalous mechanisms exist in native and modified starches. The results show that water absorption by native and modified starch granules is characterized by anomalous diffusion. This is due to the temperature conditions and differences in the chemical potential of the starches. It is verified that the diffusive characteristics of native and modified starches differ under the same hydration conditions. [ABSTRACT FROM AUTHOR]

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian (− Δ) α 2 . Numerical analysis and experiments are provided to study its performance. Our method has the same symbol | ξ | α as the fractional Laplacian (− Δ) α 2 at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This unique feature distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark 1.1). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is O (2 N log ⁡ (2 N)) , and the memory storage is O (N) with N the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems. [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]

We investigate the H-theorem for a class of generalized kinetic equations with fractional time-derivative, hyperbolic term, and nonlinear diffusion. When the H-theorem is satisfied, we demonstrate that different entropic forms may emerge due to the equation's nonlinearity. We obtain the entropy production related to these entropies and show that its form remains invariant. Furthermore, we investigate some behaviors for these equations from both numerical and analytical perspectives, showing a large class of behaviors connected with anomalous diffusion and their effects on entropy. [ABSTRACT FROM AUTHOR]

We establish a practical means for unbiased computation of the marginal probability density function of the dynamics under stochastic resetting. In contrast to conventional dynamics devoid of resetting, the marginal probability density function under resetting may exhibit cusps and, in multi-dimensions, explosions at reset positions, arising from the compelled displacement of mass. Standard numerical techniques, such as kernel density estimation, fall short in accurately reproducing those characteristics due to their inherent smoothing effects. The proposed unbiased estimation formulas are derived using advanced stochastic calculus in their general formulations, yet their implementation in specific problem settings involves only elementary numerical operations, requiring minimal mathematical expertise and marking the very first instance of a numerical method free from bias in this context. We present numerical results throughout to validate the derived estimation formulas and, more broadly, to demonstrate the effectiveness of our approach in accurately capturing the irregularities of the marginal probability density function. [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]

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]

The main objective of this manuscript is to define the concepts of F-(⋏,h)-contraction and (α , η) -Reich type interpolative contraction in the framework of orthogonal F -metric space and prove some fixed point results. Our primary result serves as a cornerstone, from which established findings in the literature emerge as natural consequences. To enhance the clarity of our novel contributions, we furnish a significant example that not only strengthens the innovative findings but also facilitates a deeper understanding of the established theory. The concluding section of our work is dedicated to the application of these results in establishing the existence and uniqueness of a solution for a fractional differential equation of anomalous diffusion. [ABSTRACT FROM AUTHOR]

We study the broadening of initially localized wave packets in a quasi one-dimensional diamond ladder with interacting, spinless fermions. The lattice possesses a flat band causing localization. We place special focus on the transition away from the flat band many-body localized case by adding very weak dispersion. By doing so, we allow propagation of the wave packet on significantly different timescales which causes anomalous diffusion. Due to the temporal separation of dynamic processes, an interaction-induced, prethermal equilibrium becomes apparent. A physical picture of light and heavy modes for this prethermal behavior can be obtained within Born–Oppenheimer approximation via basis transformation of the original Hamiltonian. This reveals a detachment between light, symmetric and heavy, anti-symmetric particle species. We show that the prethermal state is characterized by heavy particles binding together mediated by the light particles. [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]

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]

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]

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]

The measurements, recorded as time series (TS), of urban meteorology, including temperature (T), relative humidity (RH), wind speed (WS), and pollutants (PM10, PM2.5, and CO), in three different geographical morphologies (basin, mountain range, and coast) are analyzed through chaos theory. The parameters calculated at TS, including the Lyapunov exponent (λ > 0), the correlation dimension (DC < 5), Kolmogorov entropy (SK > 0), the Hurst exponent (0.5 < H < 1), Lempel–Ziv complexity (LZ > 0), the loss of information (<ΔI> < 0), and the fractal dimension (D), show that they are chaotic. For the different locations of data recording, CK is constructed, which is a proportion between the sum of the Kolmogorov entropies of urban meteorology and the sum of the Kolmogorov entropies of the pollutants. It is shown that, for the three morphologies studied, the numerical value of the CK quotient is compatible with the values of the exponent α of time t in the expression of anomalous diffusion applied to the diffusive behavior of atmospheric pollutants in basins, mountain ranges, and coasts. Through the Fréchet heavy tail study, it is possible to define, in each morphology, whether urban meteorology or pollutants exert the greatest influence on the diffusion processes. [ABSTRACT FROM AUTHOR]

We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a stochastic process, which consists of a Markov chain whose sojourn times in every state are Mittag-Leffler distributed. The resulting algorithm is able to calculate the solution at conveniently chosen points in the domain with high efficiency. In addition, we present how to generalize this algorithm in order to compute the complete solution. For several large-scale numerical problems, our method showed remarkable performance in both shared-memory and distributed-memory systems, achieving nearly perfect scalability up to 16 , 384 CPU cores. [ABSTRACT FROM AUTHOR]

The interplay of diffusion with phenomena like stochastic adsorption–desorption, absorption, and reaction–diffusion is essential for life and manifests in diverse natural contexts. Many factors must be considered, including geometry, dimensionality, and the interplay of diffusion across bulk and surfaces. To address this complexity, we investigate the diffusion process in heterogeneous media, focusing on non-Markovian diffusion. This process is limited by a surface interaction with the bulk, described by a specific boundary condition relevant to systems such as living cells and biomaterials. The surface can adsorb and desorb particles, and the adsorbed particles may undergo lateral diffusion before returning to the bulk. Different behaviors of the system are identified through analytical and numerical approaches. [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]

Abstract 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.

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]

Hyper-ballistic diffusion is shown to arise from a simple model of microswimmers moving through a porous media while competing for resources. By using a mean-field model where swimmers interact through the local concentration, we show that a non-linear Fokker–Planck equation arises. The solution exhibits hyper-ballistic superdiffusive motion, with a diffusion exponent of four. A microscopic simulation strategy is proposed, which shows excellent agreement with theoretical analysis. [ABSTRACT FROM AUTHOR]

Hydrologically mediated hot moments (HM‐HMs) of transient anomalous diffusion (TAD) denote abrupt shifts in hydraulic conditions that can profoundly influence the dynamics of anomalous diffusion for pollutants within heterogeneous aquifers. How to efficiently model these complex dynamics remains a significant challenge. To bridge this knowledge gap, we propose an innovative model termed "the impulsive, tempered fractional advection‐dispersion equation" (IT‐fADE) to simulate HM‐HMs of TAD. The model is approximated using an L1‐based finite difference solver with unconditional stability and an efficient convergence rate. Application results demonstrate that the IT‐fADE model and its solver successfully capture TAD induced by hydrologically trigged hot phenomena (including hot moments and hot spots) across three distinct aquifers: (a) transient sub‐diffusion arising from sudden shifts in hydraulic gradient within a regional‐scale alluvial aquifer, (b) transient sub‐ or super‐diffusion due to convergent or push‐pull tracer experiments within a local‐scale fractured aquifer, and (c) transient sub‐diffusion likely attributed to multiple‐conduit flow within an intermediate‐scale karst aquifer. The impulsive terms and fractional differential operator integrated into the IT‐fADE aptly capture the ephemeral nature and evolving memory of HM‐HMs of TAD by incorporating multiple stress periods into the model. The sequential HM‐HM model also characterizes breakthrough curves of pollutants as they encounter hydrologically mediated, parallel hot spots. Furthermore, we delve into discussions concerning model parameters, extensions, and comparisons, as well as impulse signals and the propagation of memory within the context of employing IT‐fADE to capture hot phenomena of TAD in aquatic systems. Plain Language Summary: Hydrologically mediated hot moments (HM‐HMs) encompass sudden shifts in water flow conditions in aquifers, exerting substantial influence on the migration of pollutants. To enhance our comprehension and modeling of these occurrences, we propose a mathematical model termed the Impulsive, Tempered Fractional Advection‐Dispersion Equation (IT‐fADE). This model employs specialized mathematical operators to effectively capture the rapid changes caused by HM‐HMs and to efficiently solve the associated equations. Our assessment of the IT‐fADE model across three distinct aquifers demonstrates its aptitude in faithfully depicting the transient anomalous diffusion of pollutants resulting from HM‐HMs. The IT‐fADE model incorporates specific terms and operators that represent the dynamic nature of HM‐HMs and their impact on pollutant mobility. In summary, the IT‐fADE model furnishes a tool for comprehending and modeling the repercussions of hydrologically triggered hot events on pollutant migration in aquifers. This framework may help researchers and scientists better quantify how pollutants will behave in these environments whose conditions can change abruptly. Key Points: An impulsive, tempered fractional model captured transient anomalous diffusion (TAD) due to hydrologically mediated hot moments (HM‐HMs)The impulsive term and fractional differential operator in the new model describe the ephemeral and dynamic nature of HM‐HMs of TADThe sequential HM‐HM model fitted the pollutant breakthrough curves affected by parallel hydrologically mediated hot spots [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]

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]

Random dropout has become a standard regularization technique in artificial neural networks (ANNs), but it is currently unknown whether an analogous mechanism exists in biological neural networks (BioNNs). If it does, its structure is likely to be optimized by hundreds of millions of years of evolution, which may suggest novel dropout strategies in large-scale ANNs. We propose that the brain serotonergic fibers (axons) meet some of the expected criteria because of their ubiquitous presence, stochastic structure, and ability to grow throughout the individuals lifespan. Since the trajectories of serotonergic fibers can be modeled as paths of anomalous diffusion processes, in this proof-of-concept study we investigated a dropout algorithm based on the superdiffusive fractional Brownian motion (FBM). The results demonstrate that serotonergic fibers can potentially implement a dropout-like mechanism in brain tissue, supporting neuroplasticity. They also suggest that mathematical theories of the structure and dynamics of serotonergic fibers can contribute to the design of dropout algorithms in ANNs.

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]

We investigate the diffusion phenomenon of particles in the vicinity of a spherical colloidal particle where particles may be adsorbed/desorbed and react on the surface of the colloidal particle. The mathematical model comprises a generalized diffusion equation to govern bulk dynamics and kinetic equations which can describe non-Debye relaxations and is used for the colloid’s surface. For the reaction processes, we also consider the presence of convolution kernels, which offer the flexibility of describing a single process or process with intermediate reactions before forming the final species. Our analysis focuses on analytical and numerical calculations to obtain the particles’ behavior on the colloidal particle’s surface and to determine how it affects the diffusion of particles around it. The solutions obtained show various behaviors that can be connected to anomalous diffusion phenomena and may be used to describe the ever-richer science of colloidal particles better.

Alzheimer’s disease (AD) is a progressive degenerative disorder that causes behavioral changes, cognitive decline, and memory loss. Currently, AD is incurable, and the few available medicines may, at best, improve symptoms or slow down AD progression. One main challenge in drug delivery to the brain is the presence of the blood–brain barrier (BBB), a semi-permeable layer around cerebral capillaries controlling the influx of blood-borne particles into the brain. In this paper, a mathematical model of drug transport to the brain is proposed that incorporates two mechanisms of BBB crossing: transcytosis and diffusion. To account for the structural damage and accumulation of harmful waste in the brain caused by AD, the diffusion is assumed to be anomalous and is modeled using spatial Riemann–Liouville fractional-order derivatives. The model’s parameters are taken from published experimental observations of the delivery to mice brains of the orally administered AD drug donepezil hydrochloride. Numerical simulations suggest that drug delivery modalities should depend on the BBB fitness and anomalous diffusion and be tailored to AD severity. These results may inspire novel brain-targeted drug carriers for improved AD therapies.

We investigate the H-theorem for a class of generalized kinetic equations with fractional time-derivative, hyperbolic term, and nonlinear diffusion. When the H-theorem is satisfied, we demonstrate that different entropic forms may emerge due to the equation’s nonlinearity. We obtain the entropy production related to these entropies and show that its form remains invariant. Furthermore, we investigate some behaviors for these equations from both numerical and analytical perspectives, showing a large class of behaviors connected with anomalous diffusion and their effects on entropy.