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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]

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]

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]

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]

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]

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 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 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]

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]

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]

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 this paper, a new fractional derivative within the exponential decay kernel is addressed for the first time. A new anomalous diffusion model is proposed to describe the heat-conduction problem. With the use of the Laplace transform, the analytical solution is discussed in detail. The presented result is as an accurate and efficient approach proposed for the heat-conduction problem in the complex phenomena. [ABSTRACT FROM AUTHOR]

Following the paradigm shift in the pharmaceutical industry from batch to continuous production, additional instrumentation and revision of control strategies to optimize material flow throughout the downstream processes are required. Tableting manufacturing is one of the most productive in terms of turnover and investment into new sensor technologies is an important decision-making step. This paper proposes a continuous solution to detect changes in material properties, and a control algorithm to aid in minimizing risk at the end-product line. Some of the sub-processes involved in tableting manufacturing perform changes in powder and liquid mixtures, granulation, density, therefore changing flow conditions of the raw material. Using impedance spectroscopy in a continuous sensing and monitoring context, it is possible to perform online identification of generalized (fractional) order parametric models where the coefficients are correlated to changes in material properties. The model parameters are then included in a self-tuning control gain used in ratio control as part of the local process control loop. The solution proposed here is easy to implement and poses a significant added value to the current state of art in pharmaceutical manufacturing technologies. [ABSTRACT FROM AUTHOR]

In this paper the stretch function resulting from solving the fractional-order Bloch equations using fractional calculus was discussed. This function has promising results to represent diffusion signal decay from MRI images. Conventional analyses of (DWI) measurements resolve the normalized magnetization decay profiles in terms of discrete and mono-exponential components with distinct lifetimes. In complex, heterogeneous biological and biophysical samples such as tissue, multi-exponential decay functions can appear to provide truer representation to normalized magnetization decay profile than the assumption of a mono-exponential decay, but the assumption of multiple discrete components is arbitrary and is often erroneous. Moreover, interactions, between both normalized magnetization and with their environment, can result in complex normalized magnetization decay profiles that represent a continuous distribution of lifetimes. The purpose in this paper is to study different factors that influence the stretch function strength, clarity, and contrast of MRI magnetization signal relaxation by manipulating the anomalous diffusion parameters Δ, δ, Gz, β and μ. of Bloch equations. Through this study, it was found that complex normalized magnetization decay profiles behave like stretch exponential function inside power law. Further developments of this study may be useful in optimizing anomalous diffusion in tissues with neurodegenerative, and ischemic diseases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate general fractional derivatives with a non-singular power-law kernel. The anomalous diffusion models with non-singular power-law kernel are discussed in detail. The results are efficient for modelling the anomalous behaviors within the frameworks of the Riemann-Liouville and Liouville-Caputo general fractional derivatives. [ABSTRACT FROM AUTHOR]

In this paper we address the general fractional calculus of Liouville-Weyl and Liouville- Caputo general fractional derivative types with non-singular power-law kernel for the first time. The Fourier transforms and the anomalous diffusions are discussed in detail. The formulations are adopted to describe complex phenomena of the heat transfer problems. [ABSTRACT FROM AUTHOR]

This paper pays attention to develop a variable-order fractal derivative model for anomalous diffusion. Previous investigations have indicated that the medium structure, fractal dimension or porosity may change with time or space during solute transport processes, results in time or spatial dependent anomalous diffusion phenomena. Hereby, this study makes an attempt to introduce a variable-order fractal derivative diffusion model, in which the index of fractal derivative depends on temporal moment or spatial position, to characterize the previous mentioned anomalous diffusion (or transport) processes. Compared with other models, the main advantages in description and the physical explanation of new model are explored by numerical simulation. Further discussions on the dissimilitude such as computational efficiency, diffusion behavior, and heavy tail phenomena of the new model, and variable-order fractional derivative model are also offered. [ABSTRACT FROM AUTHOR]

The random propagation of molecules in a fluid medium is characterized by the spontaneous diffusion law as well as the interaction between the environment and molecules. In this paper, we embody the anomalous diffusion theory for modeling and analysis in molecular communication. We employ $H$ -diffusion to model a non-Fickian behavior of molecules in diffusive channels. $H$ -diffusion enables us to model anomalous diffusion as the subordinate relationship between self-similar parent and directing processes and their corresponding probability density functions with two $H$ -variates in a unified fashion. In addition, we introduce standard $H$ -diffusion to make a bridge of normal diffusion across well-known anomalous diffusions such as space-time fractional diffusion, Erdélyi-Kober fractional diffusion, grey Brownian motion, fractional Brownian motion, and Brownian motion. We then characterize the statistical properties of uncertainty of the random propagation time of a molecule governed by $H$ -diffusion laws by introducing a general class of molecular noise—called $H$ -noise. Since $H$ -noise can be an algebraic-tailed distribution, we provide a concept of $H$ -noise power using finite logarithm moments based on zero-order statistics. Finally, we develop a unifying framework for error probability analysis in a timing-based molecular communication system with a concept of signal-to-noise power ratio. [ABSTRACT FROM AUTHOR]

This paper is a sequel of the review paper on reaction–diffusion systems with a balance law by Martin and Pierre (1990). Here, we present the results obtained after Martin and Pierre (1990) not only for classical systems but also for systems with either time fractional derivatives or space fractional derivatives accounting for anomalous diffusions. [ABSTRACT FROM AUTHOR]

In the current paper, we present an efficient direct scheme for weakly singular Volterra integro-differential equations arising in the theory of anomalous diffusion. The behavior of the system demonstrating the anomalous diffusion is significant for small times. The method is based on operational matrices of Chebyshev and Legendre polynomials with some techniques to reduce the total errors of the already existing schemes. The proposed scheme converts these equations into a linear system of algebraic equations. The main advantages of the method are high accuracy, simplicity of performing, and low storage requirement. The main focus of this study is to obtain an analytical explicit expression to estimate the error. Numerical results confirm the superiority and applicability of our scheme in comparison with other methods in the literature. [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]

In this article we propose the anomalous diffusion models with respect to monotone increasing functions. The Riesz-type fractional order derivatives operators with respect to power-law function are considered based on the extended work of Riesz. Two models for the anomalous diffusion processes are given to describe the special behaviors in the complex media. [ABSTRACT FROM AUTHOR]

In this article we propose the anomalous diffusion models with respect to monotone increasing functions. The Riesz-type fractional order derivatives operators with respect to power-law function are considered based on the extended work of Riesz. Two models for the anomalous diffusion processes are given to describe the special behaviors in the complex media. [ABSTRACT FROM AUTHOR]

In this paper, an enriched finite element method with fractional basis $$\left[ 1,x^{\alpha }\right] $$ for spatial fractional partial differential equations is proposed to obtain more stable and accurate numerical solutions. For pure fractional diffusion equation without advection, the enriched Galerkin finite element method formulation is demonstrated to simulate the exact solution successfully without any numerical oscillation, which is advantageous compared to the traditional Galerkin finite element method with integer basis $$\left[ 1,x\right] $$ . For fractional advection-diffusion equation, the oscillatory behavior becomes complex due to the introduction of the advection term which can be characterized by a fractional element Peclet number. For the purpose of addressing the more complex numerical oscillation, an enriched Petrov-Galerkin finite element method is developed by using a dimensionless fractional stabilization parameter, which is formulated through a minimization of the residual of the nodal solution. The effectiveness and accuracy of the enriched finite element method are demonstrated by a series of numerical examples of fractional diffusion equation and fractional advection-diffusion equation, including both one-dimensional and two-dimensional, steady-state and time-dependent cases. [ABSTRACT FROM AUTHOR]

In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier–Laplace transform method. We represent solutions in terms of infinite series in Fox H -functions. The fractional and second moments are derived by using Mittag-Leffler functions. We observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives. Generalized uniformly distributed order diffusion equation, as a model for strong anomalous behavior, is analyzed by using Tauberian theorem. Some previously obtained results are special cases of those presented in this paper. [ABSTRACT FROM AUTHOR]

[Display omitted] In long-term induced general anesthesia cases such as those uniquely defined by the ongoing Covid-19 pandemic context, the clearance of hypnotic and analgesic drugs from the body follows anomalous diffusion with afferent drug trapping and escape rates in heterogeneous tissues. Evidence exists that drug molecules have a preference to accumulate in slow acting compartments such as muscle and fat mass volumes. Currently used patient dependent pharmacokinetic models do not take into account anomalous diffusion resulted from heterogeneous drug distribution in the body with time varying clearance rates. This paper proposes a mathematical framework for drug trapping estimation in PK models for estimating optimal drug infusion rates to maintain long-term anesthesia in Covid-19 patients. We also propose a protocol for measuring and calibrating PK models, along with a methodology to minimize blood sample collection. We propose a framework enabling calibration of the models during the follow up of Covid-19 patients undergoing anesthesia during their treatment and recovery period in ICU. The proposed model can be easily updated with incoming information from clinical protocols on blood plasma drug concentration profiles. Already available pharmacokinetic and pharmacodynamic models can be then calibrated based on blood plasma concentration measurements. The proposed calibration methodology allow to minimize risk for potential over-dosing as clearance rates are updated based on direct measurements from the patient. The proposed methodology will reduce the adverse effects related to over-dosing, which allow further increase of the success rate during the recovery period. [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]

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]

Fractional derivatives play a very important role in modeling physical phenomena involving long-range correlation effects. However, they raise challenges of computational cost and memory storage requirements when solved using current well developed numerical methods. In this paper, the differential operator multiplication method is proposed to address the issues by considering a reaction-advection-diffusion equation with a fractional derivative in time. The linear fractional differential equation is transformed into an integer order differential equation by the proposed method, which can fundamentally fix the aforementioned issues for select fractional differential equations. In such a transform, special attention should be paid to the initial conditions for the resulting differential equation of higher integer order. Through numerical experiments, we verify the proposed method for both fractional ordinary differential equations and partial differential equations. [ABSTRACT FROM AUTHOR]

Classical diffusion theory is widely applied in natural science and has made a great achievement. However, the phenomenon of anomalous diffusion in discontinuous media (fractal, porous, etc.) shows that classical diffusion theory is no longer suitable. The differential equations with fractional order have recently been proved to be powerful tools for describing anomalous diffusion. Nevertheless, the analysis methods and numerical methods for fractional differential equations are still in the stage of exploration. In the paper, we consider the Sturm-Liouville problem and the numerical method of a fractional sub-diffusion equation with Dirichlet condition, respectively. We have given the series solution of equation and proved the stability and the convergence of the implicit numerical scheme. It is found that the numerical results are in satisfactory agreement with the analytical solution. Through the robustness analysis, it is also found that the diffusion processes on fractals are more sensitive to the spectral dimension than to the anomalous diffusion exponent. [ABSTRACT FROM AUTHOR]

Traditional diffusion tensor imaging (DTI) maps brain structure by fitting a diffusion model to the magnitude of the electrical signal acquired in magnetic resonance imaging (MRI). Fractional DTI employs anomalous diffusion models to obtain a better fit to real MRI data, which can exhibit anomalous diffusion in both time and space. In this paper, we describe the challenge of developing and employing anisotropic fractional diffusion models for DTI. Since anisotropy is clearly present in the three-dimensional MRI signal response, such models hold great promise for improving brain imaging. We then propose some candidate models, based on stochastic theory. [ABSTRACT FROM AUTHOR]