Generalized Diffusion Equation Research Articles

Analytical Investigations into Anomalous Diffusion Driven by Stress Redistribution Events: Consequences of Lévy Flights

This research is concerned with developing a generalised diffusion equation capable of describing diffusion processes driven by underlying stress-redistributing type events. The work utilises the development of an appropriate continuous time random walk framework as a foundation to consider a new generalised diffusion equation. While previous work has explored the resulting generalised diffusion equation for jump-timings motivated by stick-slip physics, here non-Gaussian probability distributions of the jump displacements are also considered, specifically Lévy flights. This work illuminates several features of the analytic solution to such a generalised diffusion equation using several known properties of the Fox H function. Specifically demonstrated are the temporal behaviour of the resulting position probability density function, and its normalisation. The reduction of the proposed form to expected known solutions upon the insertion of simplifying parameter values, as well as a demonstration of asymptotic behaviours, is undertaken to add confidence to the validity of this equation. This work describes the analytical solution of such a generalised diffusion equation for the first time, and additionally demonstrates the capacity of the Fox H function and its properties in solving and studying generalised Fokker–Planck equations.

Redefined cubic B-spine finite element method for the generalized diffusion equation with delay

Redefined cubic B-spine finite element method for the generalized diffusion equation with delay

Crust Cooling of Soft X-Ray Transients—the Uncertainties of Shallow Heating

Crust cooling of soft X-ray transients has been observed after outbursts, but an additional shallow heating during accretion in outburst is needed to explain the quiescent light curve. However, shallow heating is significantly different between sources and even within one source between different outbursts, and the source of shallow heat is as yet unknown. Using the open source code “dStar” which solves the fully general relativistic heat diffusion equation for the crust, we investigate the effect of magnitude and depth of shallow heating on crust cooling and find that some exceptional sources (Swift J174805.3-244637, MAXI J0556-332 during outburst II and GRO J1750-27) in which shallow heating may be inactive could be explained by a deeper shallow heating mechanism. We compare our results with those from previous works and find that the shallow heating is model dependent. In addition, the effects of mass and radius of a neutron star on shallow heating are studied, and it is shown that the more compact the star, the less shallow heating will be needed to fit the crust cooling light curves.

A UNIFIED THREE-DIMENSIONAL EXTENDED FRACTIONAL ANALYTICAL SOLUTION FOR AIR POLLUTANTS DISPERSION

A novel technique is presented to analytically solve the fractional diffusion equation for non-reactive air pollutants emitted from an elevated continuous source into the air. A generalized methodical solution combining first-order Wentzel–Kramers–Brillouin (WKB) approximation theory and the Sturm–Liouville problem is used to solve the air pollutants’ fractional dispersion equation. Drawing insight from previous analysis, we expanded the initial issue assuming the turbulent flow characteristics appearing in the diffusion process in a non-integer dimensional space. We solved the transformed problem and compared the solutions against data from real experiment. Physical consequences connecting to the conventional generalized diffusion equations are presented. The results indicate that the present solutions are in accordance with those obtained in literature. This report demonstrates that fractional equations can be applied in a practical prediction of pollutant dissemination in a turbulent atmosphere.

The memory function of the generalized diffusion equation of active motion

In this paper, an exact description of the statistical motion of active particles in three dimensions is presented in the framework of a generalized diffusion equation. Such a generalization contemplates a nonlocal, in time and space, connecting (memory) function. This couples the rate of change of the probability density of finding the particle at position [Formula: see text] at time [Formula: see text] with the Laplacian of the probability density at all previous times and to all points in space. Starting from the standard Fokker–Planck-like equation for the probability density of finding an active particle at position [Formula: see text] swimming along the direction [Formula: see text] at time [Formula: see text], we derive in this paper in an exact manner the connecting function that allows a description of active motion in terms of this diffusion equation.

Time and space generalized diffusion equation on graph/networks

Normal and anomalous diffusion are ubiquitous in many complex systems [1]. Here, we define a time and space generalized diffusion equation (GDE), which uses fractional-time derivatives and transformed d-path Laplacian operators on graphs/networks. We find analytically the solution of this equation and prove that it covers the regimes of normal, sub- and superdiffusion as a function of the two parameters of the model. We extend the GDE to consider a system with temporal alternancy of normal and anomalous diffusion which can be observed for instance in the diffusion of proteins along a DNA chain. We perform computational experiments on a one-dimensional system emulating a linear DNA chain. It is shown that a subdiffusive-superdiffusive alternant regime allows the diffusive particle to explore more slowly small regions of the chain with a faster global exploration, than a subdiffusive-subdiffusive regime. Therefore, an alternancy of sliding (subdiffusive) with hopping and intersegmental transfer (superdiffusive) mechanisms show important advances for protein-DNA interactions.

Li–Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

We establish a reduction principle to derive Li–Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li–Yau inequality for positive solutions u to the fractional (in space) heat equation of the form (-Delta )^{beta /2}(log u)le C/t, where beta in (0,2). We also show that this Li–Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li–Yau inequality for diffusion on a complete graph.

Third-order transport coefficients of ion swarms.

Mobility and diffusion coefficients are generally extracted from experimental measurements of ion arrival time distributions using tensors of ranks one and two, i.e., in terms of the diffusion equation that is equivalent to Fick's second law. The theory is extended here to tensors of rank three. It is shown that under customary circumstances, the generalized diffusion equation only contains a single third-order transport coefficient. This equation is used to generate synthetic data for ions moving through a pure gas. The mobility and diffusion coefficients and third-order transport coefficients inferred from these data are compared with values used to simulate the arrival time distribution. Finally, an existing computer program has been modified in order to compute one component of the third-order transport coefficient, and this program has been applied to Li+ in He.

Generalized diffusion equation with nonlocality of space-time: Analytical and numerical analysis

We present a general approach for obtaining the generalized transport equations for weakly nonequilibrium processes with fractional derivatives by using the Liouville equation with fractional derivatives for a system of classical particles and the Zubarev nonequilibrium statistical operator method. A generalized diffusion equation for a system of classical particles in fractional derivatives is obtained for weakly nonequilibrium processes. Based on the non-Markov diffusion equation, taking into account the spatial nonlocality and modeling the generalized coefficient of particle diffusion Dαα′(r,r′;t,t′)=W(t,t′)D̄αα′(r,r′) using fractional calculus, the generalized Cattaneo–Maxwell-type diffusion equation in fractional time and space derivatives is obtained. In the case of a constant diffusion coefficient, analytical and numerical studies of the frequency spectrum for the Cattaneo–Maxwell diffusion equation in fractional time and space derivatives are performed. Numerical calculations of the phase and group velocities with a change in values of characteristic relaxation time, diffusion coefficient, and indices of temporal ξ and spatial α nonlocality are carried out.

Half Way There: Theoretical Considerations for Power Laws and Sticks in Diffusion MRI for Tissue Microstructure

In this article, we consider how differing approaches that characterize biological microstructure with diffusion weighted magnetic resonance imaging intersect. Without geometrical boundary assumptions, there are techniques that make use of power law behavior which can be derived from a generalized diffusion equation or intuited heuristically as a time dependent diffusion process. Alternatively, by treating biological microstructure (e.g., myelinated axons) as an amalgam of stick-like geometrical entities, there are approaches that can be derived utilizing convolution-based methods, such as the spherical means technique. Since data acquisition requires that multiple diffusion weighting sensitization conditions or b-values are sampled, this suggests that implicit mutual information may be contained within each technique. The information intersection becomes most apparent when the power law exponent approaches a value of 12, whereby the functional form of the power law converges with the explicit stick-like geometric structure by way of confluent hypergeometric functions. While a value of 12 is useful for the case of solely impermeable fibers, values that diverge from 12 may also reveal deep connections between approaches, and potentially provide insight into the presence of compartmentation, exchange, and permeability within heterogeneous biological microstructures. All together, these disparate approaches provide a unique opportunity to more completely characterize the biological origins of observed changes to the diffusion attenuated signal.

Anomalous diffusion driven by the redistribution of internal stresses.

This article explores the mathematical description of anomalous diffusion, driven not by thermal fluctuations but by internal stresses. A continuous time random walk framework is outlined in which the waiting times between displacements (jumps), generated by the dynamics of internal stresses, are described by the generalized Γ distribution. The associated generalized diffusion equation is then identified. The solution to this equation is obtained as an integral over an infinite series of Fox H functions. The probability density function is identified as initially non-Gaussian, while at longer timescales Gaussianity is recovered. Likewise, the second moment displays a transient nature, shifting between subdiffusive and diffusive character. The potential application of this mathematical description to the quaking observed in several soft-matter systems is discussed briefly.

Recovering the weight function in distributed order fractional equation from interior measurement

Consider the recovery of the weight function in distributed time-fractional diffusion system using the interior measurement, which arises in some ultra-slow diffusion phenomena. Due to the nonlinear and nonlocal dependance of the measurement data on the weight function, such an inverse problem is novel and important. Based on the regularities of the direct problems for general diffusion equations with distributed time fractional derivative shown in this paper, we establish the theoretical framework for the optimization version of the inverse problem, including existence of the minimizer, the differentiability of the cost functional, as well as the gradient of the cost functional, in suitable functional spaces. Then an iteration process of descent type is realized efficiently for minimizing the regularizing cost functional in terms of the gradient type iteration, with numerical examples showing the validity of the proposed scheme.

Relation between generalized diffusion equations and subordination schemes.

Generalized (non-Markovian) diffusion equations with different memory kernels and subordination schemes based on random time change in the Brownian diffusion process are popular mathematical tools for description of a variety of non-Fickian diffusion processes in physics, biology, and earth sciences. Some of such processes (notably, the fluid limits of continuous time random walks) allow for either kind of description, but other ones do not. In the present work we discuss the conditions under which a generalized diffusion equation does correspond to a subordination scheme, and the conditions under which a subordination scheme does possess the corresponding generalized diffusion equation. Moreover, we discuss examples of random processes for which only one, or both kinds of description are applicable.

Conditional Lie–Bäcklund symmetries, differential constraints and new explicit solutions for the general nonlinear diffusion equations

In this paper, based on the concepts of the conditional Lie–Bäcklund symmetry and the linear determining equations, the general nonlinear diffusion equations is discussed. And the results of classifying are displayed for such equations, it found that these equations preserve the second and third-order conditional Lie–Bäcklund symmetries. As a result, some new explicit solutions of these equations are presented.

Understanding the ethanol and acetone sensing behaviour of CuO thin films through elements of gas diffusion theory

A general gas diffusion equation was used to predict the variation of sensor response with operating temperature and thickness for p-type CuO thin films for ethanol and acetone vapours. Assuming Langmuir-Hinshelwood mechanism and non-linear variation of sensor conductance with gas concentration, the response transients were modelled for a wide concentration range under isothermal conditions and fitted with extended Freundlich and Langmuir isotherms for ethanol and acetone respectively. Maximum responses of 144% and 168% were obtained for 300 ppm ethanol and acetone respectively at 300 °C for 240 nm CuO thin films. For CuO thin films of thickness 120–240 nm, response (S) versus temperature (T) exhibited a bell-shaped dependence with peak maxima of ethanol and acetone varying only slightly. Irreversible type gas response was obtained for a concentration range of 300–50 ppm for both ethanol and acetone vapours wherein response time systematically decreases with an increase in target gas concentration.

Analytical solution of the steady-state atmospheric fractional diffusion equation in a finite domain

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

Numerical and experimental investigation of the thermal and electrical characteristics of a lithium ion cell

In this study, an electrochemical-thermal coupled model was developed to investigate the electrical and thermal behaviors of the commercial NCR18650b Li-ion cell during three different discharge rates. The 1-dimensional electrochemical model consists of a positive electrode, electrolyte, and a negative electrode and employs the related mass and charge transfer equations for both solid and liquid phases predicting the cell's voltage variation. The 3-dimensional thermal model involves a mandrel, an active battery part, and a shell. The thermal model solves the general heat diffusion equation and predicts the temperature variation of the cell. The results show that the predicted temperature-voltage profiles follow the same trend with experimental data and are consistent. The maximum calculated root mean square errors are obtained as 0.11 V for voltage, and 0.96 °C for temperature predictions. On the other hand, the maximum temperature differences within the cell was found to be 0.16 °C, 0.43 °C, and 1.29 °C after the 0.5 C, 1C and 1.5 C rate discharging processes, respectively. Finally, the results from the 3-dimensional thermal model reveal that the type of mandrel affects the temperature variation within the cell. However, the average surface temperature of the cell remains comparable for the investigated C rates.

Correlated continuous-time random walk in a velocity field: Anomalous bifractional crossover.

The diffusion of space-time correlated continuous-time random walk moving in the velocity field, which includes the fluid flowing freely and the fluid flowing through porous media, is investigated in this paper. Results reveal that it presents anomalous diffusion merely caused by space-time correlation in the freely flowing fluid, and the bias from the velocity field only supplies a standard advection, which is verified by the corresponding generalized diffusion equation which includes a standard advection term. However, the diffusion in the fluid flowing through porous media, i.e., the mean squared displacement, can display a bifractional form of which one originates from space-time correlation and the other one originates from dispersive bias caused by sticking of the porous media. The fractional advection term emerging in the corresponding generalized diffusion equation confirms the results. Moreover, the coexistence of correlation and dispersive bias result in crossover phenomenon in-between the diffusive process at an intermediate timescale, but just as the definition of diffusion, the one owning the largest diffusion exponent always prevails at large timescales. However, since the two fractional diffusions originate from a different mechanism, even if it owns the smaller diffusion exponent, that one can dominate the diffusion if it fluctuates much stronger than the other one, which no longer obeys the previous conclusion.

The renormalization group and the diffusion equation

Abstract We study the relationship between the renormalization group and the diffusion equation. We consider the exact renormalization group equation for a scalar field that includes an arbitrary cutoff function and an arbitrary quadratic seed action. As a generalization of the result obtained by Sonoda and Suzuki, we find that the correlation functions of diffused fields with respect to the bare action agree with those of bare fields with respect to the effective action, where the diffused field obeys a generalized diffusion equation determined by the cutoff function and the seed action and agrees with the bare field at the initial time.

Well-posedness for the backward problems in time for general time-fractional diffusion equation

In this article, we consider an evolution partial differential equation with Caputo time-derivative with the zero Dirichlet boundary condition: \partial^{\alpha}_tu + Au = F where 0 < \alpha < 1 and the principal part -A , is a non-symmetric elliptic operator of the second order. Given a source F , we prove the well-posedness for the backward problem in time and our result generalizes the existing results assuming that -A is symmetric. The key is a perturbation argument and the completeness of the generalized eigenfunctions of the elliptic operator A .