Generalized Diffusion Equation Research Articles

Numerical algorithm for a general fractional diffusion equation

In this manuscript, the analytical solution to the general time fractional diffusion equation followed by the regularity analysis of its solution is presented. Further, a hybrid efficient numerical algorithm for a general fractional derivative of order α∈(0,1) is designed. The physical application of the developed approximation formula is employed to design a hybrid numerical algorithm to determine the numerical solution of the corresponding general time fractional diffusion equation. The proposed numerical techniques are subjected to solvability, numerical stability, and convergence analysis. Numerical example is utilized to verify that (3−α)-th order convergence is attained in time which is higher than the existing scheme (Xu et al., 2013). In the spatial direction, the fourth order convergence is obtained utilizing the compact finite difference method. Besides, such a hybrid numerical algorithm is also used in the stochastic model.

A Generalized Neural Diffusion Framework on Graphs

Recent studies reveal the connection between GNNs and the diffusion process, which motivates many diffusion based GNNs to be proposed. However, since these two mechanisms are closely related, one fundamental question naturally arises: Is there a general diffusion framework that can formally unify these GNNs? The answer to this question can not only deepen our understanding of the learning process of GNNs, but also may open a new door to design a broad new class of GNNs. In this paper, we propose a general diffusion equation framework with the fidelity term, which formally establishes the relationship between the diffusion process with more GNNs. Meanwhile, with this framework, we identify one characteristic of graph diffusion networks, i.e., the current neural diffusion process only corresponds to the first-order diffusion equation. However, by an experimental investigation, we show that the labels of high-order neighbors actually appear monophily property, which induces the similarity based on labels among high-order neighbors without requiring the similarity among first-order neighbors. This discovery motives to design a new high-order neighbor-aware diffusion equation, and derive a new type of graph diffusion network (HiD-Net) based on the framework. With the high-order diffusion equation, HiD-Net is more robust against attacks and works on both homophily and heterophily graphs. We not only theoretically analyze the relation between HiD-Net with high-order random walk, but also provide a theoretical convergence guarantee. Extensive experimental results well demonstrate the effectiveness of HiD-Net over state-of-the-art graph diffusion networks.

Anomalous heat transport and universality in macroscopic diffusion models

AbstractAnomalous diffusion is ubiquitous in nature and relevant for a wide range of applications, including energy transport, especially in bio- and nano-technologies. Numerous approaches have been developed to describe it from a microscopic point of view, and recently, it has been framed within universality classes, characterized by the behaviour of the moments and auto-correlation functions of the transported quantities. It is important to investigate whether such universality applies to macroscopic models. Here, the spectrum of the moments of the solutions of the transport equations is investigated for three continuous PDE models featuring anomalous diffusion. In particular, we consider the transport described by: (i) a generalized diffusion equation with time-dependent diffusion coefficient; (ii) the Porous Medium Equation and (iii) the Telegrapher Equation. For each model, the key features of the source-type solution as well as the analytical results for the moment analysis are revisited and extended via both analytical and numerical approaches. Equivalence of the asymptotic behaviour of the corresponding heat transport is confirmed within the realm of weak anomalous diffusion.

Identifying temperature distribution and source term for generalized diffusion equation with arbitrary memory kernel

A diffusion equation involving integral convolution in time variable with arbitrary kernel and nonlocal boundary conditions is considered. The existence and uniqueness results for two inverse problems of determining source terms (space‐ and time‐dependent sources) along with diffusion concentration from appropriate over‐specified conditions are presented. A bi‐orthogonal system of functions is used to have series representation of the solutions of the inverse problems. Several special cases such as standard diffusion, multi‐term diffusion, and tempered diffusion equations are discussed, and some examples are provided.

Feynman-Kac formula for tempered fractional general diffusion equations with nonautonomous external potential

Feynman-Kac formula for tempered fractional general diffusion equations with nonautonomous external potential

On the solvability of direct and inverse problems for a generalized diffusion equation

This paper delves into both direct and two inverse source problems associated with a diffusion equation featuring integral convolution over time, while considering non-classical boundary conditions. The inverse source problems are shown to exhibit ill-posed characteristics in accordance with Hadamard’s definition. A bi-orthogonal function system is employed to express series solutions for the inverse source problems. By imposing specific conditions on the provided data, we establish the existence of unique series solutions. Several special cases of the diffusion equation are presented, depending on the nature of the memory kernel. Furthermore, to illustrate the findings regarding inverse source problems, we provide specific examples.

General rigid bead-rod theory for steady-shear flow

General rigid bead-rod theory yields uniquely the relation between macromolecular architecture and complex viscosity. For this, it relies on the analytical solution of the general diffusion equation for small-amplitude oscillatory shear flow of Bird et al. [Dynamics of Polymeric Liquids, 2nd ed. (Wiley, New York, 1987), Vols. 1–2]. Unfortunately, this general diffusion equation has yet to be solved for any other flow field. In this paper, we do so for steady-shear material functions, namely, viscosity and first normal stress coefficient. We, thus, explain the non-Newtonian behaviors of macromolecular suspensions of any axisymmetric design in steady-shear flow.

Large-amplitude oscillatory shear flow from general rigid bead-rod theory

Oscillatory shear flow, performed at small-amplitude, interrogates polymeric liquids in their equilibrium states. The fluid responds in sinusoidal shear stress waves whose amplitude and phase lead depend on the dimensionless frequency (called the Deborah number). By contrast, this same flow field, performed at large-amplitude, probes departures from the equilibrium state, and the fluid responds with shear stress in the form of a Fourier series, whose component amplitudes and phase leads depend on both the dimensionless frequency (called the Deborah number) and the dimensionless shear rate amplitude (called the Weissenberg number). The physics of these departures from equilibrium in an oscillatory shear flow may be explained by (i) chain disentanglement or (ii) motion along the polymer chain axes (called reptation) or (iii) macromolecular orientation. Of these radically different and yet otherwise equally effective approaches, only (iii) allows the macromolecular structure to be varied arbitrary so that the effect of molecular architecture on the rheology can be explored. Though much has been written about a large-amplitude oscillatory shear flow, we understand little about the role of molecular structure on the measured behaviors, and this has limited its usefulness. In this work, we explain the higher harmonics of both the shear stress (first and third), the first normal stress differences (zeroth, second, and fourth), and the second normal stress differences (zeroth and second) arriving at analytical expressions for all three. These expressions, written in dimensionless form, express the dimensionless rheological responses in large-amplitude oscillatory shear flow in terms of the ratio of the two principal macromolecular moments of inertia. To get these expressions, we derive the first five terms of the orientation distribution function, by solving the general diffusion equation in Euler coordinates. We then integrate in phase space with this orientation result to arrive at our expression for the first seven terms of the polymer contribution to the extra stress tensor. From this tensor, we next write down the Fourier coefficients for the shear stress responses, and the normal stress difference responses, in large-amplitude oscillatory shear flow for a suspension of macromolecules sculpted from a rigid bead-rod structure of any arbitrary axisymmetric shape.

Diffusion Equation Generalized for Modeling of Chladni Patterns

Random walk of particles during Chladni pattern formation is macroscopically treated as a diffusion process. The corresponding generalized diffusion equation is formulated based upon the generator of vibration driven random walk by following Einstein’s treatment of Brownian motion.

Dynamics of stochastic-constrained particles

Prior studies have focused on the overall behavior of randomly moving particle swarms. However, the characteristics of the stochastic-constrained particles that form ubiquitously within these swarms remain oblivious. This study demonstrates a generalized diffusion equation for stochastic-constrained particles that considers the velocity and location aggregation effects observed from their parent particle swarm (i.e., a completely random particle swarm). This equation can be approximated as the form of Schrödinger equation in the microcosmic case (low relative density) and describe the dynamics of the total mass distribution in the macrocosmic case (high relative density). The predicted density distribution of the particle swarm in the stable aggregation state is consistent with the total mass distribution of massive, relaxed galaxy clusters (at least in the range of r<r_{textrm{s}}), preventing cuspy problems in the empirical Navarro–Frenk–White profile. This study opens a window to observe the dynamics of stochastic-constrained particles from a third perspective, from which the aggregation effect of particles without gravitation can be saw.

General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency is systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator is decomposed into the sum of a maximal monotone operator and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper is also applied to the Finsler porous medium and fast diffusion equations, which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation.

On the existence of solutions to a general mean field equation of nonlinear diffusion with the Newtonian potential pressure

On the existence of solutions to a general mean field equation of nonlinear diffusion with the Newtonian potential pressure

A generalised diffusion equation corresponding to continuous time random walks with coupling between the waiting time and jump length distributions

This work outlines a transiently coupled continuous time random walk framework. The coupling is between the displacement probability density function (PDF) and the elapsed waiting time, and is of the form . Coupling of this kind generates larger displacements for longer waiting times, however, decouples on longer timescales. Such coupling is proposed to be physically relevant to systems in which diffusion is driven by the development of internal stresses which release and develop cyclically. This article outlines the associated generalised diffusion equation (GDE), which describes the time evolution of the position PDF, . The solution for is obtained using the properties of the Fox H function, both in terms of its transform properties but also its expansion theorems. The second moment and the asymptotic features of the solution are extracted. The relaxation of back to the solution of a decoupled-type GDE is highlighted.

Сопряженный массоперенос и иммобилизация паров кремния при пропитке пористой среды на основе углеродного волокна

We constructed a conjugate physical and mathematical model that describes a high-temperature mass transfer of silicon vapor from the melt mirror to a porous carbon material, the process of its filtration, and deposition inside the sample. In the outer region, the distribution of vapor is deter-mined by a general nonlinear diffusion equation that takes into account the convective mass trans-fer in the working space of the retort. Inside the porous sample, the behavior of gaseous silicon is described by a system of equations in the framework of the MIM-approach. It is assumed that the silicon deposition mainly depends on the temperature distribution inside the sample. For simplicity, a one-dimensional problem is considered. The boundary conditions are chosen taking into account the significantly different permeabilities of the carrier medium and the porous carbon fiber. The temperature outside the material is assumed to be constant, while inside a non-uniform distribution is maintained. The constructed boundary value problem was solved numerically with the use of the finite difference method. The distributions of silicon vapor outside and inside the sample were ob-tained. It is shown that the non-linearity in the mass transfer equation leads to a distortion of the distribution from the melt mirror to the sample, which indicates a significant contribution of the convective mass transfer to the silicon vapor flow in this region.

Well-Posedness of General Time-Fractional Diffusion Equations Involving Atangana-Baleanu Derivative

Well-Posedness of General Time-Fractional Diffusion Equations Involving Atangana-Baleanu Derivative

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.