Standard Diffusion Equation Research Articles

On the Accuracy of Transport Spatial Discretization Methods for Diffusive Regimes

An infinite-medium analysis is performed for neutron transport spatial discretization methods in planar geometry. Angular flux solutions of the spatially continuous transport equation, which are driven by a linear (or quadratic) source, are shown to vary linearly (or quadratically) in space and angle; these are used to assess whether the discretized transport equations preserve certain cell-averaged and edge quantities. Each of the continuous angular flux solutions has a scalar flux that satisfies the standard diffusion equation; our analysis predicts whether the transport discretizations yield an accurate diffusion coefficient and (diffusion) spatial differencing scheme. The linear moment–based discretization methods under consideration, which are found to preserve certain features of the linear (or quadratic) infinite-medium angular flux solutions, are the familiar linear discontinuous (LD), lumped linear discontinuous (LLD), and linear characteristic (LC) schemes. The step characteristic scheme, which yields an unphysically large diffusion coefficient, is revisited and shown to possess, for diffusive problems, a solution error that would occur if an unphysical anisotropic scattering term had been included in the starting discretized S N transport equations. The numerical results verify the theoretical predictions and demonstrate the accuracy of the LD, LLD, and LC schemes in highly scattering problems that are optically thick. Our numerical results also illustrate the impact of inaccuracies in the diffusion coefficient on the numerical solutions of eigenvalue problems. The analysis in this paper has practical implications in the choice of spatial schemes used to solve realistic eigenvalue problems.

Analytical expressions for the first passage time distribution and hit distribution in two and three dimensions

The distribution of the time elapsed before a random variable reaches a threshold value for the first time, called the first passage time (FPT) distribution, is a fundamental characteristic of stochastic processes. Here, by solving the standard macroscopic diffusion equation, we derive analytical expressions for the FPT distribution of a diffusing particle hitting a spherical object in two dimensions (2D) and three dimensions (3D) in the course of unrestricted diffusion in open space. In addition, we calculate, analytically, the angular dependence of the FPT, known as the hit distribution. The analytical results are also compared to simulations of the motions of a random walker on a discrete lattice. This topic could be of wide pedagogical interest because the FPT is important not only in physics but also in chemistry, biology, medicine, agriculture, engineering, and finance. Additionally, the central equations often appear in physics and engineering with only trivial variations, making the solution techniques widely applicable.

Modelling of surface reactions and diffusion mediated by bulk diffusion.

We develop a continuum framework applicable to solid-state hydrogen storage, cell biology and other scenarios where the diffusion of a single constituent within a bulk region is coupled via adsorption/desorption to reactions and diffusion on the boundary of the region. We formulate content balances for all relevant constituents and develop thermodynamically consistent constitutive equations. The latter encompass two classes of kinetics for adsorption/desorption and chemical reactions-fast and Marcelin-De Donder, and the second class includes mass action kinetics as a special case. We apply the framework to derive a system consisting of the standard diffusion equation in bulk and FitzHugh-Nagumo type surface reaction-diffusion system of equations on the boundary. We also study the linear stability of a homogeneous steady state in a spherical region and establish sufficient conditions for the occurrence of instabilities driven by surface diffusion. These findings are verified through numerical simulations which reveal that instabilities driven by diffusion lead to the emergence of steady-state spatial patterns from random initial conditions and that bulk diffusion can suppress spatial patterns, in which case temporal oscillations can ensue. We include an extension of our framework that accounts for mechanochemical coupling when the bulk region is occupied by a deformable solid. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Drift phase resolved diffusive radiation belt model: 2. implementation in a case of random electric potential fluctuations

In the first part of this work, we highlighted a drift-diffusion equation capable of resolving the magnetic local time dimension when describing the effects of trapped particle transport on radiation belt intensity. Here, we implement these general considerations in a special case. Specifically, we determine the various transport and diffusion coefficients required to solve the drift-diffusion equation for equatorial electrons drifting in a dipole magnetic field in the presence of a specific model of time-varying electric fields. Random electric potential fluctuations, described as white noise, drive fluctuations of trapped particle drift motion. We also run a numerical experiment that consists of tracking trapped particles’ drift motion. We use the results to illustrate the validity of the drift-diffusion equation by showing agreement in the solutions. Our findings depict how a structure initially localized in magnetic local time generates drift-periodic signatures that progressively dampen with time due to the combined effects of radial and azimuthal diffusions. In other words, we model the transition from a drift-dominated regime, to a diffusion-dominated regime. We also demonstrate that the drift-diffusion equation is equivalent to a standard radial diffusion equation once the distribution function is phase-mixed. The drift-diffusion equation will allow for radiation belt modeling with a better spatiotemporal resolution than radial diffusion models once realistic inputs, including localized transport and diffusion coefficients, are determined.

Numeric Fem’s Solution for Space-Time Diffusion Partial Differential Equations with Caputo–Fabrizion and Riemann–Liouville Fractional Order’s Derivatives

Abstract In this paper, we use the finite element method to solve the fractional space-time diffusion equation over finite fields. This equation is obtained from the standard diffusion equation by replacing the first temporal derivative with the new fractional derivative recently introduced by Caputo and Fabrizion and the second spatial derivative with the Riemann–Liouville fractional derivative. The existence and uniqueness of the numerical solution and the result of error estimation are given. Numerical examples are used to support the theoretical results.

Numerical solution of two‐dimensional nonlinear Riesz space‐fractional reaction–advection–diffusion equation using fast compact implicit integration factor method

AbstractIn the present article, a finite domain is considered to find the numerical solution of a two‐dimensional nonlinear fractional‐order partial differential equation (FPDE) with Riesz space fractional derivative (RSFD). Here two types of FPDE–RSFD are considered, the first one is a two‐dimensional nonlinear Riesz space‐fractional reaction–diffusion equation (RSFRDE) and the second one is a two‐dimensional nonlinear Riesz space‐fractional reaction‐advection‐diffusion equation (RSFRADE). SFRDE is obtained by simply replacing second‐order derivative term of the standard nonlinear diffusion equation by the Riesz fractional derivative of order whereas the SFRADE is obtained by replacing the first‐order and second‐order space derivatives from the standard order advection–dispersion equation with the Riesz fractional derivatives of order . A numerical method is provided to deal with the RSFD with the weighted and shifted Grünwald–Letnikov (WSGD) approximations, for the spatial discretization. The SFRDE and SFRADE are transformed into a system of ordinary differential equations (ODEs), which have been solved using a fast compact implicit integration factor (FcIIF) with nonuniform time meshes. Finally, the demonstration of the validation and effectiveness of the numerical method is given by considering some existing models.

Analysis of 3D transient heat conduction in functionally graded materials using a local semi-analytical space-time collocation scheme

A local semi-analytical space-time collocation scheme, combined the localized space-time method of fundamental solutions (LSTMFS) and transformed function, is proposed for simulating 3D transient heat transfer in functionally graded materials. In this computation, the governing equation of the transient heat transfer in functionally graded materials is reduced to a standard diffusion equation by using transformed function. The transformed equation has a fundamental solution satisfying the governing equation, and thus the LSTMFS is used in estimating the solution of the degraded diffusion equation. As a local semi-analytical meshless method, the proposed LSTMFS has the advantages of a simple mathematical expression, high precision, and easy implementation. In comparison with the conventional space-time method of fundamental solutions, the developed approach forms a sparse linear system and is more suitable for high-dimensional problems and large-scale complex structures. Three benchmark numerical examples are provided to confirm the accuracy and feasibility of the proposed scheme.

An approximate solution for the time-fractional diffusion equation

In this paper, a numerical method based on a finite difference scheme is proposed for solving the time-fractional diffusion equation (TFDE). The TFDE is obtained from the standard diffusion equation by replacing the first-order time derivative with Caputo fractional derivative. At first, we introduce a time discrete scheme. Then, we prove the proposed method is unconditionally stable and the approximate solution converges to the exact solution with order O(Δt2−α)O(Δt2−α), where ΔtΔt is the time step size and αα is the order of Caputo derivative. Finally, some examples are presented to verify the order of convergence and show the application of the present method.

Three Approximations of Numerical Solution's by Finite Element Method for Resolving Space-Time Partial Differential Equations Involving Fractional Derivative's Order

In this paper, we apply to a class of partial differential equation the finite element method when the problem is involving the Riemann-Liouville fractional derivative for time and space variables on a bounded domain with bounded conditions. The studied equation is obtained from the standard time diffusion equation by replacing the first order time derivative by  for 0<<1 and for the second standard order space derivative by  for 1<<2 respectively. The existence of the unique solution is proved by the Lax-Milgram Lemma. We present here three schemes to approximate numerically the time derivative and use the finite element method for the space derivative using the Hadamard finite part integral and the Diethlem's first degree compound quadrature formula, the second approach is based on the link between Riemann-Liouville and Caputo fractional derivative, when the third method was based on the approximation of the Riemann-Liouville by the Grunwald-Letnikov fractional derivative. For the approximation of the space fractional derivative, the finite element method is introduced for all the three approaches. Finally, to check the effectiveness of the three methods, a numerical example was given.

Asymptotic Diffusion Limits of the Multigroup Neutron Transport Equations

In this paper, the standard multigroup neutron diffusion equations are derived as an asymptotic approximation to the multigroup neutron transport equations. The asymptotic analysis employs a scaling that (1) is suggested by the multigroup neutron diffusion equations themselves and (2) generalizes the long-known asymptotic scaling for monoenergetic transport problems. Two other asymptotic scalings of the multigroup transport equations are also considered, both of which lead to a new “group-collapsed” (monoenergetic) “equilibrium” diffusion approximation. The standard multigroup and equilibrium diffusion approximations are shown to preserve certain nonasymptotic properties of the multigroup transport equations. Generalizations of the analyses in this paper, and possible practical applications, are discussed.

Emergent traveling waves in spatially extended system with finite memory of transport

Standard diffusion equation is formalized on the basis of Brownian movement of dispersing species in absence of any persistence in the transport motion of individuals. As a result, this description accounts for instantaneous spreading of dispersing species over an arbitrarily large distance from their original location predicting infinite velocities. This feature appears to be unrealistic, particularly while considering invasion dynamics in biological systems. Therefore, a better description demands the consideration of dispersal with inertia. Here, we investigate the spatiotemporal dynamics of a reaction-transport system including Cattaneo's modification of flux and a source term with cubic nonlinearity, reminiscent of population dynamics or flame propagation models. We show that the spatially extended system admits a traveling wave solution. The presence of a small finite relaxation time of the diffusive flux modifies the speed of the traveling wave, specifically, resulting decay in speed with an increase of relaxation time of the flux. Our investigation reveals a complex interplay of reaction parameters and the finite memory of transport. The observed traveling wave solution reveals that there are conditions inbuilt in the choice of the reaction rate parameters, which governs the evolution of the wave solution. Our analytical predictions are qualitatively in good agreement with the numerically simulated results.

Foundations and interpretations of the pulsed-Townsend experiment

The pulsed-Townsend (PT) experiment is a well known swarm technique used to measure transport properties from a current in an external circuit, the analysis of which is based on the governing equation of continuity. In this paper, the Brambring representation (1964 Z. Phys. 179 532) of the equation of continuity often used to analyse the PT experiment, is shown to be fundamentally flawed when non-conservative processes are operative. The Brambring representation of the continuity equation is not derivable from Boltzmann’s equation and consequently transport properties defined within the framework are not clearly representable in terms of the phase-space distribution function. We present a re-analysis of the PT experiment in terms of the standard diffusion equation which has firm kinetic theory foundations, furnishing an expression for the current measured by the PT experiment in terms of the universal bulk transport coefficients (net ionisation rate, bulk drift velocity and bulk longitudinal diffusion coefficient). Furthermore, a relationship between the transport properties previously extracted from the PT experiment using the Brambring representation, and the universal bulk transport coefficients is presented. The validity of the relationship is tested for two gases Ar and SF6, highlighting also estimates of the differences.

Caputo Fractional Derivative and Quantum-Like Coherence.

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

Time fractional diffusion equation for shipping water events simulation

Shipping water (SW) events occur when waves, usually during extreme weather conditions, overtop the deck of vessels or structures. Since SW events take place independently but close enough in time to interact with one another, their simulation requires strong nonlinear equations to achieve an appropriate precision. To avoid using equations with high complexity and computational cost, we explore the use of the time fractional diffusion equation (TFDE) for modeling SW events. The idea behind this proposal is to keep the simplicity of the widely used standard diffusion equation, but employing a generalized derivative of order α∈[1,2]. This order can be used to describe an intermediate behavior between diffusion and wave propagation. For the time derivative, we adopted the Caputo fractional derivative. To demonstrate that the TFDE is a suitable model to describe SW events, we present its results compared against data sets of water free surface elevation recorded during SW experiments carried out in a wave flume. The best agreement was found for a fractional order derivative between 1.69 and 1.75.

Anomalous fractional diffusion equation for magnetic losses in a ferromagnetic lamination

During a full magnetization cycle and under a collinearity situation, the magnetic losses in a ferromagnetic are observable by plotting the average magnetic flux density or magnetization as a function of the tangent magnetic excitation. This highly frequency-dependent magnetic signature is called hysteresis cycle, and its area is equal to the energy consumed during the magnetization cycle. The physical mechanisms behind this energy conversion are complex as they interfere and take place at different geometrical scales. Microscopic Eddy currents due to domain wall variations play an important role, as well as the macroscopic Eddy currents due to the excitation field time variations and ruled by the magnetic field diffusion equation. From the literature on this topic, researchers have been proposing simulation models to reproduce and understand those complex observations. Even if all these losses contributions are physically interconnected, most of the simulation models available in the literature are based on the magnetic losses separation principle where each contribution is considered separately. Physically, the Weiss domains distribution and movements distort the diffusion process which becomes anomalous. In this article, the standard magnetic field diffusion equation is modified to take into account such anomaly. The first-order time derivation of the magnetic induction diffusion is replaced by a fractional-order time derivation. This change offers flexibility in the simulation scheme as the fractional order can be considered as an additional degree of freedom. By adjusting precisely this order, very accurate simulation results can be obtained on a very broad frequency bandwidth for the prediction of the iron losses in a ferromagnetic material.

Multi-dimensional spectral tau methods for distributed-order fractional diffusion equations

The distributed-order fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking power-law scaling over the whole time-domain. An important application of distributed-order diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate. To broaden the range of applicability of distributed-order fractional diffusion models, efficient numerical methods are needed to solve the model equation. In this work, we develop spectral tau schemes to discretize the fractional diffusion equation with distributed-order fractional derivative in time and Dirichlet boundary conditions. The model solution is expanded in multi-dimensions in terms of Legendre polynomials and the discrete equations are obtained with the tau method. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. The proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth. Our results confirm that nonlocal numerical methods are best suited to discretize distributed-order fractional differential equations as they naturally take the global behavior of the solution into account.

Identification of thermal conductivity for orthotropic FGMs by DT-DRBEM and L-M algorithm

ABSTRACTThe differential transformation dual reciprocity boundary element method (DT-DRBEM) combined with the Levenberg–Marquardt (L-M) algorithm is proposed to identify the thermal conductivity for orthotropic functionally gradient materials (FGMs). The original governing equation of two-dimensional transient heat conduction problem is transformed into a time-independent and recursive form equation by the differential transformation method (DTM). The analog equation method is used to convert the partial differential equation to a standard thermal diffusion equation. Then, the DRBEM is applied to solve the direct problem and the measured temperature is obtained. After that, the L-M algorithm is used to minimize the objective function and recover the desired thermal conductivity. The DT-DRBEM is compared with the finite difference DRBEM (FD-DRBEM). The L-M algorithm is also compared with the conjugate gradient method (CGM). The influences of different initial guesses, number of measurement points and random errors on the inverse results are also investigated. The initial guesses have little effect on the number of iteration. With the increase of measurement points and with the decrease of measurement errors, the results are more accurate.

On Optimal Tempered Lévy Flight Foraging

Optimal random foraging strategy has gained increasing attention. It is shown that L{\'e}vy flight is more efficient compared with the Brownian motion when the targets are sparse. However, standard L{\'e}vy flight generally cannot be followed in practice. In this paper, we assume that each flight of the forager is possibly interrupted by some uncertain factors, such as obstacles on the flight direction, natural enemies in the vision distance, and restrictions in the energy storage for each flight, and introduce the tempered L{\'e}vy distribution $p(l)\sim {\rm e}^{-\rho l}l^{-\mu}$. It is validated by both theoretical analyses and simulation results that a higher searching efficiency can be achieved when a smaller $\rho$ or $\mu$ is chosen. Moreover, by taking the flight time as the waiting time, the master equation of the random searching procedure can be obtained. Interestingly, we build two different types of master equations: one is the standard diffusion equation and the other one is the tempered fractional diffusion equation.

Atrial Rotor Dynamics Under Complex Fractional Order Diffusion.

The mechanisms of atrial fibrillation (AF) are a challenging research topic. The rotor hypothesis states that the AF is sustained by a reentrant wave that propagates around an unexcited core. Cardiac tissue heterogeneities, both structural and cellular, play an important role during fibrillatory dynamics, so that the ionic characteristics of the currents, their spatial distribution and their structural heterogeneity determine the meandering of the rotor. Several studies about rotor dynamics implement the standard diffusion equation. However, this mathematical scheme carries some limitations. It assumes the myocardium as a continuous medium, ignoring, therefore, its discrete and heterogeneous aspects. A computational model integrating both, electrical and structural properties could complement experimental and clinical results. A new mathematical model of the action potential propagation, based on complex fractional order derivatives is presented. The complex derivative order appears of considering the myocardium as discrete-scale invariant fractal. The main aim is to study the role of a myocardial, with fractal characteristics, on atrial fibrillatory dynamics. For this purpose, the degree of structural heterogeneity is described through derivatives of complex order γ = α + jβ. A set of variations for γ is tested. The real part α takes values ranging from 1.1 to 2 and the imaginary part β from 0 to 0.28. Under this scheme, the standard diffusion is recovered when α = 2 and β = 0. The effect of γ on the action potential propagation over an atrial strand is investigated. Rotors are generated in a 2D model of atrial tissue under electrical remodeling due to chronic AF. The results show that the degree of structural heterogeneity, given by γ, modulates the electrophysiological properties and the dynamics of rotor-type reentrant mechanisms. The spatial stability of the rotor and the area of its unexcited core are modulated. As the real part decreases and the imaginary part increases, simulating a higher structural heterogeneity, the vulnerable window to reentrant is increased, as the total meandering of the rotor tip. This in silico study suggests that structural heterogeneity, described by means of complex order derivatives, modulates the stability of rotors and that a wide range of rotor dynamics can be generated.

In Russian

For the first time, it is demonstrated that the methanol diffusion in the H-ZSM-5/Al 2 O 3 catalyst pellet is anomalous and is described by the time-fractional diffusion equation. The regime of the methanol transport is sub-diffusive. The aim of the present work is a description of the experimental data of the methanol transport through the pellet of the zeolite-containing catalyst for olefins synthesis form the methanol based on the solutions of the diffusion equation of the second Fick’s law of diffusion and the time-fractional diffusion equation. In this work, the mesoporous catalyst based on zeolite H-ZSM-5 and alumina with zeolite/alumina ratio 3/1 by mass is used. The methanol transport has been studied using the developed method of mass transfer process investigation in the porous solid media in flow regime. The method is based on the porous sample saturation by a pulse of the adsorbate. The porous sample is installed to the diffusion cell. Adsorbate quantity evolution versus time is chromatographically analyzed. The porous sample is installed to the diffusion cell so that half of its surface is impermeable for adsorbate which allows applying the von Neumann boundary conditions to sole the diffusion equation. The investigation resulted in the obtaining the relative concentration decay versus time on the boundary of the catalyst pellet. The obtained experimental dependences are analyzed on the whole temporal scale using the analytic solution of the diffusion equation based on the second Fick’s law of diffusion. However, the correspondence between the theoretical solution and the experimental data is very poor. The asymptotic analysis of the experimental data in the long-time range linearized in the logarithmic coordinates according to the solution of the standard diffusion equation has demonstrated that the equation for the second Fick’s law of diffusion is inapplicable to a description of the obtained experimental data because the slope of the experimental data is far from the theoretical one which is equal to unity. On the other, an hand analysis of the experimental data in the long-time range linearized in the logarithmic coordinates according to the solution of the time-fractional diffusion equation revealed the high correlation between the theoretical solution and the experimental data. The calculated values of the fractional order and the fractional diffusion coefficient are independent on the experimental conditions. This means that these characteristics are individual for each pair of the porous media and diffusate and may be associated with the methanol adsorption on the active sites on the surface of the catalyst. The fractional order value is lower than unity, which reveals the presence of the sub-diffusive regime of transport, which is slower comparing to the standard diffusion. Based on the analysis of the methanol mass transfer in the pellet of the zeolite-containing catalyst, it is found that the solution of the time-fractional diffusion equation gives good fit to the experimental data comparing to the solution of the standard diffusion equation. The values of the diffusion coefficients and the fractional orders calculated on the long times are equal to the values estimated for the whole temporal range. It is found that the methanol transfer in the catalyst pellet occurs in the slow sub-diffusive regime. The experimental evidence of the presence of an anomalous diffusion is fundamental for the theoretical understanding the mass transfer process and modeling as well as for application during solving the engineering problems.