Diffusion-like Equations Research Articles

Stochastic resetting and linear reaction processes: A continuous time random walk approach

We investigate a diffusion process by considering simultaneously stochastic resetting and linear reaction kinetics in the continuous time random walk approach. We first consider the formalism for a single species and then extend it to multiple species. We perform the analysis by considering a general probability density function for the random walk, allowing us to obtain various behaviors for the waiting time and jumping probability distributions. The behavior of these distributions has implications for the diffusion-like equations which emerge from this approach and can be connected to different fractional operators with singular or nonsingular kernels. We also show that diffusion-like equations can exhibit a large class of behaviors related to different processes, particularly anomalous diffusion.

Generalized solution and eventual smoothness in a logarithmic Keller–Segel system for criminal activities

This paper focuses on a simplified variant of the Short et al. model, which is originally introduced by Rodríguez, and consists of a system of two coupled reaction–diffusion-like equations — one of which models the spatio-temporal evolution of the density of criminals and the other of which describes the dynamics of the attractiveness field. Such model is apparently comparable to the logarithmic Keller–Segel model for aggregation with the signal production and the cell proliferation and death. However, it is surprising that in the two-dimensional setting, the model shares some essential ingredients with the classical logarithmic Keller–Segel model with signal absorption rather than that with signal production, due to its special mechanism of proliferation and death for criminals. Precisely, it indicates that for all reasonably regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution which is akin to that established for the classical logarithmic Keller–Segel system with signal absorption; however, it is different from the generalized framework for the counterpart with signal production. Furthermore, it demonstrates that such generalized solution becomes bounded and smooth at least eventually, and the long-time asymptotic behaviors of such solution are discussed as well.

Uniform Blow-up Rate for Nonlocal Diffusion-like Equations with Nonlocal Nonlinear Source

We present new blow-up results for nonlocal reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of an unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. We first show the existence and uniqueness of the solution to problem relying on contraction mapping fixed point theorem. Then, the comparison principles for problem are established through a standard method. Finally, for the radially symmetric and non-increasing initial data, we give a complete classification in terms of global and single point blow-up according to the parameters. Moreover, the blow-up rates are also determined in each case.

Interplay between continuous dislocations and disclinations in elasto-plasticity

A new elasto-plastic constitutive model is proposed to describe the behaviour of crystalline materials with microstructural defects which account for the interplay between dislocations and disclinations. The defects are defined in terms of the incompatible tensor fields and related to the anholonomic configurations. Based on the formulated free energy imbalance principle, involving the free energy which depends on the torsion of the so-called plastic connection as a measure of the interplay disclinations–dislocations, the non-local diffusion-like evolution equations describing the coupling between dislocations and disclinations are derived.In the case of small distortions, the wedge disclination and edge dislocations problems are defined. The evolution equations for the plastic distortion and disclination tensor are derived starting from the finite distortion case. The initial and boundary value problem concerning the tensile test of a rectangular sheet is solved numerically in the hypothesis that, at the initial moment, a single disclination dipole or edge dislocations describe the heterogeneity of defects.This paper aims to give a comprehensive procedure within the constitutive framework of continuous mechanics, which is able to solve problems related to the presence of defects such as dislocations and disclinations under various aspects, for initial preexisting dislocations or disclinations, as well as problems with pure dislocations or a dislocation–disclination interplay.

Formula omitted] and [formula omitted] formulations of the linear transport equation in heterogeneous media

The aim of the paper is to provide a contribution for the extension to heterogeneous media of the A∞ and AN formulations for the linear transport equation. A new approach is proposed, in which the exponential of the optical length is decomposed in a sum of pure exponential terms, as in a homogeneous medium; in this way, a set of coupled diffusion-like equations is obtained. Different approximations are presented and their validity is discussed.

On radiative transfer using synthetic kernel and simplified spherical harmonics methods in linearly anisotropically scattering media

The Synthetic Kernel (SKN) method is employed to a 3D absorbing, emitting and linearly anisotropically scattering inhomogeneous medium. Standard SKN approximation is applied only to the diffusive components of the radiative transfer equations. An alternative SKN (SKN⁎) method is also derived in full 3-D generality by extending the approximation to the direct wall contributions. Complete sets of boundary conditions for both SKN approaches are rigorously obtained. The simplified spherical harmonics (P2N−1 or SP2N−1) and simplified double spherical harmonics (DPN−1 or SDPN−1) equations for linearly anisotropically scattering homogeneous medium are also derived. Resulting full P2N−1 and DPN−1 (or SP2N−1 and SDPN−1) equations are cast as diagonalized second order coupled diffusion-like equations. By this analysis, it is shown that the SKN method is a high-order approximation, and simply by the selection of full or half range Gauss–Legendre quadratures, SKN⁎ equations become identical to P2N−1 or DPN−1 (or SP2N−1 or SDPN−1) equations. Numerical verification of all methods presented is carried out using a 1D participating isotropic slab medium. The SKN method proves to be more accurate than SKN⁎ approximation, but it is analytically more involved. It is shown that the SKN⁎ with proposed BCs converges with increasing order of approximation, and the BCs are applicable to SPN or SDPN methods.

Synaptic boutons sizes are tuned to best fit their physiological performances

To truly appreciate the myriad of events which relate synaptic function and vesicle dynamics, simulations should be done in a spatially realistic environment. This holds true in particular in order to explain as well the rather astonishing motor patterns which we observed within in vivo recordings which underlie peristaltic contractionsas well as the shape of the EPSPs at different forms of long-term stimulation, presented both here, at a well characterized synapse, the neuromuscular junction (NMJ) of the Drosophila larva (c.f. Figure ​Figure1).1). To this end, we have employed a reductionist approach and generated three dimensional models of single presynaptic boutons at the Drosophila larval NMJ. Vesicle dynamics are described by diffusion-like partial differential equations which are solved numerically on unstructured grids using the uG platform. In our model we varied parameters such as bouton-size, vesicle output probability (Po), stimulation frequency and number of synapses, to observe how altering these parameters effected bouton function. Hence we demonstrate that the morphologic and physiologic specialization maybe a convergent evolutionary adaptation to regulate the trade off between sustained, low output, and short term, high output, synaptic signals. There seems to be a biologically meaningful explanation for the co-existence of the two different bouton types as previously observed at the NMJ (characterized especially by the relation between size and Po), the assigning of two different tasks with respect to short- and long-time behaviour could allow for an optimized interplay of different synapse types. We can present astonishing similar results of experimental and simulation data which could be gained in particular without any data fitting, however based only on biophysical values which could be taken from different experimental results. As a side product, we demonstrate how advanced methods from numerical mathematics could help in future to resolve also other difficult experimental neurobiological issues. Figure 1 Simulation of a bouton of the Drosophila NMJ

Some properties of the Green's function of simplified elastodynamic problems

It is now widely accepted that resulting displacement field within elastic, inhomogeneous, anisotropic solids subjected to equipartitioned, uniform illumination from uncorrelated sources, has intensities that follow diffusion-like equations. Typically, coda waves are invoked to illustrate this concept. These waves arrive later as a consequence of multiple scattering and appear at the tail (coda, in Latin) of seismograms and are usually considered an example of diffuse field. It has been demonstrated that average correlations of motions within a diffuse field, in frequency domain, is proportional to imaginary part of Green`s function tensor. If only one station is available, average autocorrelation is equal to average squared amplitudes or average power spectrum and this gives Green`s function at source itself. Several works address this point from theoretical and experimental point of view. However, a complete and explicit analytical description is lacking. In this work we study analytically some properties of Green`s function, specifically imaginary part of Green`s function for 2D antiplane problems. This choice is guided by fact that these scalar problems have a closed analytical solution (Kausel 2006). We assume diffusiveness of field and explore its analytical consequences.

Continuous Time Random Walk and different diffusive regimes

We investigate how it is possible to obtain different diffusive regimes from the Continuous Time Random Walk (CTRW) approach performing suitable changes for the waiting time and jumping distributions in order to get two or more regimes for the same diffusive process. We also obtain diffusion-like equations related to these processes and investigate the connection of the results with anomalous diffusion.

A theory for microtremor H/V spectral ratio: application for a layered medium

Microtremors are produced by multiple random sources close to the surface of the Earth. They may include the effects of multiple scattering, which suggests that their intensities could be well described by diffusion-like equations. Within this theoretical framework, the average autocorrelation of the motions at a given receiver, in the frequency domain, measures average energy density and is proportional to the imaginary part of the Green's function (GF) when both source and receiver are the same. Assuming the seismic field is diffuse we compute the H/V ratio for a surface receiver on a horizontally layered medium in terms of the imaginary part of the GF at the source. This theory links average energy densities with the GF in 3-D and considers the H/V ratio as an intrinsic property of the medium. Therefore, our approach naturally allows for the inversion of H/V, the well-known Nakamura's ratio including the contributions of Rayleigh, Love and body waves. Broad-band noise records at Texcoco, a soft soil site near Mexico City, are studied and interpreted using this theory.

Effects of thermal motion on electromagnetically induced absorption

We describe the effect of thermal motion and buffer-gas collisions on a four-level closed N system interacting with strong pump(s) and a weak probe. This is the simplest system that experiences electromagnetically induced absorption (EIA) due to transfer of coherence via spontaneous emission from the excited to ground state. We investigate the influence of Doppler broadening, velocity-changing collisions (VCC), and phase-changing collisions (PCC) with a buffer gas on the EIA spectrum of optically active atoms. In addition to exact expressions, we present an approximate solution for the probe absorption spectrum, which provides physical insight into the behavior of the EIA peak due to VCC, PCC, and wave-vector difference between the pump and probe beams. VCC are shown to produce a wide pedestal at the base of the EIA peak, which is scarcely affected by the pump-probe angular deviation, whereas the sharp central EIA peak becomes weaker and broader due to the residual Doppler-Dicke effect. Using diffusion-like equations for the atomic coherences and populations, we construct a spatial-frequency filter for a spatially structured probe beam and show that Ramsey narrowing of the EIA peak is obtained for beams of finite width.

Stopping Criteria for Anisotropic PDEs in Image Processing

A number of nonlinear diffusion-like equations have been proposed for filtering noise, removing blurring and other applications. These equations are usually developed as time independent equations. An artificial time is introduced to change these equations to parabolic type equations which are then marched to a steady state. In practice the time iteration is stopped before the steady state is reached. The time when to stop the iteration is usually determined manually for each case. In this study we develop a more automatic procedure for stopping the time integration.

Optimum Interparticle Porosity for Charge Storage in a Packed Bed of Nanoporous Particles

Analytical and numerical methods are used to investigate ion transport and storage within the pore network of a porous bed of nanoporous particles. Under simplifying assumptions that electromigration dominates ion transport, that the pore capacitance is constant, and that the electrolyte exhibits ideal behavior, transport within this hierarchical pore network is modeled using a coupled pair of one-dimensional equations describing net charge motion into the bed by way of the interparticle void and subsequently into smaller pores within particles. These diffusion-like equations are solved analytically via Laplace transforms, and the results are used to determine the optimum interparticle porosity yielding the maximum energy deliverable in a specified time. Both step and periodic boundary conditions are considered, and closed-form expressions for the optimum porosity are derived for the periodic case. We find that the optimum porosity is remarkably similar for the periodic and step boundary conditions when the normalized bed thickness is very large or very small. We also find that the optimum porosity increases at least linearly with bed thickness when the thickness is small but is independent of both the thickness and prescribed time for delivery when the bed thickness is large. Sample results are presented and discussed.

Dynamic correlations in an orderedc(2×2)lattice gas

We obtain the dynamic correlation function of two-dimensional lattice gas with nearest-neighbor repulsion in ordered $c(2\ifmmode\times\else\texttimes\fi{}2)$ phase (antiferromagnetic ordering) under the condition of low concentration of structural defects. It is shown that displacements of defects of the ordered state are responsible for the particle number fluctuations in the probe area. The corresponding set of kinetic equations is derived and solved in linear approximation on the defect concentration. Three types of strongly correlated complex jumps are considered and their contribution to fluctuations is analyzed. These are jumps of excess particles, vacancies and flip-flop jumps. The kinetic approach is more general than the one based on diffusionlike equations used in our previous papers. Thus, it becomes possible to adequately describe correlations of fluctuations at small times, where our previous theory fails to give correct results. Our analytical results for fluctuations of particle number in the probe area agree well with those obtained by Monte Carlo simulations.

Transport via mass transportation

Weak topology implicit schemes based on Monge-Kantorovich or Wasserstein metrics have become prominent for their ability to solve a variety of diffusion and diffusion-like equations. They are very flexible, encompassing a wide range of nonlinear effects. They have interesting interpretations as descent algorithms in an infinite dimensional manifold setting or as dissipation principles for motion in a highly viscous environment. Transport plays a fundamental role in these schemes, as noted by Brenier and Benamou and reviewed below. The reverse implication is less explored and, at least at the outset, less obvious. Here we discuss the simplest situations in the context of systems of transport equations. We show how arbitrary Fokker-Planck Equations in one dimension conform to the mass transport paradigm. Finally, we provide some additional examples, including a simple existence result for velocity-jump processes.

Time-dependent crack behavior in an integrated structure

Devices in modern technologies often have complex architectures, dissimilar materials, and small features. Their long-term reliability relates to inelastic, time-dependent mechanical behavior of such structures. This paper analyzes a three-layer structure consisting of, from top to bottom, an elastic film, a power-law creep underlayer, and a rigid substrate. The layers are bonded. Initially, the film is subject to a uniform biaxial tensile stress. A channel crack is introduced in the elastic film. As the underlayer creeps, the stress field in the film relaxes in the crack wake, but intensifies around the crack tip. We formulate nonlinear diffusion-like equations that evolve the displacement field. When the crack is stationary, the region in which the stress field relaxes increases with time. We identify the length scale of the region as a function of time. The stress intensity factor is proportional to the square-root of the length scale. For the power-law creep underlayer, this newly identified length depends on the film stress, and corrects an error in a previous paper by Huang, Prevost and Suo (Acta Materialia 50, 4137, 2002). When the crack advances, its velocity can reach a steady state. We identify the scaling law for the steady velocity. An extended finite element method (X-FEM) is used to simultaneously evolve the creep strain and crack length. Numerical results are presented for the stress intensity factors of stationary cracks, and the steady velocities of advancing cracks.

Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations.

We propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, cannot be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish their relation to the continuous-time random walk theory. We show that the distributed-order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the Wiener process and whose diffusion exponent decreases in time (retarding subdiffusion). This process may lead to superslow diffusion, with the mean square displacement growing logarithmically in time. We also demonstrate that the distributed-order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time (accelerating superdiffusion).

Meshless method for linear elastostatics based on a path integral formulation

A Path Integral (PI) formulation of linear elastostatics is first presented. For this, Navier equations are modified by adding a fictitious ‘time’ derivative of displacements so that equilibrium corresponds to the steady state of the resulting diffusion-like equations. The evolution of displacement is then represented as the propagation, through the fictitious time co-ordinate, of an initial displacement field corresponding to the unloaded state. The resulting procedure somehow mimics the well-known Feynman path integral of quantum mechanics, which is equivalent to the differential formulation embodied in Schrödinger equation. However, the path integral for elastostatics is formulated in terms of infinitesimal propagators of local support. In its simplest form, the formulation can be used as a relaxation method of solution, by updating displacements until convergence. This may be advantageous for problems involving a very large number of unknowns. On the other hand, by equating the updated displacement field to the actual one a direct method of solution is obtained, which leads to non-symmetric (but sparse and banded) discrete equations. Unlike variational principles this formulation does not require integration over the whole domain, effectively eliminating the need of a background mesh for integration. Also, it only requires continuity of the displacement field on the propagator's support. As a consequence, the formulation lends itself to very flexible meshless implementations. To demonstrate this we describe a simple numerical method in which displacements around each node are approximated by quadratic bivariate polynomials, which is the simplest approximation technique. The feasibility of the method is assessed through a number of numerical examples and comparisons with analytical solutions and other meshless methods. Copyright © 2000 John Wiley & Sons, Ltd.

The asymptotic behaviour of Ramanujan's integral and its application to two-dimensional diffusion-like equations

The large-time behaviour of a large class of solutions to the two-dimensional linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrödinger's equation. We examine the asymptotic expansion of Ramanujan's integral for large values of its argument over the whole complex plane by considering the analytic continuation of Ramanujan's integral to the left half-plane. The resulting expansions are compared to accurate numerical computations of the integral. The large-time behaviour derived from Ramanujan's integral of the solution to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resulting form of the solution combines the inverse logarithmic decay in time typical of Ramanujan's integral with spatial dependence on the usual similarity variable for the diffusion equation.

99/03161 On a rigorous resolution of the transport equation into a system of diffusion-like equations

99/03161 On a rigorous resolution of the transport equation into a system of diffusion-like equations