Classical Linear Diffusion Research Articles

NUMERICAL SIMULATION OF FRACTIONAL POWER DIFFUSION BIOSENSORS

The main aim of this paper is to propose new mathematical models for simulation of biosensors and to construct and investigate discrete methods for the efficient solution of the obtained systems of nonlinear PDEs. The classical linear diffusion operators are substituted with nonlocal fractional powers of elliptic operators. The splitting type finite volume scheme is used as a basic template for the introduction of new mathematical models. Therefore the accuracy of the splitting scheme is investigated and compared with the symmetric Crank-Nicolson scheme. The dependence of the approximation error on the regularity of the solution is investigated. Results of computational experiments for different values of fractional parameters are presented and analysed.

Interpreting how nonlinear diffusion affects the fate of bistable populations using a discrete modelling framework.

Understanding whether a population will survive or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction–diffusion equations, where migration is usually represented as a linear diffusion term, and birth–death is represented with a nonlinear source term. While linear diffusion is most commonly employed to study migration, there are several limitations of this approach, such as the inability of linear diffusion-based models to predict a well-defined population front. One way to overcome this is to generalize the constant diffusivity, , to a nonlinear diffusivity function , where is the population density. While the choice of affects long-term survival or extinction of a bistable population, working solely in a continuum framework makes it difficult to understand how the choice of affects survival or extinction. We address this question by working with a discrete simulation model that is easy to interpret. This approach provides clear insight into how the choice of either encourages or suppresses population extinction relative to the classical linear diffusion model.

Randomized nonlinear software-oriented MDS diffusion layers

Abstract MDS diffusion layers are critical components in the design of symmetric ciphers. In this paper, after introducing some new algebraic structures, we provide new MDS matrices over special types of R-modules. With the help of the proposed methodology, we have more flexibility in designing software-oriented diffusion layers. Most notably, we construct randomized and/or nonlinear MDS diffusion layers, based upon the presented theoretical results, and discuss the resistance of the presented diffusion layers against various kinds of cryptanalysis, compared with classical linear diffusion layers.

Analytical solutions to a nonlinear diffusion–advection equation

In this paper, we construct some exact and analytical solutions to a nonlinear diffusion and advection model (Pudasaini in Eng Geol 202: 62–73, 2016) using the Lie symmetry, travelling wave, generalized separation of variables, and boundary layer methods. The model in consideration can be viewed as an extension of viscous Burgers equation, but it describes significantly different physical phenomenon. The nonlinearity in the model is associated with the quadratic diffusion and advection fluxes which are described by the sub-diffusive and sub-advective fluid flow in general porous media and debris material. We also observe that different methods consistently produce similar analytical solutions. This highlights the intrinsic characteristics of the flow of fluid in porous material. The nonlinear diffusion and advection is characterized by a gradually thinning tail that stretches to the rear of the fluid and the evolution of forward advecting frontal bore head, in contrast to the classical linear diffusion and advection. Additionally, we compare solutions for the linear and nonlinear diffusion and advection models highlighting the similarities and differences. The analytical solutions constructed in this paper and the existing high-resolution numerical solution presented previously for the nonlinear diffusion and advection model independently support each other. This implies that the exact and analytical solutions constructed here are physically meaningful and can potentially be applied to calculate the complex nonlinear re-distribution of fluid in porous landscape, and debris and porous materials.

Coexistence states in a cross-diffusion system of a predator–prey model with predator satiation term

In this paper, we study the existence and non-existence of coexistence states for a cross-diffusion system arising from a prey–predator model with a predator satiation term. We use mainly bifurcation methods and a priori bounds to obtain our results. This leads us to study the coexistence region and compare our results with the classical linear diffusion predator–prey model. Our results suggest that when there is no abundance of prey, the predator needs to be a good hunter to survive.

A novel description of fluid flow in porous and debris materials

Based on a generalized two-phase mass flow model (Pudasaini, 2012) as a mixture of solid particles and interstitial fluid, here, we derive a novel dynamical model equation for sub-diffusive and sub-advective fluid flow in general porous media and debris material in which the solid matrix is stationary. We construct some exact analytical solutions to the new model. The complete exact solutions are derived for the full sub-diffusive fluid flows. Solutions for the classical linear diffusion and the new sub-diffusion with quadratic fluxes are compared, and the similarities and differences are discussed. We show that the solution to sub-diffusive fluid flow in porous and debris material is fundamentally different from the diffusive fluid flow. In the sub-diffusive process, the fluid diffuses slowly in time, and thus, the flow (substance) is less spread. Furthermore, we construct some analytical solutions for the full sub-diffusion and sub-advection equation by transforming it into classical diffusion and advection structure. High resolution numerical solutions are presented for the full sub-diffusion and sub-advection model, which is then compared with the solution of the classical diffusion and advection model. Solutions to the sub-diffusion and sub-advection model reveal very special flow behavior, namely, the evolution of forward advecting frontal bore head followed by a gradually thinning tail that stretches to the original rear position of the fluid. However, for the classical diffusion–advection model, the fluid simply advects and diffuses. Moreover, the full sub-diffusion and sub-advection model solutions are presented both for the linear and quadratic drags, which show that the generalized drag plays an important role in generating special form and propagation speed of the sub-diffusion–advection waves. We also show that the long time solution to sub-diffusive and sub-advective fluid flow through porous media is largely independent of the initial fluid profile. These exact, analytical and numerical solutions reveal many essential physical phenomena, and thus may find applications in modeling and simulation of environmental, engineering and industrial fluid flows through general porous media and debris materials.

Turing Instability and Pattern Formation for the Lengyel–Epstein System with Nonlinear Diffusion

In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel--Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; %favors the mechanism of pattern formation with respect to the classical linear diffusion case; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator's one. In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude of the stationary pattern, both in the supercritical and in the subcritical case. Moreover, we compute the complex Ginzburg-Landau equation in the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal modulation of the phase and amplitude of the homogeneous oscillatory solution.

Turing pattern formation in the Brusselator system with nonlinear diffusion

In this work we investigate the effect of density-dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern formation. We study the process of pattern formation both in one-dimensional and two-dimensional spatial domains. Through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations shows the occurrence of a number of different phenomena, including stable supercritical and subcritical Turing patterns with multiple branches of stable solutions leading to hysteresis. Moreover, we consider traveling patterning waves: When the domain size is large, the pattern forms sequentially and traveling wave fronts are the precursors to patterning. We derive the Ginzburg-Landau equation and describe the traveling front enveloping a pattern which invades the domain. We show the emergence of radially symmetric target patterns, and, through a matching procedure, we construct the outer amplitude equation and the inner core solution.

ON A MACROSCOPIC LIMIT OF A KINETIC MODEL OF ALIGNMENT

In the present paper we study the macroscopic limits of a kinetic model for interacting entities (individuals, organisms, cells). The kinetic model is one-dimensional and the entities are characterized by their position and orientation (+/-) with swarming interaction controlled by a sensitivity parameter. The macroscopic limits of the model are considered for solutions close either to the diffusive (isotropic) or to the aligned (swarming) equilibrium states for various values of that parameter. In the former case the classical linear diffusion equation results whereas in the latter a traveling wave solution does both in the zeroth ("Euler") and first ("Navier–Stokes") order of approximation.

Diffusion Impedance in a Thin-Layer Cell: Experimental and Theoretical Study on a Large-Disk Electrode

Using an accurate thin-layer cell, the electrochemical impedance corresponding to the diffusion of ferro-cyanide ions was measured on a large-disk electrode covered by thin films of electrolyte. Experimental impedance diagrams obtained in a large frequency range (100 kHz−5 mHz) were in good agreement with those expected in the case of a spatially restricted linear diffusion impedance. For electrolyte film thicknesses lower than 100 μm, an increase of the low-frequency limit of the real part of the impedance and a decrease of the high-frequency phase angle were however observed experimentally as the electrolyte film thickness decreases. This specific behavior, in disagreement with the classical linear diffusion theory, was explained by the potential distribution existing in the thin-layer cell and was modeled using an original transmission line. In a very low-frequency range (lower than 1 mHz), an additional time constant was also evidenced. In consistency with previous work of Gabrielli et al.1 on microelectrodes, this feature was ascribed to the radial contribution of diffusion processes existing within a thin-layer cell.

Single-particle, particle-pair, and multiparticle dispersion of fluid particles in forced stably stratified turbulence

The dispersion of fluid particles in statistically stationary stably stratified turbulence is studied by means of direct numerical simulations. Due to anisotropy of the flow, horizontal and vertical dispersion show different behavior. Single-particle dispersion in horizontal direction is similar to that in isotropic turbulence for short times, but shows a long-time growth rate proportional to t2.1±0.1, larger than the classical linear diffusion limit. In vertical direction, three successive regimes can be identified: a classical t2-regime, a plateau that scales as N−2, and a diffusion limit where dispersion is proportional to t. By forcing the flow and performing long-time simulations, we are able to observe this last regime, which was predicted but not observed before in stratified turbulence. This diffusive regime is caused by molecular diffusion of the active scalar (density). The mean squared separation of particle pairs (relative dispersion) in vertical direction shows two plateaus that are not present in isotropic turbulence. They can be associated with the characteristic layered structure of the flow. In the long-time limit again a linear regime is found as for single-particle dispersion. Pair dispersion in horizontal direction behaves similar to that in isotropic turbulence except for long times. Finally, the study of multiparticle statistics in stably stratified turbulent flows is reported. The evolution of tetrads gives an impression of the shape of particle clouds. It is found that with increasing stratification, the volume of the tetrads decreases, and they become flatter and more elongated.

Some diffusion equations with finite propagation speed

AbstractWe summarize some of our recent results on diffusion equations with finite speed of propagation. These equations have been introduced to correct the infinite speed of propagation predicted by the classical linear diffusion theory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solid-state diffusion under a large driving force and the sintering of nanosized particles

A nonlinear kinetic law and its corresponding variational principle are developed for solid-state diffusion under a large thermodynamic driving force. These are used to model the sintering of spherical particles. It is demonstrated that the classical linear diffusion law significantly underpredicts the sintering rate for nanosized particles.

Nonlinear diffusion and Nelson-Brown movement

The nonlinear diffusion process, which can be described as the Nelson-Brown motion, is considered. The obtained equation becomes the classical linear diffusion equation for small relaxation times, and for long relaxation times it is transferred into the Schrodinger-like equation. The possible nonequilibrium stationary states are discussed.

The oxygen micro-electrode.

The reduction of oxygen in a watery solution at a noble metal surface by a low voltage was first observed by Danneel in Nernst’s laboratory in 1897. The application of this technique for biological purposes was instigated by Blinks and Skow (1938) in suspensions of plant cells and was developed for use in mammalian tissues by Davies and Brink (1942). These latter authors made a variety of electrodes which included small elec- trodes of 25 gm platinum wire enclosed in glass insulation which protruded a short distance beyond the end of the platinum. They called these ‘recessed’ micro-electrodes and successfully used them for measuring oxygen gradients on the surface of the brain of the cat. This type of electrode corresponded in its behaviour to the law of classical linear diffusion during its initial polarisation and was able to take its oxygen from very small regions. The disadvantage of this system was that the recess of the electrode was relatively long and readings could only be made at intervals of many minutes. However the readings that were obtained had real meaning and indicated that