Diffusion Differential Equations Research Articles

Electrical Conductivity Relaxation in the Nonlinear Regime

Electrical Conductivity Relaxation (ECR) is an experimental procedure used to assess the chemical diffusion and reaction coefficients (i.e. Dchem and kchem) of mixed ionic electronic conductors (MIECs). The analytical model usually employed to fit the ECR data is based on linear physics. In fact, it is obtained from a linear partial differential equation for the oxygen diffusion with a linearized surface reaction rate as the boundary condition. In this article, we extend the linear approach and develop a novel nonlinear ECR simulation framework. The latter is based on a nonlinear partial differential equation that models the transport of oxygen defects and nonlinear surface reaction rates. Our approach also considers the influence of extrinsic factors, such as the testing chamber's size and the gas-feed flow-rates, on the ECR response. We show that the nonlinear model is able to model ECR well beyond the limits of the typical linear analytical models, i.e., when the final partial pressures inside the testing chamber are low. Further, we also show that, under certain conditions, the exchange of oxygen between the MIEC and the testing chamber can significantly affect the ECR response.

An approximate solution to the transient space-fractional diffusion equation: Integral-balance approach, optimization problems and analyzes

This paper presents approximate analytical solutions of an initial-boundary value problem of fractional partial differential diffusion equation with spatial Riemann-Liouville fractional derivative. The proposed approximate solutions are based on the concept of a finite penetration depth with the integral-balance method and series expansions of the assumed parabolic profile with undefined exponent. Optimization problems referring to optimal exponents of the assumed parabolic have been developed.

Diskretizatsiya differentsialnogo uravneniya konvektsii i diffuzii na osnove metoda kontrolnogo obyema

Diskretizatsiya differentsialnogo uravneniya konvektsii i diffuzii na osnove metoda kontrolnogo obyema

Stochastic differential equation for Brox diffusion

This paper studies the weak and strong solutions to the stochastic differential equation dX(t)=−12Ẇ(X(t))dt+dB(t), where (B(t),t≥0) is a standard Brownian motion and W(x) is a two sided Brownian motion, independent of B. It is shown that the Itô–McKean representation associated with any Brownian motion (independent of W) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term ∫0TẆ(X(t))dt, the main idea is to use the concept of local time together with the polygonal approximation Wπ. Some new results on the local time of Brownian motion needed in our proof are established.

A theory of synchrony for active compartments with delays coupled through bulk diffusion

We extend recent work on the analysis of synchronization in a pair of biochemical oscillators coupled by linear bulk diffusion, in order to explore the effects of discrete delays. More specifically, we consider two well-mixed, identical compartments located at either end of a bounded, one-dimensional domain. The compartments can exchange signaling molecules with the bulk domain, within which the signaling molecules undergo diffusion. The concentration of signaling molecules in each compartment is modeled by a delay differential equation (DDE), while the concentration in the bulk medium is modeled by a partial differential equation (PDE) for diffusion. Coupling in the resulting PDE–DDE system is via flux terms at the boundaries. Using linear stability analysis, numerical simulations and bifurcation analysis, we investigate the effect of diffusion on the onset of a supercritical Hopf bifurcation. The direction of the Hopf bifurcation is determined by numerical simulations and a winding number argument. Near a Hopf bifurcation point, we find that there are oscillations with two possible modes: in-phase and anti-phase. Moreover, the critical delay for oscillations to occur increases with the diffusion coefficient. Our numerical results suggest that the selection of the in-phase or anti-phase oscillation is sensitive to the diffusion coefficient, time delay and coupling strength. For slow diffusion and weak coupling both modes can coexist, while for fast diffusion and strong coupling, only one of the modes is dominant, depending on the explicit choice of DDE.

Parameter-uniform numerical methods for general nonlinear singularly perturbed reaction diffusion problems having a stable reduced solution

A general nonlinear singularly perturbed reaction diffusion differential equation with solutions exhibiting boundary layers is analysed in this paper. The problem is considered as having a stable (attractive) reduced solution that satisfies any one of a comprehensive set of conditions for stable reduced solutions of reaction diffusion problems. A numerical method is presented consisting of a finite difference scheme to be solved over a Shishkin mesh. It is shown that suitable transition points for the Shishkin mesh and the error of the numerical method depend on which stability condition the reduced solution satisfies. Moreover, we show that the error may be affected adversely depending on the stability condition satisfied. Numerical experiments are presented to demonstrate the convergence rate established.

Metabolic modeling of a chronic wound biofilm consortium predicts spatial partitioning of bacterial species.

BackgroundChronic wounds are often colonized by consortia comprised of different bacterial species growing as biofilms on a complex mixture of wound exudate. Bacteria growing in biofilms exhibit phenotypes distinct from planktonic growth, often rendering the application of antibacterial compounds ineffective. Computational modeling represents a complementary tool to experimentation for generating fundamental knowledge and developing more effective treatment strategies for chronic wound biofilm consortia.ResultsWe developed spatiotemporal models to investigate the multispecies metabolism of a biofilm consortium comprised of two common chronic wound isolates: the aerobe Pseudomonas aeruginosa and the facultative anaerobe Staphylococcus aureus. By combining genome-scale metabolic reconstructions with partial differential equations for metabolite diffusion, the models were able to provide both temporal and spatial predictions with genome-scale resolution. The models were used to analyze the metabolic differences between single species and two species biofilms and to demonstrate the tendency of the two bacteria to spatially partition in the multispecies biofilm as observed experimentally. Nutrient gradients imposed by supplying glucose at the bottom and oxygen at the top of the biofilm induced spatial partitioning of the two species, with S. aureus most concentrated in the anaerobic region and P. aeruginosa present only in the aerobic region. The two species system was predicted to support a maximum biofilm thickness much greater than P. aeruginosa alone but slightly less than S. aureus alone, suggesting an antagonistic metabolic effect of P. aeruginosa on S. aureus. When each species was allowed to enhance its growth through consumption of secreted metabolic byproducts assuming identical uptake kinetics, the competitiveness of P. aeruginosa was further reduced due primarily to the more efficient lactate metabolism of S. aureus. Lysis of S. aureus by a small molecule inhibitor secreted from P. aeruginosa and/or P. aeruginosa aerotaxis were predicted to substantially increase P. aeruginosa competitiveness in the aerobic region, consistent with in vitro experimental studies.ConclusionsOur biofilm modeling approach allows the prediction of individual species metabolism and interspecies interactions in both time and space with genome-scale resolution. This study yielded new insights into the multispecies metabolism of a chronic wound biofilm, in particular metabolic factors that may lead to spatial partitioning of the two bacterial species. We believe that P. aeruginosa lysis of S. aureus combined with nutrient competition is a particularly relevant scenario for which model predictions could be tested experimentally.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-016-0334-8) contains supplementary material, which is available to authorized users.

Influence of resonance radiation transfer on ionization balance in a positive column plasma

A method of self-consistent solution of charged particles balance equation, which is described by a differential equation of ambipolar diffusion, and an equation of resonance atom balance, which is described by an integral equation of radiation transfer, is proposed. The method is related to a replacement of an integral operator and a differential operator by a system of linear algebraic equations. The difference between a precise solution and a solution in the approximation of the effective resonance transition probability is shown. The influence of highest diffusion and radiation modes becomes apparent during transition to a contracted state.

Diffusion and lifetime modeling for slurry aluminide coating on P92 at 650 °C with a computational and experimental approach

The present work is devoted to the estimation of the coating lifetime by modeling aluminum diffusion in a slurry‐applied coating on P92 at 650 °C employing the computational and experimental approach. This coating protects ferritic steels from both steam and fireside corrosion on new coal‐fired power plants operating under super‐supercritical steam parameters. The computer model of the diffusion and oxidation based on a numerical solving of the differential equations of diffusion was developed, and its unknown parameters, namely the concentration‐dependent aluminum diffusion coefficient, were identified from relatively short‐term experimental tests up to 8000 h using the inverse problem solution approach. The model was then used for coating lifetime estimation and furthermore for obtaining the dependency of the lifetime on the initial coating thickness and on the critical beneath‐surface aluminum content defined for the estimation. The resulting simulated concentration profiles were compared with experimental data yielding an adequate fit.

A maximum principle for fractional diffusion differential equations

A maximum principle for fractional diffusion differential equations

Dynamics of a delayed diffusive predator–prey model with hyperbolic mortality

This paper is devoted to consider a time-delayed diffusive prey–predator model with hyperbolic mortality. We focus on the impact of time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively, and we investigate the similarities and differences between them. Our conclusions show that when time delay continues to increase and crosses through some critical values, a family of homogenous and inhomogeneous periodic solutions emerge. Particularly, we find the minimum value of time delay, which is often hard to be found. We also consider the nonexistence and existence of steady state solutions to the reaction–diffusion model without time delay.

Difference scheme of highest accuracy order for a singularly perturbed reaction–diffusion equation based on the solution decomposition method

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction–diffusion equation. For this problem, a new approach is developed in order to construct difference schemes whose solutions converge in the maximum norm uniformly with respect to the perturbation parameter e, e ∈ (0, 1] (i.e., e-uniformly) with order of accuracy significantly greater than the ultimate achievable accuracy order for the Richardson method on piecewise uniform grids. The main point of this approach is that uniform grids are used to solve grid subproblems for the regular and singular components of the discrete solution. Using the asymptotic construction technique, a basic difference scheme of the solution decomposition method is constructed that converges e-uniformly in the maximum norm at the rate O(N −2 ln2 N), where N + 1 is the number of nodes in the uniform grids used. The Richardson extrapolation technique on three embedded grids is applied to the basic scheme of the solution decomposition method. As a result, we have constructed the Richardson scheme of the solution decomposition method with highest accuracy order. The solution of this scheme converges e-uniformly in the maximum norm at the rate O(N −6 ln6 N).

Effective moisture diffusivity in wheat kernels during adsorption

Wheat samples with low (4.97–6.08%), normal (10.11–11.04%), and high (20.7–22.72%) initial moisture content (IMC, wet basis) were used to determine the rate of moisture adsorption by the gravimetric method at 10, 20, 25, 30, and 35°C under 65%, 86%, and 100% relative humidity (RH), respectively. A moisture diffusion equation was modified to fit the relationship between the moisture ratio of samples and exposure time. From 65% to 100% RH, the IMC of wheat samples was inversely related to the moisture adsorption rate at temperatures from 10 to 35°C. Moisture adsorption rates of samples increased with increasing temperatures. The moisture adsorption rate of samples with the same IMC increased with increasing RH at a given temperature. A single wheat kernel was considered geometrically to be a finite homogeneous slab shape, and the analytical solution of the partial differential equation for moisture diffusion was given. The effective moisture diffusivity was calculated using the slope method by plotting the experimental data in terms of ln (MR) versus rewetting time. In the range of 10–35°C, the effective moisture adsorption diffusivity of wheat kernels with normal moisture was 1.681×10−8–1.516×10−7m2h−1, and their activation energy was 23.651–28.434kJmol−1. For the same IMC, the effective moisture diffusivity of wheat kernels tended to increase with increasing temperature at a given RH, but decreased with an increase in RH at a given temperature. Activation energy and the pre-exponential factor of Arrhenius equation (D0) tended to decrease with increasing RH. With similar initial moisture content, winter wheat and spring wheat exhibited similar effective moisture diffusivity.

A geometrical multi-scale numerical method for coupled hygro-thermo-mechanical problems in photovoltaic laminates

A comprehensive computational framework based on the finite element method for the simulation of coupled hygro-thermo-mechanical problems in photovoltaic laminates is herein proposed. While the thermo-mechanical problem takes place in the three-dimensional space of the laminate, moisture diffusion occurs in a two-dimensional domain represented by the polymeric layers and by the vertical channel cracks in the solar cells. Therefore, a geometrical multi-scale solution strategy is pursued by solving the partial differential equations governing heat transfer and thermo-elasticity in the three-dimensional space, and the partial differential equation for moisture diffusion in the two dimensional domains. By exploiting a staggered scheme, the thermo-mechanical problem is solved first via a fully implicit solution scheme in space and time, with a specific treatment of the polymeric layers as zero-thickness interfaces whose constitutive response is governed by a novel thermo-visco-elastic cohesive zone model based on fractional calculus. Temperature and relative displacements along the domains where moisture diffusion takes place are then projected to the finite element model of diffusion, coupled with the thermo-mechanical problem by the temperature and crack opening dependent diffusion coefficient. The application of the proposed method to photovoltaic modules pinpoints two important physical aspects: (i) moisture diffusion in humidity freeze tests with a temperature dependent diffusivity is a much slower process than in the case of a constant diffusion coefficient; (ii) channel cracks through Silicon solar cells significantly enhance moisture diffusion and electric degradation, as confirmed by experimental tests.

Spatiotemporal Metabolic Modeling of a Chronic Wound Biofilm Consortium

Abstract: Chronic wounds are often colonized by consortia comprised of different bacterial species growing as biofilms on a complex mixture of wound exudate. The spatial organization of biofilm consortia cause bacteria to exhibit phenotypes distinct from planktonic growth and often render the application of antibacterial compounds ineffective. We developed a spatiotemporal model to analyze the multispecies metabolism of a biofilm consortium comprised of two common chronic wound isolates: the aerobe Pseudomonas aeruginosa and the facultative anaerobe Staphylococcus aureus. By combining genome-scale metabolic reconstructions with partial differential equations for metabolite diffusion, the model was able to provide both temporal and spatial predictions with genome-scale resolution. The model was used to analyze the metabolic differences between single species and multispecies biofilms and to demonstrate the tendency of the two bacteria to spatially partition due to nutrient gradients. The model predicted that S. aureus would dominant throughout the biofilm, especially in the anaerobic region where P. aeruginosa had very low growth rates. Lactate secreted by S. aureus and consumed by both species was predicted to further diminish P. aeruginosa competitiveness due to its lower cell densities. Lysis of S. aureus by inhibitors secreted from P. aeruginosa was predicted to enhance spatial partitioning of the two species and to substantially enhance the competitiveness of P. aeruginosa in partially aerobic regions of the biofilm interior.

Solving Singularly Perturbed Problems Using Multi-quadric/Inverse Multi-quadric Radial Basis Function Method

This work focuses on the implementation of a multi-quadric/inverse multi-quadric (MQ/IMQ) radial basis function method on Singularly Perturbed Problems (SPP). Elliptic equation and convection–diffusion differential equation are solved using MQ/IMQ RBF methods and results are compared with analytical results. Numerical results are computed using stationary approximation and non-stationary approximation for SPP problems. Condition number of the system matrix in both cases is tuned with shape parameters. Maximum error is reduced for small shape parameter although it is depending on the perturbation parameter. Accuracy and CPU time are computed with increasing the number of distinct centres with a given shape parameter also. Better accuracy is obtained in the case of elliptic SPP as compared to convective–diffusion SPP for a very small perturbation parameter.

Analytical Solutions to Bloch NMR Flow Equation in Porous System: Future and Emerging Magnetic Resonance Computational Imaging for Medical and Biomedical Engineering

Background: In most brain tissue architecture, water diffusion is anisotropic or directional and can be treated as a restricted geometry. This requires the specification of diffusion coefficients as function of space coordinate(s) and such a phenomenon is an indication of non-uniform compartmental conditions observed in porous media. Since the diffusion coefficient becomes directional, the imaging process becomes very complex and the analytical solutions to the NMR diffusion tensor equation with variable coefficients for non-uniform tissue composition can open a new field of research in diffusion tensor imaging. Metohod: Detailed analytical solutions to the Bloch NMR flow differential equation in cylindrical and spherical restricted geometries needed in the study of general motion of pathological tissues are presented. The system considered was treated as a rigid body which is fundamental to the formulation of the concept of stress in a viscous fluid (blood flow). We show how the analytical solutions to the Bloch NMR flow equation can provide fundamental computational tools that can stimulate interest for future research at the molecular and atomic scales for conceptualization, characterization and development of emerging magnetic resonance imaging theory and applications. Results and Conclusion: Diffusion tensor imaging is treated theoretically by the analytical solutions to fundamental Bloch NMR equations for Magnetic Resonance Imaging. We derived a differential equation for diffusion in both cylindrical and spherical geometries. The general strategy is to consider the diffusion equation together with a forcing function, which is due to the time dependent magnetic field. First the diffusion equation is considered in terms of the instantaneous change to obtain the spatial and time dependence of magnetization. Then the time dependent magnetization is integrated over time to obtain the steady state solutions for the diffusion tensor equation derived from the Bloch NMR flow equations. It is very interesting to note that the derived experimental parameters are very much related to magnetic resonance imaging parameters where the distribution and nature of axonal injuries can be quantified in terms of refined measurements of tortuosity, permeability, porosity and formation factor. Keywords: Apparent diffusion coefficient, cylindrical and spherical geometries, diffusion tensor, diffusion NMR equation, permeability, porosity, tortuosity.

Smallness of the number of incompatibility loci can facilitate parapatric speciation

We studied the time to speciation by geographic isolation for a species living on two islands connected by infrequent migration. Assumptions were that incompatibility was controlled by a finite number of quantitative loci, and individuals differing in loci of more than some threshold fraction do not mix genetically with each other. We also assumed sexual haploid species, each population being nearly monomorphic, and free recombination between loci for within-population processes. The genetic distance (defined as the fraction of loci differing between populations) followed stochastic processes, which were analyzed by means of stochastic differential equations, diffusion equations, and individual-based simulations. The distance increases by the accumulation of novel mutations but decreases by migration and hybridization. It may converge to a quasi-equilibrium around which it fluctuates thereafter. If the threshold fraction of speciation is controlled, the smallness of the number of incompatibility loci enhanced the magnitude of fluctuation around the quasi-equilibrium and shortened the time to speciation considerably. Novel species were created by mutation accumulation and repeated infrequent migration, and the rate of species creation was the fastest for an intermediate rate of migration. A smaller number of loci increased the optimal migration rate and the species creation rate.

Predictive Modeling of Gas-Phase Ignition of Products of the Thermal Decomposition of Coal

A predictive mathematical model for estimating the fire hazard presented by the interaction of a mixture of the products of the thermal decomposition of coal with atmospheric air at relatively low (up to 500 K) temperatures is developed. Heat and mass transfer processes are described by the differential equations of thermal conductivity and diffusion as well as by the balance equation of the concentrations of the components of the gas mixture. Exothermic reaction in the zone of the oxidation reaction is described by the Arrhenius equation. The limiting temperature of the medium formed by the oxidizing agent and the ignition delay time corresponding to these temperatures are established as a result of a numerical investigation of the integral characteristics of “low-temperature” ignition of the gaseous products of thermal decomposition of a single particle of coal of low dimensions (50–500 μm). Recommendations for reducing the number of uncontrolled ignitions of coal in the interaction with “low-temperature” flows of air under the conditions of industrial production are formulated.

Finite Element Simulation of Shelf Life Prediction of Moisture-Sensitive Crackers in Permeable Packaging under Different Storage Conditions

A finite element method (FEM) approach using COMSOL Multiphysics software was presented to solve unsteady-state moisture diffusion differential equations and predict the shelf life of crackers based on critical moisture content obtained by sensory evaluation. The crackers' moisture contents were given as a function of storage time, environment conditions (temperature and relative humidity) and the package materials. Classical Oswin model was used to fit the moisture sorption data at four different temperatures. Effective moisture diffusivity of crackers as a function of moisture content and temperature were investigated. In addition, the relationship of moisture permeability coefficient and temperature of two selected packaging films, low-density polyethylene (LDPE) and cast polypropylene (CPP) film, was determined based on Arrhenius equation. The crackers were packaged in LDPE and CPP packaging pouches, and stored at four different conditions to simulate actual storage conditions and determine the shelf life experimentally. There was a good agreement between experimental results and finite element predictions for moisture-sensitive crackers subjected to different storage and packaging conditions. Practical Applications This paper describes the shelf life prediction of crackers using a simulation model based on unsteady-state Fickian diffusion of moisture through the food. The temperature effect on the moisture sorption, moisture diffusivity of foods and the moisture permeability coefficient of packaging films are considered. The food product's moisture content was simulated using COMSOL Multiphysics as a function of storage time under different storage conditions and packaging film. The finite element simulation provides a powerful support tool for shelf life estimation and packaging development.