Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative and the second order space derivative by the symmetric fractional derivative. Firstly, a method of separating variables is used to express the analytical solution of the tss-fde in terms of the Mittag–Leffler function. Secondly, we propose two numerical methods to approximate the Caputo time fractional derivative, namely, the finite difference method and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results are presented to demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Modeling the stepped potential discharge of primary alkaline battery cathodes

A novel model for the potentiostatic discharge of primary alkaline battery cathodes is presented. The model is used to simulate discharges resulting from the stepped potential electrochemical spectroscopy (SPECS) of primary alkaline battery cathodes cathodes, and the results are validated with experimental data. We show that a model based on a single (or mean) reaction framework can be used to simulate multi-reaction discharge behaviour and we develop a consistent functional modification to the kinetic equation of the model that allows for this to occur. The model is used to investigate the effects that the initial exchange current density, i00, and the diffusion coefficient for protons in electrolytic manganese dioxide (EMD), DH+, have on SPECS discharge. The behaviour observed is consistent with the idea that individual reduction reactions, within the multi-reaction, reduction behaviour of EMD, have distinct i00 and DH+ values.

Nonlinear diffusion and exclusion processes with contact interactions

Exclusion processes on a regular lattice are used to model many biological and physical systems at a discrete level. The average properties of an exclusion process may be described by a continuum model given by a partial differential equation. We combine a general class of contact interactions with an exclusion process. We determine that many different types of contact interactions at the agent-level always give rise to a nonlinear diffusion equation, with a vast variety of diffusion functions D(C). We find that these functions may be dependent on the chosen lattice and the defined neighborhood of the contact interactions. Mild to moderate contact interaction strength generally results in good agreement between discrete and continuum models, while strong interactions often show discrepancies between the two, particularly when D(C) takes on negative values. We present a measure to predict the goodness of fit between the discrete and continuous model, and thus the validity of the continuum description of a motile, contact-interacting population of agents. This work has implications for modeling cell motility and interpreting cell motility assays, giving the ability to incorporate biologically realistic cell-cell interactions and develop global measures of discrete microscopic data.

Predictions of road traffic noise on residential balconies using a specular and diffusion model

A combined specular reflection and diffusion model using the radiosity technique was developed to calculate road traffic noise level on residential balconies. The model is capable of numerous geometrical configurations for a single balcony situated in the centre of a street canyon. The geometry of the balcony and the street can be altered with width,length and height. The model was used to calculate for three different geometrical and acoustic absorption characteristics for a balcony. The calculated results are presented in this paper.

Models of collective cell spreading with variable cell aspect ratio : a motivation for degenerate diffusion models

Continuum diffusion models are often used to represent the collective motion of cell populations. Most previous studies have simply used linear diffusion to represent collective cell spreading, while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles. In the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations. Furthermore, there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate. Here we provide a link between individual-based and continuum models using a multi-scale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice-based exclusion process. For round agents that occupy a single lattice site, we find that the relevant continuum description of the system is a linear diffusion equation, whereas for elongated rod-shaped agents that occupy L adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation (pme). The exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents. Our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the pme to represent cell density profiles. Results suggest that when using continuum models to represent cell population spreading, we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models.

Digital processing of diffusion-tensor images of avascular tissues

Diffusion is the process that leads to the mixing of substances as a result of spontaneous and random thermal motion of individual atoms and molecules. It was first detected by the English botanist Robert Brown in 1827, and the phenomenon became known as ‘Brownian motion’. More specifically, the motion observed by Brown was translational diffusion – thermal motion resulting in random variations of the position of a molecule. This type of motion was given a correct theoretical interpretation in 1905 by Albert Einstein, who derived the relationship between temperature, the viscosity of the medium, the size of the diffusing molecule, and its diffusion coefficient. It is translational diffusion that is indirectly observed in MR diffusion-tensor imaging (DTI). The relationship obtained by Einstein provides the physical basis for using translational diffusion to probe the microscopic environment surrounding the molecule.

Diffusion of business improvement processes in manufacturing companies in India

Manufacturing organisations spend more on Business Process Improvement initiatives to make them more competitive in growing global market. This paper presents a Rapid Improvement Workshop (RIW) framework which companies can used to identify the critical factors regulating the diffusion of business process improvement in their company. The framework can then be used address how process improvement can be efficiently implemented. We use the results from case studies at Caterpillar India. The paper identifies the critical factors that contribute to the successful implementation of process improvement programs in manufacturing organisations. We further identify certain technological and cultural barriers to the implementation of process improvement programs and how Indian manufacturing companies can overcome these barriers to attain competitive advantage in the global markets.

Distinguishing forced versus discretionary replacements in new product diffusion models

In many product categories of durable goods such as TV, PC, and DVD player, the largest component of sales is generated by consumers replacing existing units. Aggregate sales models proposed by diffusion of innovation researchers for the replacement component of sales have incorporated several different replacement distributions such as Rayleigh, Weibull, Truncated Normal and Gamma. Although these alternative replacement distributions have been tested using both time series sales data and individual-level actuarial “life-tables” of replacement ages, there is no census on which distributions are more appropriate to model replacement behaviour. In the current study we are motivated to develop a new “modified gamma” distribution by two reasons. First we recognise that replacements have two fundamentally different drivers – those forced by failure and early, discretionary replacements. The replacement distribution for each of these drivers is expected to be quite different. Second, we observed a poor fit of other distributions to out empirical data. We conducted a survey of 8,077 households to empirically examine models of replacement sales for six electronic consumer durables – TVs, VCRs, DVD players, digital cameras, personal and notebook computers. This data allows us to construct individual-level “life-tables” for replacement ages. We demonstrate the new modified gamma model fits the empirical data better than existing models for all six products using both a primary and a hold-out sample.

An analytical method to solve a general class of nonlinear reactive transport models

Three recent papers published in Chemical Engineering Journal studied the solution of a model of diffusion and nonlinear reaction using three different methods. Two of these studies obtained series solutions using specialized mathematical methods, known as the Adomian decomposition method and the homotopy analysis method. Subsequently it was shown that the solution of the same particular model could be written in terms of a transcendental function called Gauss’ hypergeometric function. These three previous approaches focused on one particular reactive transport model. This particular model ignored advective transport and considered one specific reaction term only. Here we generalize these previous approaches and develop an exact analytical solution for a general class of steady state reactive transport models that incorporate (i) combined advective and diffusive transport, and (ii) any sufficiently differentiable reaction term R(C). The new solution is a convergent Maclaurin series. The Maclaurin series solution can be derived without any specialized mathematical methods nor does it necessarily involve the computation of any transcendental function. Applying the Maclaurin series solution to certain case studies shows that the previously published solutions are particular cases of the more general solution outlined here. We also demonstrate the accuracy of the Maclaurin series solution by comparing with numerical solutions for particular cases.

Reinforced hydrogels for silicone copolymer delivery for scar remediation

Hypertrophic scars are formed by collagen overproduction in wounded areas and often occur in victims of severe burns. There are several methods for hypertrophic scar remediation and silicone gel therapy is one of the more successful methods. Research by others has shown that the activity of these gels may be due to migration of amphiphilic silicone oligomers from the gel and into the dermis, down-regulating production of collagen by fibroblasts. Normal silicone oil (PDMS) does not produce the same effect on fibroblasts. The main purpose of this project is the introduction of a particular amphiphilic silicone rake copolymer into an appropriate network which can absorb and release the silicone copolymer on the scarred area. Hydrogels are polymeric crosslinked networks which can swell in water or a drug solution, and gradually release the drug when applied to the skin. The application of gel enhances the effectiveness of the therapy, reduces the period of treatment and can be comfortable for patients to use. Polyethylene glycol (PEG) based networks have been applied in this research, because the amphiphilic silicone rake copolymer to be used as a therapy has polyethylene oxide (PEO) as a side chain. These PEO side chains have very similar chemical structure to a PEG gel chain so enhancing both the compatibility and the diffusion of the amphiphilic silicone rake copolymer into and out of the gel. Synthesis of PEG-based networks has been performed by two methods: in situ silsesquioxane formation as crosslink with a sol-gel reaction under different conditions and UV curing. PEG networks have low mechanical properties which is a fundamental limitation of the polymer backbone. For mechanical properties enhancement, composite networks were synthesized using nano-silica with different surface modification. The chemical structure of in situ silsesquioxane in the dry network has been examined by Solid State NMR, Differential Scanning Calorimetry (DSC) and swelling measurements in water. Mechanical properties of dry networks were tested by Dynamic Mechanical Thermal Analysis (DMTA) to determine modulus and interfacial interaction between silica and the network. In this way a family of self-reinforced networks has been produced that have been shown to absorb and deliver the active amphiphilic silicone- PEO rake copolymer.

Drug diffusion from polymeric delivery devices : a problem with two moving boundaries

An existing model for solvent penetration and drug release from a spherically-shaped polymeric drug delivery device is revisited. The model has two moving boundaries, one that describes the interface between the glassy and rubbery states of polymer, and another that defines the interface between the polymer ball and the pool of solvent. The model is extended so that the nonlinear diffusion coefficient of drug explicitly depends on the concentration of solvent, and the resulting equations are solved numerically using a front-fixing transformation together with a finite difference spatial discretisation and the method of lines. We present evidence that our scheme is much more accurate than a previous scheme. Asymptotic results in the small-time limit are presented, which show how the use of a kinetic law as a boundary condition on the innermost moving boundary dictates qualitative behaviour, the scalings being very different to the similar moving boundary problem that arises from modelling the melting of an ice ball. The implication is that the model considered here exhibits what is referred to as ``non-Fickian'' or Case II diffusion which, together with the initially constant rate of drug release, has certain appeal from a pharmaceutical perspective.

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy