Approximations of Sturm-Liouville Eigenvalues Using Sinc-Galerkin and Differential Transform Methods

In this paper, we present a comparative study of Sinc-Galerkin method and differential transform method to solve Sturm-Liouville eigenvalue problem. As an application, a comparison between the two methods for various celebrated Sturm-Liouville problems are analyzed for their eigenvalues and solutions. The study outlines the significant features of the two methods. The results show that these methods are very efficient, and can be applied to a large class of problems. The comparison of the methods shows that although the numerical results of these methods are the same, differential transform method is much easier, and more efficient than the Sinc-Galerkin method

Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation

We study a reaction diffusion model recently proposed in [5] to describe the spatiotemporal evolution of the bacterium Bacillus subtilis on agar plates containing nutrient. An interesting mathematical feature of the model, which is a coupled pair of partial differential equations, is that the bacterial density satisfies a degenerate nonlinear diffusion equation. It was shown numerically that this model can exhibit quasi-one-dimensional constant speed travelling wave solutions. We present an analytic study of the existence and uniqueness problem for constant speed travelling wave solutions. We find that such solutions exist only for speeds greater than some threshold speed giving minimum speed waves which have a sharp profile. For speeds greater than this minimum speed the waves are smooth. We also characterise the dependence of the wave profile on the decay of the front of the initial perturbation in bacterial density. An investigation of the partial differential equation problem establishes,via a global existence and uniqueness argument, that these waves are the only long time solutions supported by the problem. Numerical solutions of the partial differential equation problem are presented and they confirm the results of the analysis

Monte Carlo modelling of Case I and Case II solvent diffusion in polymers.

The development of two original Monte Carlo models of solvent diffusion into a polymer is described. Employing a coarse grained model of a polymer solution on a regular lattice, the dynamic properties of both the solvent and polymer molecules can be observed. The "Simple" Monte Carlo model reliably reproduces Case I dynamics, but no departure from this is seen for any reasonable model parameters. This "Simple" Monte Carlo model is unable to reproduce Case II diffusion dynamics. One reason for this is that in this Monte Carlo model the processes of solvent diffusion and polymer relaxation are entirely independent processes. In this thesis it is suggested that a simple Monte Carlo model of this type will always produce Case I diffusion dynamics. The dynamic algorithm described in this work relies on simple instantaneous molecular motions between neighbouring lattice sites. It is shown that a diffusion process based on these motions is purely concentration dependent, relying only on the current state of the system. To use the Monte Carlo method to simulate Case II diffusion dynamics, the diffusion process is made time dependent by incorporating a history dependent model of diffusion first proposed by Crank (CRANK 1953). In this "History Dependent" Monte Carlo model the motions of both the solvent and the polymer are no longer instantaneous, but occur at a rate that approaches equilibrium by a first order process governed by a relaxation time characteristic of the viscoelastic relaxation of the polymer. This "History Dependent" Monte Carlo model successfully simulates most of the features of Case II diffusion and also demonstrates a return to Case I diffusion in the limit of long times. Unlike many models of Case II diffusion, this Monte Carlo model is able to simultaneously model the microscopic motions of both the solvent and the polymer molecules. This novel feature demonstrates the formation of a discontinuous moving boundary between the rubbery polymer and the glassy polymer that is typical of Case II diffusion dynamics

Theoretical studies in support of the 3M-vapor transport (PVTOS-) experiments

Results are reported for a preliminary theoretical study of the coupled mass-, momentum-, and heat-transfer conditions expected within small ampoules used to grow oriented organic solid (OS-) films, by physical vapor transport (PVT) in microgravity environments. It is show that previous studies made restrictive assumptions (e.g., smallness of delta T/T, equality of molecular diffusivities) not valid under PVTOS conditions, whereas the important phenomena of sidewall gas creep, Soret transport of the organic vapor, and large vapor phase supersaturations associated with the large prevailing temperature gradients were not previously considered. Rational estimates are made of the molecular transport properties relevant to copper-phthalocyanine monomeric vapor in a gas mixture containing H2(g) and Xe(g). Efficient numerical methods have been developed and are outlined/illustrated here to making steady axisymmetric gas flow calculations within such ampoules, allowing for realistic realistic delta T/T(sub)w-values, and even corrections to Navier-Stokes-Fourier 'closure' for the governing continuum differential equations. High priority follow-on studies are outlined based on these new results

Modelling and simulation framework for reactive transport of organic contaminants in bed-sediments using a pure java object - oriented paradigm

Numerical modelling and simulation of organic contaminant reactive transport in the environment is being increasingly relied upon for a wide range of tasks associated with risk-based decision-making, such as prediction of contaminant profiles, optimisation of remediation methods, and monitoring of changes resulting from an implemented remediation scheme. The lack of integration of multiple mechanistic models to a single modelling framework, however, has prevented the field of reactive transport modelling in bed-sediments from developing a cohesive understanding of contaminant fate and behaviour in the aquatic sediment environment. This paper will investigate the problems involved in the model integration process, discuss modelling and software development approaches, and present preliminary results from use of CORETRANS, a predictive modelling framework that simulates 1-dimensional organic contaminant reaction and transport in bed-sediments

Predicting Material Properties: Applications of Multi-Scale Multiphysics Numerical Modeling to Transport Problems in Biochemical Systems and Chemical Process Engineering

Material properties are used in a wide variety of theoretical models of material behavior. Descriptive properties quantify the nature, structure, or composition of the material. Behavioral properties quantify the response of the material to an imposed condition. The central question of this work concerns the prediction of behavioral properties from previously determined descriptive properties through hierarchical multi-scale, multiphysics models implemented as numerical simulations. Applications covered focus on mass transport models, including sequential enzyme-catalyzed reactions in systems biology, and an industrial chemical process in a common reaction medium

A modeling and simulation study of siderophore mediated antagonism in dual-species biofilms

<p>Abstract</p> <p>Background</p> <p>Several bacterial species possess chelation mechanisms that allow them to scavenge iron from the environment under conditions of limitation. To this end they produce siderophores that bind the iron and make it available to the cells later on, while rendering it unavailable to other organisms. The phenomenon of siderophore mediated antagonism has been studied to some extent for suspended populations where it was found that the chelation ability provides a growth advantage over species that do not have this possibility. However, most bacteria live in biofilm communities. In particular <it>Pseudomonas fluorescens </it>and <it>Pseudomonas putida</it>, the species that have been used in most experimental studies of the phenomenon, are known to be prolific biofilm formers, but only very few experimental studies of iron chelation have been published to date for the biofilm setting. We address this question in the present study.</p> <p>Methods</p> <p>Based on a previously introduced model of iron chelation and an existing model of biofilm growth we formulate a model for iron chelation and competition in dual species biofilms. This leads to a highly nonlinear system of partial differential equations which is studied in computer simulation experiments.</p> <p>Conclusions</p> <p>(i) Siderophore production can give a growth advantage also in the biofilm setting, (ii) diffusion facilitates and emphasizes this growth advantage, (iii) the magnitude of the growth advantage can also depend on the initial inoculation of the substratum, (iv) a new mass transfer boundary condition was derived that allows to a priori control the expect the expected average thickness of the biofilm in terms of the model parameters.</p