Analytic shock-fronted solutions to a reaction-diffusion equation with negative diffusivity

Reaction-diffusion equations (RDEs) model the spatiotemporal evolution of a density field $u(\vec{x},t)$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of $u$, which causes the density to disperse. However, RDEs with negative diffusivity can model aggregation, which is the preferred behaviour in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity $D(u) = (u - a)(u - b)$ that is negative for $u\in(a,b)$. We use a non-classical symmetry to construct analytic receding time-dependent, colliding wave, and receding travelling wave solutions. These solutions are initially multi-valued, and we convert them to single-valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan-like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the $u = 0$ and $u=1$ constant solutions, and prove for certain $a$ and $b$ that receding travelling waves are spectrally stable. Additionally, we introduce an new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well-known equal-area rule, but for non-symmetric diffusivity it results in a different shock position.Comment: 35 pages, 10 figure

Nonlinear Fluid Flow and Heat Transfer

Please cite as follows: Makinde, O. D., Moitsheki, R. J., Jana, R. N., Bradshaw-Hajek, B. H. & Khan, W. A. 2014. Nonlinear fluid flow and heat transfer. Advances in Mathematical Physics, 2014:1-2 (Article ID 719102), doi:10.1155/2014/719102.The original publication is available at http://www.hindawi.com/journals/ampNo abstract.http://www.hindawi.com/journals/amp/2014/719102/Publisher's versio

Ion size effects on the electrokinetics of salt-free concentrated suspensions in ac fields

We analyze the influence of finite ion size effects in the response of a salt-free concentrated suspension of spherical particles to an oscillating electric field. Salt-free suspensions are just composed of charged colloidal particles and the added counterions released by the particles to the solution, that counterbalance their surface charge. In the frequency domain, we study the dynamic electrophoretic mobility of the particles and the dielectric response of the suspension. We find that the Maxwell-Wagner-O'Konski process associated with the counterions condensation layer, is enhanced for moderate to high particle charges, yielding an increment of the mobility for such frequencies. We also find that the increment of the mobility grows with ion size and particle charge. All these facts show the importance of including ion size effects in any extension attempting to improve standard electrokinetic models.Comment: J. Colloid Interface Sci., in press, 13 pages, 9 figure

Finishing the euchromatic sequence of the human genome

The sequence of the human genome encodes the genetic instructions for human physiology, as well as rich information about human evolution. In 2001, the International Human Genome Sequencing Consortium reported a draft sequence of the euchromatic portion of the human genome. Since then, the international collaboration has worked to convert this draft into a genome sequence with high accuracy and nearly complete coverage. Here, we report the result of this finishing process. The current genome sequence (Build 35) contains 2.85 billion nucleotides interrupted by only 341 gaps. It covers ∼99% of the euchromatic genome and is accurate to an error rate of ∼1 event per 100,000 bases. Many of the remaining euchromatic gaps are associated with segmental duplications and will require focused work with new methods. The near-complete sequence, the first for a vertebrate, greatly improves the precision of biological analyses of the human genome including studies of gene number, birth and death. Notably, the human enome seems to encode only 20,000-25,000 protein-coding genes. The genome sequence reported here should serve as a firm foundation for biomedical research in the decades ahead

An analysis of organic carbon removal in a two-reactor cascade with recycle and a two-reactor step-feed cascade with recycle

2019 Curtin University and John Wiley & Sons, Ltd. We analyse the steady-state operation of two process configurations that employ a reactor cascade with recycle. The first process configuration is a two-reactor cascade in which the feed stream enters into the first reactor. The second process configuration is a two-reactor step-feed cascade in which the feed stream enters the second reactor. In each process configuration, part of the effluent stream is recycled back into the first reactor. The reactors in the cascade need not be of equal volume. The reaction is assumed to be a biological process governed by Monod growth kinetics with a decay coefficient for the microorganisms. The stability of the washout solution is determined analytically for both process configurations. It follows from the stability analysis that that there is a range of residence times over which the effluent concentration leaving a step-reactor cascade is lower than that leaving a standard reactor cascade. Steady-state diagrams are presented showing the effluent concentration as a function of the residence time. We find that if the desired effluent concentration is not too low, then the process configuration that minimises the residence time is the step-feed reactor cascade. However, for much cleaner waste streams, the process configuration that minimises the residence time is the standard reactor cascade

Reaction-diffusion equations for population genetics

In this thesis, we reinforce the validity of using reaction-diffusion equations with cubic source terms to describe the change in frequency of alleles in a gene pool. In a population with two possible alleles at the locus in question, the Fitzhugh-Nagumo equation is shown to be appropriate when there is no dominance, whereas the Huxley equation is appropriate when one of the alleles is completely dominant. The difference between the Huxley equation (with cubic source term) and the Fisher-Kolmogorov equation (with quadratic source term) is explained numerically and analytically. Using the method of nonclassical symmetry analysis, we construct some practical analytic solutions to the Fitzhugh-Nagumo and Huxley equations. The solutions satisfy specific boundary conditions and are different from previously derived travelling wave solutions. We derive a system of reaction-diffusion equations describing the case of three possible alleles at the locus in question. By introducing a nonlinear transformation, we are able to construct an exact travelling wave solution. We also extend the model to include the case of spatially dependent reproductive success rates. We use classical and nonclassical symmetry methods to discover what forms of explicit spatial variability will enable us to find exact solutions to our equations. A number of solutions are constructed for various forms of spatial variability. Finally, we demonstrate the benefits of systematic symmetry analysis by studying two related systems of reaction-diffusion equations

Nonclassical Symmetry Solutions for Non-Autonomous Reaction-Diffusion Equations

The behaviour of many systems in chemistry, combustion and biology can be described using nonlinear reaction diffusion equations. Here, we use nonclassical symmetry techniques to analyse a class of nonlinear reaction diffusion equations, where both the diffusion coefficient and the coefficient of the reaction term are spatially dependent. We construct new exact group invariant solutions for several forms of the spatial dependence, and the relevance of some of the solutions to population dynamics modelling is discussed

The evolution of a viscous thread pulled with a prescribed speed

We examine the extension of an axisymmetric viscous thread that is pulled at both ends with a prescribed speed such that the effects of inertia are initially small. After neglecting surface tension, we derive a particularly convenient form of the long-wavelength equations that describe long and thin threads. Two generic classes of initial thread shape are considered as well as the special case of a circular cylinder. In these cases, we determine explicit asymptotic solutions while the effects of inertia remain small. We further show that inertia will ultimately become important only if the long-time asymptotic form of the pulling speed is faster than a power law with a critical exponent. The critical exponent can take two possible values depending on whether or not the initial minimum of the thread radius is located at the pulled end. In addition, we obtain asymptotic expressions for the solution at large times in the case in which the critical exponent is exceeded and hence inertia becomes important. Despite the apparent simplicity of the problem, the solutions exhibit a surprisingly rich structure. In particular, in the case in which the initial minimum is not at the pulled end, we show that there are two very different types of solution that exhibit very different extension mechanics. Both the small-inertia solutions and the large-time asymptotic expressions compare well with numerical solutions.J.J. Wylie, B.H. Bradshaw-Hajek and Y.M. Stoke