Ron Wiltshire

Systems of reaction diffusion equations and their symmetry properties

A constructive algorithm is proposed for the investigation of symmetries of partial differential equations. The algorithm is used to present classical Lie symmetries of systems of two nonlinear reaction diffusion equations.

Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations

In this paper, we propose upper bounds for the sum of the maximal eigenvalues of the solutions of the continuous coupled algebraic Riccati equation (CCARE) and the discrete coupled algebraic Riccati equation (DCARE), which are then used to infer upper bounds for the maximal eigenvalues of the solutions of each Riccati equation. By utilizing the upper bounds for the maximal eigenvalues of each equation, we then derive upper matrix bounds for the solutions of the CCARE and DCARE. Following the development of each bound, an iterative algorithm is proposed which can be used to derive tighter upper matrix bounds. Finally, we give numerical examples to demonstrate the effectiveness of the proposed results, making comparisons with existing results.

The use of Lie transformation groups in the solution of the coupled diffusion equation

The solutions of coupled, linear and nonlinear, diffusion equations in a semi-infinite medium, are derived using the method of continuous one-parameter point symmetry group invariants. Initially, conditions for both classical and non-classical Lie group invariants are developed and solutions are derived corresponding to linear coupled diffusion. The non-classical invariants are shown to be expressible in terms of a linear parabolic operator. In addition, solutions corresponding to constant, impulse and sinusoidally time varying condition are derived from the equations determining classical group invariance. Nonlinear cases are also considered. The classical symmetry groups for any form of diffusion matrix are presented. In addition, the point source solution for a power-law form governing the diffusion matrix is derived. It is a natural extension of the scalar analogue.

Lie symmetry classification analysis for nonlinear coupled diffusion

The group properties of the (1 + 1)-dimensional matrix diffusion equation of the form with respect to point symmetries are given. It is shown that in particular cases the group properties of this equation are similar to those of the corresponding scalar equation. Namely, the Lie algebra is extended for power and exponential functions, with additional extensions when powers of and -2 exist. However, a specific form of is shown to exist for cases admitting both infinite and finite groups. The result obtained is used to construct new group-invariant solutions for impulsive boundary inputs. Using two similarity variables, a reduction is applied to the coupled diffusion of temperature and volumetric moisture content in porous media under periodic boundary conditions. A perturbation method is employed to obtain an explicit solution for the case when can be expressed exponentially in terms of the similarity variables.

Beam Propagation Analysis of Polarization Rotation in Soft Glass Nematic Liquid Crystal Photonic Crystal Fibers

In this letter, a novel design of high tunable polarization rotator (PR) based on nematic liquid crystal (NLC) photonic crystal fiber is proposed and analyzed. The tunability of the proposed PR results from the change of the optical properties of the NLC with the temperature or external electric field. The simulation results are obtained using the full vectorial finite-difference method as well as the full vectorial finite-difference beam propagation method. The numerical results reveal that the suggested PR can provide a strong polarization conversion ratio of 99.81% with a device length of 1072 ¿m.

Non-classical and potential symmetry analysis of Richard's equation for moisture flow in soil

This paper focuses upon the derivation of the non-classical symmetries of Bluman and Cole as they apply to Richards equation for water flow in an unsaturated uniform soil. It is shown that the determining equations for the non-classical case lead to four highly non-linear equations which have been solved in five particular cases. In each case the corresponding similarity ansatz has been derived and Richards equation is reduced to an ordinary differential equation. Explicit solutions are produced when possible. Richards equation is also expressed as a potential system and in reviewing the classical Lie solutions a new symmetry is derived together with its similarity ansatz. Determining equations are then produced for the potential system using the non-classical algorithm. This results in an under-determined set of equations and an example symmetry that reveals a missing classical case is presented. An example of a classical and a non-classical symmetry reduction applied to the infiltration of moisture in soil is presented. The condition for surface invariance is used to demonstrate the equivalence of a classical Lie and a potential symmetry.

Automated object recognition of blue-green algae for measuring water quality—A preliminary study

In this paper a computer algorithm for the automated detection of blue-green algae is presented. Samples of seven species of blue-green algae and two species of green algae were examined under a microscope and transferred to a computer. The microscope pictures were stored as digital images. In order to locate the organisms Image Segmentation routines were applied. Image Enhancement improved the quality and appearance of the species in the images. With the aid of shape algorithms (Fourier Descriptors, Moment Invariants) and textural algorithms (Cell finding algorithm, Statistical measures) certain features were identified, extracted and then fed into a classifier. The classifier, using Discriminant Analysis, was then able to predict 155 out of 158 samples correctly.

The automated detection of cyanobacteria using ddigital image processing techniques

Abstract The aim of this paper is to show how the process of detection of cyanobacteria can be automated by means of a digital image processing package. In Britains lakes and reservoirs, the occurrence of nine different species of cyanobacteria is of interest This paper determines a way to detect and distinguish two of them, Anabaena and Oscillatoria , automatically. Water samples containing cyanobacteria were obtained and examined under a microscope Through an attached camera, microscopic pictures of algae were recorded on a videotape and transferred to the computer, a SUN Workstation. Image processing techniques were applied to the image to improve the quality and enhance particular features. In particular, the Logarithm of the Gaussian (LoG) operator proved to the effective for the enhancement of biological organisms. The shape and texture were analyzed and differences in the algae were obtained using features given in he biological key. With the aid of the textural features, it was possible to distinguish between Anabaena and Oscillatoria with an accuracy of over 90%.

Slowly, Rotating Non-Stationary, Fluid Solutions of Einstein's Equations and Their Match to Kerr Empty Space-Time

A general class of solutions of Einsteins equation for a slowly rotating fluid source, with supporting internal pressure, is matched using Lichnerowicz junction conditions, to the Kerr metric up to and including first order terms in angular speed parameter. It is shown that the match applies to any previously known non-rotating fluid source made to rotate slowly for which a zero pressure boundary surface exists. The method is applied to the dust source of Robertson-Walker and in outline to an interior solution due to McVittie describing gravitational collapse. The applicability of the method to additional examples is transparent. The differential angular velocity of the rotating systems is determined and theinduced rotation of local inertial frame is exhibited.

Letter: Slowly Rotating, Compact Fluid Sources Embedded in Kerr Empty Space-Time

Spherically symmetric static fluid sources are endowed with rotation and embedded in Kerr empty space-time up to and including quadratic terms in an angular velocity parameter using Darmois junction conditions. The boundary behaviour of the metric tensor and partial derivatives is used to develop a series solution of Einsteins equations for the rotating fluid. The boundary of the rotating source is expressed explicitly in terms of sinusoidal functions of the polar angle. As an example of the analysis the Schwarzschild interior solution is endowed with rotation and the equation of the fluid boundary is generated together with surface behaviour of the fluid density and angular velocity.