Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem yk+1 - 2yk + yk-1 + f (k, yk, vk) = 0, for k = 1,..., n - 1, y0 = 0 = yn,, where f is continuous and vk = yk - yk-1, for k = 1,..., n. In the special case f (k, t, p) = f (t) greater than or equal to 0, we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue. (C) 2002 Elsevier Science Ltd. All rights reserved.

Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.

Numerical investigations of flow and energy fields near a thermoacoustic couple

The flow field and the energy transport near thermoacoustic couples are simulated using a 2D full Navier-Stokes solver. The thermoacoustic couple plate is maintained at a constant temperature; plate lengths, which are short and long compared with the particle displacement lengths of the acoustic standing waves, are tested. Also investigated are the effects of plate spacing and the amplitude of the standing wave. Results are examined in the form of energy vectors, particle paths, and overall entropy generation rates. These show that a net heat-pumping effect appears only near the edges of thermoacoustic couple plates, within about a particle displacement distance from the ends. A heat-pumping effect can be seen even on the shortest plates tested when the plate spacing exceeds the thermal penetration depth. It is observed that energy dissipation near the plate increases quadratically as the plate spacing is reduced. The results also indicate that there may be a larger scale vortical motion outside the plates which disappears as the plate spacing is reduced. (C) 2002 Acoustical Society of America.

Effects of the solute ionization on the adsorption of aromatic compounds from dilute aqueous solutions by activated carbon

Adsorption of four dissociating aromatic compounds and one nondissociating compound on a commercial activated carbon is investigated systematically. All adsorption experiments were carried out in pH-controlled aqueous solutions. The adsorption isotherms are fitted to the binary homogeneous Langmuir model, where the concentrations of the molecular and the ionic species in the liquid phase are expressed in terms of the sum of the two and the degree of solute ionization. Examination of the relationships between the solution pH, the degree of ionization of the solutes, and the model parameters is found to give new insights into the adsorption process. Furthermore, this is used to correlate the variation of the monolayer capacity with the solution pH.

The strength of concentrated Mg-Zn solid solutions

Solid solution effects on the hardness and flow stress have been studied for zinc contents between 0.2 and 2.4 at% (0.5 and 6.9 wt%) in Mg. The alloys were grain refined with 0.6 wt% zirconium to ensure a similar grain size at all compositions. The hardness increases with the zinc content as Hv(10) (kg mm(-2)) = 9 Zn (at%) + 33. At low solute concentrations the (0.2%) proof strength does not change significantly with concentration. At concentrations above 0.7 at%, within the supersaturated solid solution region, the rate of solid solution hardening is high, following a c(2) rule, where c is the atom fraction of Zn. It is suggested that short-range order may account for most of the observed strengthening in concentrated Mg-Zn alloys.

Numerical modelling of contaminant transport in coastal aquifers

This paper employs a two-dimensional variable density flow and transport model to investigate the transport of a dense contaminant plume in an unconfined coastal aquifer. Experimental results are also presented to show the contaminant plume in a freshwater-seawater flow system. Both the numerical and experimental results suggest that the neglect of the seawater interface does not noticeably affect the horizontal migration rate of the plume before it reaches the interface. However, the contaminant will travel further seaward and part of the solute mass will exit under the sea if the higher seawater density is not included. If the seawater density is included, the contaminant will travel upwards towards the beach along the freshwater-saltwater interface as shown experimentally. Neglect of seawater density, therefore, will result in an underestimate of solute mass rate exiting around the coastline. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.

An Analytic Simplification for the Reconstruction Problem of Electrical Impedance Tomography

This paper introduces a new reconstruction algorithm for electrical impedance tomography. The algorithm assumes that there are two separate regions of conductivity. These regions are represented as eccentric circles. This new algorithm then solves for the location of the eccentric circles. Due to the simple geometry of the forward problem, an analytic technique using conformal mapping and separation of variables has been employed. (C) 2002 John Wiley Sons, Inc.

Finite difference time domain (FDTD) method for modeling the effect of switched gradients on the human body in MRI

Numerical modeling of the eddy currents induced in the human body by the pulsed field gradients in MRI presents a difficult computational problem. It requires an efficient and accurate computational method for high spatial resolution analyses with a relatively low input frequency. In this article, a new technique is described which allows the finite difference time domain (FDTD) method to be efficiently applied over a very large frequency range, including low frequencies. This is not the case in conventional FDTD-based methods. A method of implementing streamline gradients in FDTD is presented, as well as comparative analyses which show that the correct source injection in the FDTD simulation plays a crucial rule in obtaining accurate solutions. In particular, making use of the derivative of the input source waveform is shown to provide distinct benefits in accuracy over direct source injection. In the method, no alterations to the properties of either the source or the transmission media are required. The method is essentially frequency independent and the source injection method has been verified against examples with analytical solutions. Results are presented showing the spatial distribution of gradient-induced electric fields and eddy currents in a complete body model.

Effects of surface chemistry on aromatic compound adsorption from dilute aqueous solutions by activated carbon

Adsorption of one nondissociating and four dissociating aromatic compounds onto three untreated activated carbons from dilute aqueous solutions were investigated. All adsorption experiments were preformed in pH-controlled solutions. The experimental isotherms were analyzed using the homogeneous Langmuir model. The surface chemical properties of the activated carbons were characterized using a combination of water adsorption, X-ray photoemission spectroscopy, and mass titration. These data give rise to a new insight into the adsorption mechanism of aromatic solutes, in their molecular and ionic forms, onto untreated activated carbons. It was found that, for the hydrophilic activated carbons, the dominant adsorption forces were observed to be dipolar interactions when the solutes were in their molecular form whereas dispersive forces, such as pi-pi interactions, were most likely dominant in the case of the basic hydrophobic carbons. However, when the solutes were in their ionic form adsorption occurs in all cases through dispersive forces.

Aerosolization of protein solutions using thermal inkjet technology

Vapotronics Inc. is developing the thermal inkjet (TIJ) technology used extensively in the printer industry to create a digital aerosol inhaler for the inhalation of therapeutics for local and systemic delivery. The operation of thermal inkjet printers requires generation of high temperatures and vaporization of the liquid formulation to effect droplet ejection. A study was conducted to develop formulations that would permit the generation of aerosols of therapeutic proteins without damage to the inkjet system or degradation of the proteins. Two proteins, human growth hormone and insulin, were formulated and aerosolized. The aerosol was collected and subjected to assays to compare the physicochemical and biological activities of these proteins before and after aerosolization. In each case, there was no significant changes to the proteins as a result of the aerosolization, providing evidence that TIJ can be used for aerosolizing solutions of protein therapeutics.

The nonexistence of spurious solutions to discrete, two-point boundary value problems

We investigate difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations. We formulate conditions under which all solutions to the discrete problem satisfy certain a priori bounds which axe independent of the step-size. As a result, the nonexistence of spurious solutions are guaranteed. Some existence and convergence theorems for solutions to the discrete problem are also presented. (C) 2002 Elsevier Science Ltd. All rights reserved.

NEW Solutions: An Australian health promotion programme for people with mental illness

The prevalence of people who are overweight or obese is continuing to rise. This is a key risk indicator of preventable morbidity and mortality resulting from many diseases. The increase in the number of obese people is associated with dietary practices and a reduction in physical activity. There is a need to implement strategies for the reduction of obesity in at-risk groups. People with a mental illness are at high risk of many physical illnesses related to behavioural factors, which include poor diet and lack of exercise. Health promotion programmes need to be incorporated into mental-health services to improve the general health and wellbeing of people with a mental illness. An Australian psychiatric rehabilitation service developed and implemented a health promotion programme, NEW Solutions, which aimed to address issues related to weight gain, dietary practice and physical inactivity.

Existence of multiple solutions for finite difference approximations to second-order boundary value problems

Error condition detected We consider discrete two-point boundary value problems of the form D-2 y(k+1) = f (kh, y(k), D y(k)), for k = 1,...,n - 1, (0,0) = G((y(0),y(n));(Dy-1,Dy-n)), where Dy-k = (y(k) - Yk-I)/h and h = 1/n. This arises as a finite difference approximation to y" = f(x,y,y'), x is an element of [0,1], (0,0) = G((y(0),y(1));(y'(0),y'(1))). We assume that f and G = (g(0), g(1)) are continuous and fully nonlinear, that there exist pairs of strict lower and strict upper solutions for the continuous problem, and that f and G satisfy additional assumptions that are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. Under these assumptions we show that there are at least three distinct solutions of the discrete approximation which approximate solutions to the continuous problem as the grid size, h, goes to 0. (C) 2003 Elsevier Science Ltd. All rights reserved.