A multi-microprocessor architecture for solving partial differential equations

This paper presents the architecture of a fault-tolerant, special-purpose multi-microprocessor system for solving Partial Differential Equations (PDEs). The modular nature of the architecture allows the use of hundreds of Processing Elements (PEs) for high throughput. Its performance is evaluated by both analytical and simulation methods. The results indicate that the system can achieve high operation rates and is not sensitive to inter-processor communication delay.

Exact solutions of coupled multispecies linear reaction–diffusion equations on a uniformly growing domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

Derivation of the solution of certain singular integral equations

It is shown that the a;P?lication of the Poincare-Bertrand fcm~ulaw hen made in a suitable manner produces the s~lutiano f certain singular integral equations very quickly, thc method of arriving at which, otherwise, is too complicaled. Two singular integral equations are considered. One of these quaiions is with a Cauchy-tyge kcrnel arid the other is an equalion which appears in the a a w guide theory and the theory of dishcations. Adifferent approach i? alw made here to solve the singular integralquation> of the waveguide theor? ind this i ~ v o l v eth~e use of the inversion formula of the Cauchy-type singular integral equahn and dudion to a system of TIilberl problems for two unknowns which can be dwupled wry easily to obi& tbe closed form solutim of the irilegral equatlou at band. The methods of the prescnt paper avoid all the complicaled approaches of solving the singular integral equaticn of the waveguide theory knowr todate.

On a linear steady-state system of equations in microcontinuum fluid mechanics

Galerkin representations and integral representations are obtained for the linearized system of coupled differential equations governing steady incompressible flow of a micropolar fluid. The special case of 2-dimensional Stokes flows is then examined and further representation formulae as well as asymptotic expressions, are generated for both the microrotation and velocity vectors. With the aid of these formulae, the Stokes Paradox for micropolar fluids is established.

Generalized Burgers equations and Euler-Painlev transcendents. I

Initial-value problems for the generalized Burgers equation (GBE) ut+u betaux+lambdaualpha =(delta/2)uxx are discussed for the single hump type of initial data both continuous and discontinuous. The numerical solution is carried to the self-similar ``intermediate asymptotic'' regime when the solution is given analytically by the self-similar form. The nonlinear (transformed) ordinary differential equations (ODE's) describing the self-similar form are generalizations of a class discussed by Euler and Painlevé and quoted by Kamke. These ODE's are new, and it is postulated that they characterize GBE's in the same manner as the Painlev equations categorize the Kortweg-de Vries (KdV) type. A connection problem for some related ODE's satisfying proper asymptotic conditions at x=±[infinity], is solved. The range of amplitude parameter is found for which the solution of the connection problem exists. The other solutions of the above GBE, which display several interesting features such as peaking, breaking, and a long shelf on the left for negative values of the damping coefficient lambda, are also discussed. The results are compared with those holding for the modified KdV equation with damping. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Oscillation criteria for differential equations of second order

In this article, we give sufficient condition in the form of integral inequalities to establish the oscillatory nature of non linear homogeneous differential equations of the form where r, q, p, f and g are given data. We do this by separating the two cases f is monotonous and non monotonous.

Simultaneous prediction of acidity parameters (pH and titratable acidity) in Kefir using near infrared reflectance spectroscopy.

Acidity in terms of pH and titratable acids influences the texture and flavour of fermented dairy products, such as Kefir. However, the methods for determining pH and titratable acidity (TA) are time consuming. Near infrared (NIR) spectroscopy is a non-destructive method, which simultaneously predicts multiple traits from a single scan and can be used to predict pH and TA. The best pH NIR calibration model was obtained with no spectral pre-treatment applied, whereas smoothing was found to be the best pre-treatment to develop the TA calibration model. Using cross-validation, the prediction results were found acceptable for both pH and TA. With external validation, similar results were found for pH and TA, and both models were found to be acceptable for screening purposes.

Simultaneous adsorption of Cd(II) and phosphate on Al13 pillared montmorillonite

Al13 pillared montmorillonites (AlPMts) prepared with different Al/clay ratios were used to remove Cd(II) and phosphate from aqueous solution. The structure of AlPMts was characterized by X-ray diffraction (XRD), Thermogravimetric analysis (TG), and N2 adsorption–desorption. The basal spacing, intercalated amount of Al13 cations, and specific surface area of AlPMts increased with the increase of the Al/clay ratio. In the single adsorption system, with the increase of the Al/clay ratio, the adsorption of phosphate on AlPMts increased but that of Cd(II) decreased. Significantly enhanced adsorptions of Cd(II) and phosphate on AlPMts were observed in a simultaneous system. For both contaminants, the adsorption of one contaminant would increase with the increase of the initial concentration of the other one and increase in the Al/clay ratio. The enhancement of the adsorption of Cd(II) was much higher than that of phosphate on AlPMt. This suggests that the intercalated Al13 cations are the primary co-adsorption sites for phosphate and Cd(II). X-ray photoelectron spectroscopy (XPS) indicated comparable binding energy of P2p but a different binding energy of Cd3d in single and simultaneous systems. The adsorption and XPS results suggested that the formation of P-bridge ternary surface complexes was the possible adsorption mechanism for promoted uptake of Cd(II) and phosphate on AlPMt.

Optimal simultaneous maximum a posterioriestimation of states, noise statistics and parameters II. Numerical performance

A very general and numerically quite robust algorithm has been proposed by Sastry and Gauvrit (1980) for system identification. The present paper takes it up and examines its performance on a real test example. The example considered is the lateral dynamics of an aircraft. This is used as a vehicle for demonstrating the performance of various aspects of the algorithm in several possible modes.

Invariance group properties and exact solutions of equations describing time-dependent free surface flows under gravity

Using the method of infinitesimal transformations, a 6-parameter family of exact solutions describing nonlinear sheared flows with a free surface are found. These solutions are a hybrid between the earlier self-propagating simple wave solutions of Freeman, and decaying solutions of Sachdev. Simple wave solutions are also derived via the method of infinitesimal transformations. Incomplete beta functions seem to characterize these (nonlinear) sheared flows in the absence of critical levels.

A Fredholm-type theory for third-kind linear integral equations

The third-kind linear integral equation Image where g(t) vanishes at a finite number of points in (a, b), is considered. In general, the Fredholm Alternative theory [[5.]] does not hold good for this type of integral equation. However, imposing certain conditions on g(t) and K(t, t′), the above integral equation was shown [[1.], 49–57] to obey a Fredholm-type theory, except for a certain class of kernels for which the question was left open. In this note a theory is presented for the equation under consideration with some additional assumptions on such kernels.

Application of partition method to vibration problems of plates

The “partition method” or “sub-domain method” consists of expressing the solution of a governing differential equation, partial or ordinary, in terms of functions which satisfy the boundary conditions and setting to zero the error in the differential equation integrated over each of the sub-domains into which the given domain is partitioned. In this paper, the use of this method in eigenvalue problems with particular reference to vibration of plates is investigated. The deflection of the plate is expressed in terms of polynomials satisfying the boundary conditions completely. Setting the integrated error in each of the subdomains to zero results in a set of simultaneous, linear, homogeneous, algebraic equations in the undetermined coefficients of the deflection series. The algebraic eigenvalue problem is then solved for eigenvalues and eigenvectors. Convergence is examined in a few typical cases and is found to be satisfactory. The results obtained are compared with existing results based on other methods and are found to be in very good agreement.

A simple, one-tube assay for the simultaneous detection and diagnosis of ten Australian poultry Eimeria

Coccidiosis is a costly worldwide enteric disease of chickens caused by parasites of the genus Eimeria. At present, there are seven described species that occur globally and a further three undescribed, operational taxonomic units (OTUs X, Y, and Z) that are known to infect chickens from Australia. Species of Eimeria have both overlapping morphology and pathology and frequently occur as mixed-species infections. This makes definitive diagnosis with currently available tests difficult and, to date, there is no test for the detection of the three OTUs. This paper describes the development of a PCR-based assay that is capable of detecting all ten species of Eimeria, including OTUs X, Y, and Z in field samples. The assay is based on a single set of generic primers that amplifies a single diagnostic fragment from the mitochondrial genome of each species. This one-tube assay is simple, low-cost, and has the capacity to be high throughput. It will therefore be of great benefit to the poultry industry for Eimeria detection and control, and the confirmation of identity and purity of vaccine strains.