Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: proofs

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters $0< \epsilon \le \mu \le 1$, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have \emph{boundary layers} which overlap and interact, based on the relative size of $\epsilon$ and $$. We construct full asymptotic expansions together with error bounds that cover the complete range $0 < \epsilon \leq \mu \leq 1$. For the present case of analytic input data, we derive derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order

Analytical model of conductive graphite foam based sensors characteristics

Sensors play an important role in the control systems, because they provide the necessary information from surroundings to the controller of an automated systems. Today’s sensors are very sophisticated, with high accuracy, fast acquisition rate and good signal-to-noise ratio. But most of these sensors are too much expensive. Low cost sensor for measuring the force (pressure) or the displacement could be realized by utilizing conductive elastomer that exhibits property of changing the electrical resistance when the elastomer is deformed. This paper introduced a novel conductive graphite foam based sensors. The sensors are formed by inserting two thin copper wires within conductive foam, parallel to each other at the two opposite sides. The main problem of conductive foam based sensors is that the force-electrical resistance characteristic, or the displacement-electrical resistance characteristic, of conductive foam is highly nonlinear. This paper presents the analytical model of the conductive graphite foam sensors for measurement of the displacement. By measuring the changes in the electric resistance between two points of the foam and using the developed analytical model it should be possible to accurately estimate the displacement when the conductive foam is deformed

A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems

This paper treats a time-dependent singularly perturbed reaction-diffusion problem. We semidiscretize the problem in time by means of the classical backward Euler method. We develop a fitted operator finite difference method (FOFDM) to solve the resulting set of linear problems (one at each time level). We prove that the underlying fitted operator satisfies the maximum principle. This result is then used in the error analysis of the FOFDM. The method is shown to be first order convergent in time and second order convergent in space, uniformly with respect to the perturbation parameter. We test the method on several numerical examples to confirm our theoretical findings.Web of Scienc