Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach

The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a non-local operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite difference with numerical quadrature, to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solution with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure

Channel characterization for 1D molecular communication with two absorbing receivers

This letter develops a one-dimensional (1D) diffusion-based molecular communication system to analyze channel responses between a single transmitter (TX) and two fully-absorbing receivers (RXs). Incorporating molecular degradation in the environment, rigorous analytical formulas for i) the fraction of molecules absorbed, ii) the corresponding hitting rate, and iii) the asymptotic fraction of absorbed molecules as time approaches infinity at each RX are derived when an impulse of molecules are released at the TX. By using particle-based simulations, the derived analytical expressions are validated. Simulations also present the distance ranges of two RXs that do not impact molecular absorption of each other, and demonstrate that the mutual influence of two active RXs reduces with the increase in the degradation rate

Parameter Estimation in a Noisy 1D Environment via Two Absorbing Receivers

This paper investigates the estimation of different parameters, e.g., propagation distance and flow velocity, by utilizing two fully-absorbing receivers (RXs) in a one-dimensional (1D) environment. The time-varying number of absorbed molecules at each RX and the number of absorbed molecules in a time interval as time approaches infinity are derived. Noisy molecules in this environment, that are released by sources in addition to the transmitter, are also considered. A novel estimation method, namely difference estimation (DE), is proposed to eliminate the effect of noise by using the difference of received signals at the two RXs. For DE, the Cramer-Rao lower bound (CRLB) on the variance of estimation is derived. Independent maximum likelihood estimation is also considered at each RX as a benchmark to show the performance advantage of DE. Aided by particle-based simulation, the derived analytical results are verified. Furthermore, numerical results show that DE attains the CRLB and is less sensitive to the change of noise than independent estimation at each RX.Comment: 7 pages, 5 figures, accepted by Globecom 202

IC-integrated flexible shear-stress sensor skin

This paper reports the successful development of the first IC-integrated flexible MEMS shear-stress sensor skin. The sensor skin is 1 cm wide, 2 cm long, and 70 /spl mu/m thick. It contains 16 shear-stress sensors, which are arranged in a 1-D array, with on-skin sensor bias, signal-conditioning, and multiplexing circuitry. We further demonstrated the application of the sensor skin by packaging it on a semicylindrical aluminum block and testing it in a subsonic wind tunnel. In our experiment, the sensor skin has successfully identified both the leading-edge flow separation and stagnation points with the on-skin circuitry. The integration of IC with MEMS sensor skin has significantly simplified implementation procedures and improved system reliability

Solid State Inflation Balloon Active Deorbiter: Scalable Low-Cost Deorbit System for Small Satellites

The goal of the Solid State Inflation Balloon Active Deorbiter project is to develop and demonstrate a scalable, simple, reliable, and low-cost active deorbiting system capable of controlling the downrange point of impact for the full-range of small satellites from 1 kg to 180 kg. The key enabling technology being developed is the Solid State Gas Generator (SSGG) chip, generating pure nitrogen gas from sodium azide (NaN3) micro-crystals. Coupled with a metalized nonelastic drag balloon, the complete Solid State Inflation Balloon (SSIB) system is capable of repeated inflation/deflation cycles. The SSGG minimizes size, weight, electrical power, and cost when compared to the current state of the art