Numerical Approximation of two dimensional Singularly Perturbed Problems

Thesis (PhD) - Indiana University, Mathematics, 2006We demonstrate how one can improve the numerical solution of singularly perturbed problems involving multiple boundary layers by using a combination of analytic and numerical tools. Incorporating the so-called boundary layer elements (BLE), which absorb the singularities due to the boundary layers, into finite element spaces can improve the accuracy of approximate solutions and result in significant simplifications. We discuss here convection-diffusion equations in the case where both ordinary and parabolic boundary layers are present. We also revise the BLE so that it has a small compact support and hence the resulting linear system becomes sparse, more precisely, block tridiagonal. We prove the validity of the revised element for some singularly perturbed convection-diffusion equations via numerical simulations and via the H^1- approximation error analysis. Furthermore due to the compact structure of the BLE we are able to prove the L^2- stability analysis of the scheme and derive the L^2- error approximations

Analysis of mixed elliptic and parabolic boundary layers with corners

We study the asymptotic behavior at small diffusivity of the solutions, u??, to a convection-diffusion equation in a rectangular domain. The diffusive equation is supplemented with a Dirichlet boundary condition, which is smooth along the edges and continuous at the corners. To resolve the discrepancy, on ???, between u?? and the corresponding limit solution, u0, we propose asymptotic expansions of u?? at any arbitrary, but fixed, order. In order to manage some singular effects near the four corners of , the so-called elliptic and ordinary corner correctors are added in the asymptotic expansions as well as the parabolic and classical boundary layer functions. Then, performing the energy estimates on the difference of u?? and the proposed expansions, the validity of our asymptotic expansions is established in suitable Sobolev spaces.open

Exact inversion of the cone transform arising in an application of a compton camera consisting of line detectors

A Compton camera has been suggested for use in single photon emission computed tomography because a conventional gamma camera has low efficiency. Here we consider a cone transform brought about by a Compton camera with line detectors. A cone transform takes a given function on the 3-dimensional space and assigns to it the surface integral of the function over cones determined by the 1-dimensional vertex space, the 1-dimensional central axis, and the 1-dimensional opening angle. We generalize this cone transform to n-dimensional space and provide an inversion formula. Also, numerical simulations are presented to demonstrate our suggested algorithm in three dimensions.clos

Review on the current trends in tongue diagnosis systems

AbstractTongue diagnosis is an essential process to noninvasively assess the condition of a patient's internal organs in traditional medicine. To obtain quantitative and objective diagnostic results, image acquisition and analysis devices called tongue diagnosis systems (TDSs) are required. These systems consist of hardware including cameras, light sources, and a ColorChecker, and software for color correction, segmentation of tongue region, and tongue classification. To improve the performance of TDSs, various types TDSs have been developed. Hyperspectral imaging TDSs have been suggested to acquire more information than a two-dimensional (2D) image with visible light waves, as it allows collection of data from multiple bands. Three-dimensional (3D) imaging TDSs have been suggested to provide 3D geometry. In the near future, mobile devices like the smart phone will offer applications for assessment of health condition using tongue images. Various technologies for the TDS have respective unique advantages and specificities according to the application and diagnostic environment, but this variation may cause inconsistent diagnoses in practical clinical applications. In this manuscript, we reviewed the current trends in TDSs for the standardization of systems. In conclusion, the standardization of TDSs can supply the general public and oriental medical doctors with convenient, prompt, and accurate information with diagnostic results for assessing the health condition

Common fixed point theorems and applications

The purpose of this paper is to discuss the existence of common fixed points for mappings in general quasi-metric spaces. As applications, some common fixed point theorems for mappings in probabilistic quasi-metric spaces are given. The results presented in this paper generalize some recent results

SINGULAR PERTURBATIONS AND BOUNDARY LAYER THEORY FOR CONVECTION-DIFFUSION EQUATIONS IN A CIRCLE: THE GENERIC NONCOMPATIBLE CASE

We study the boundary layers and singularities generated by a convection-diffusion equation in a circle with noncompatible data. More precisely, the boundary of the circle has two characteristic points where the boundary conditions and the external data $f$ are not compatible. Very complex singular behaviors are observed, and we analyze them systematically for highly noncompatible data. The problem studied here is a simplified model for problems of major importance in fluid mechanics and thermohydraulics and in physics.open4

Effects of Textural Properties on the Response of a SnO2-Based Gas Sensor for the Detection of Chemical Warfare Agents

The sensing behavior of SnO2-based thick film gas sensors in a flow system in the presence of a very low concentration (ppb level) of chemical agent simulants such as acetonitrile, dipropylene glycol methyl ether (DPGME), dimethyl methylphosphonate (DMMP), and dichloromethane (DCM) was investigated. Commercial SnO2 [SnO2(C)] and nano-SnO2 prepared by the precipitation method [SnO2(P)] were used to prepare the SnO2 sensor in this study. In the case of DCM and acetonitrile, the SnO2(P) sensor showed higher sensor response as compared with the SnO2(C) sensors. In the case of DMMP and DPGME, however, the SnO2(C) sensor showed higher responses than those of the SnO2(P) sensors. In particular, the response of the SnO2(P) sensor increased as the calcination temperature increased from 400 °C to 800 °C. These results can be explained by the fact that the response of the SnO2-based gas sensor depends on the textural properties of tin oxide and the molecular size of the chemical agent simulant in the detection of the simulant gases (0.1–0.5 ppm)