Analytical and Numerical Studies of Direct and Inverse Problems for Parabolic Initial-Boundary Value Problems

University Of Richmond, Richmond VA

Investigators

Abstract

This project focuses on the study of direct and inverse problems for parabolic partial differential equations. The PI will utilize both analytical and numerical methods to address a number of challenging questions related to two important problems in domain identification. The first problem involves the detection and identification of an inhomogeneity inside an otherwise known domain. The second problem involves determination of an unknown portion of the boundary of a domain. In each case, information is provided in the form of overly posed boundary values on a portion of the boundary of the domain. Specific attention will be focused on the development of realistic models for cracks and inclusions inside a thermally conductive body, theoretical and numerical questions related to the inverse problem of crack and inclusion detection from thermal data, the development of stable and reliable algorithms for the determination of an unknown portion of the boundary of a domain from thermal data, and the investigation of a number of issues related to the mathematical modeling of corrosion. When studying physical objects or processes, many properties cannot be measured directly, either because it is impractical to do so, or because unwanted damage to the object would result. Well-known examples include the detection of tumors inside the human body, the detection of oil reservoirs or mineral deposits deep underground, and the detection of hidden damage inside an object such as an aircraft wing or fuselage. The field of inverse problems focuses on determining or estimating these non-measurable properties by measuring other properties, then using a mathematical model (usually a partial differential equation) to connect these measurements to the desired properties. This project investigates the use of thermal methods to determine properties of the inside of objects without damaging them. A portion of the outer surface is heated, then the temperature on the surface is measured over a period of time. The relationship between the measured temperature and the structure of the inside of the object can be modeled mathematically. This project focuses on the analysis of this mathematical model, and on the development of a practical algorithm which will permit the extraction of information about the interior of the object from these temperature measurements. This analysis will provide a theoretical foundation to develop stable and reliable algorithms for use in a wide variety of important applications, including the analysis of corrosion damage inside aircraft wings and fuselages, the inspection of newly manufactured objects for interior cracks or voids (quality control), and in medical diagnostics.