Reconstruction Of Parabolic Partial Differential Equations In The Unknown Region Inverse Problem And Its Algorithm

In this thesis, we discuss a mathematical study of non-destructive testing of media. Usually, acoustic waves, electromagnetic waves, electric current and heat are used for the non-destructive testing to find cracks, cavities and inclusions inside the medium. We will mainly focus on the non-destructive testing of an unknown inclusion of heat conductor by ejecting heat from the boundary of the medium and measuring the temperature on the boundary.In Chapter 1, we present the background of our problems, which originated from the study of non-destructive testing of media. Finally, we also introduce the some main inclusions about this.In Chapter 2, we introduce some definitions of basic space and their relational properties. And, we also introduce the regularization method, which will be used in our numerical realization.In Chapter 3, in this part, we introduce the mathematic model for heat equation, and list some basic properties for heat equation. Finally, we give the algorithm for the numerical solution of heat equation, by using finite element method.In Chapter 4, we consider an inverse problem for identifying an inclusion inside an isotropic homogeneous heat conductive medium from Neumann-to-Dirichlet map, by using probe method. Roughly speaking, this method is to construct a function I called indicator function based on a needle, when the needle approaches boundary of inclusion, the indicator function will blow up, using this property, we can reconstruct the boundary of the inclusion. For one and two dimensional space case, we will show the blow-up properties and carry out numerical implementation of this method.In Chapter 5, we present the size estimate approach in the thermal imaging problem, which consists of applying a heat flux to the surface of a medium and observing the temperature over time. We estimate the Lebesgue measure |D| of inclusion in term of the boundary data provided D is compactly contained inΩ, and also perform our numerical tests with two different choice of boundary data to check how this effects the accuracy of our results.