On an inverse problem for the heat equation that models the detection of defect in metallic plate whose lower part is embedded

摘要

In this work, we will study an inverse problem to determine corrosion in an inaccessible location of a metallic plate. Our study area is inside the plate metallic plate whose lower part is embedded, therefore inaccessible. We will perform measurements on the upper part of the plate, which is not in contact with the ground. For this, we will make measurements in this part. This problem is modeled by a mixed heat problem with presence of an unknown term in the boundary and initial conditions; this term is an unknown function which can take several forms. It is this function that we will detect the presence or absence of defect that occurs inside the plate. We will then follow our steps to the top edge of the field information on the evolution of this corrosion. We will first formulate our problem which is an inverse problem and we will make a theoretical study. We will show that this problem has a unique solution also this solution is stable. After, we will solve this problem by constructing an iterative algorithm which gives a series of cross problems which give the values of unknown functions which determine the rate of defect. Finally we study the convergence and make a numerical application.