On an Inverse Problem that Models the Detection of Corrosion in Metallic Plate whose Lower Part is Embedded

In this work, we will study an inverse problem to determine the corrosion in an inaccessible location of a metallic plate. Our study area is inside a 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 send an electric field on this part, and take measurements. This problem is modeled by a mixed Laplace problem with presence of an unknown term in the boundary conditions; this term is an unknown function which can take several forms. This function detects the presence or absence of corrosion inside the plate. For this, we make electrical measurements on different parts of the plate on different time intervals, this gives us information about detection and evolution of the corrosion on this part of the plate. 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 approximate values of impedance functions, which determine the rate of corrosion. Finally we study the convergence and make a numerical application.