Source identification for a 1D no-linear transportation equation

GUILLERMO FEDERICO UMBRICHT; DIANA RUBIO

Los Polvorines, Buenos Aires

Workshop; Primer workshop en la UNGS: Fenómenos de Transporte y Procesos fuera del Equilibrio; 2018

Universidad Nacional General Sarmiento

The problem of source identification has been widely studied andanalyzed due to its dissimilar and multiple applications found in thedierent fields of science and engineering, such as heat conduction,crack identification, electromagnetic theory, geophysical prospecting,detection of contaminants and detection of tumor cells.In this paper, we are concerned with the identification of a timeindependentsource for a one-dimensional transport equation from noisymeasurements taken at a fixed time instant. This problem has directapplications in heat transfer processes with dissipation due to convectionand the presence of lateral ow. It is also related to onedimensionalpollution control problems, such as the detection of pollutionsource location and its intensity in groundwater, that is of greatinterest in environmental science.This is an ill-posed inverse problem in the sense of Hadamard, sinceif there is a solution, it is not stable or does not depend continuouslyon the data. Diferent approaches can be found in the literature thatpropose techniques to deal with the ill-posedness of the problem. Inthis work, we consider a regularization method to approximate theexact solution. Specifically, we consider a modification to the modelingequation that consists in the addition of a penalty term that dependson a regularization parameter.We show the stability of the method and found a Holder type boundfor the error between the exact and the regularized solutions. Numericalexamples are considered that illustrate the performance of themethod.method.