A Regularization Operator for the Source Approximation of a Transport Equation

Abstract

Source identification problems have multiple applications in engineering such as the identificationof fissures in materials, determination of sources in electromagnetic fields or geophysical applications, detection of contaminant sources, among others. In this work we are concerned with thedetermination of a time-dependent source in a transport equation from noisy data measured at a fixedposition. By means of Fourier techniques can be shown that the problem is ill-posed in the sense that thesolution exists but it does not vary continuously with the data. A number of different techniques weredeveloped by other authors to approximate the solution. In this work, we consider a family of parametricregularization operators to deal with the ill-posedness of the problem. We proposed a way to select theregularization parameter as a function of noise level in data in order to obtain a regularized solution thatapproximate the unknown source. We find a Hölder type bound for the error of the approximated sourcewhen the unknown function is considered to be bounded in a given norm. Numerical examples illustratethe convergence and stability of the method.