Reconstruction of time-dependent right-hand side in parabolic equations on disjoint domains

Abstract

We study inverse problems of reconstructing the time-dependent right-hand side from point observation in a one dimensional parabolic equation on disjoint intervals. These problems are ill-possed, i.e. very slight errors in the additional input may cause relatively significant errors in the output of the left and right internal right-hand side. In this work, we construct computational algorithms, using the loaded equation method. First, we perform a decomposition with respect to the unknown source of the inverse problem solutions. Then the inverse problems are reduced to a loaded parabolic equation problems. The well-posedness of the inverse problems is studied on the base of loaded equation ones. The numerical performance of the approach is realized by finite difference schemes, solved with decomposition algorithms. Computational experiments show the efficiency of the method.