On the regularization of the ill-posed logarithmic kernel integral equation of the first kind

The logarithmic kernel integral equation of the first kind is investigated as an improperly posed problem by considering its right-hand side as an observed quantity in a suitable space with a weaker norm. The improperly posed problem is decomposed into a well-posed one, extensively studied in the literature, and an ill-posed imbedding problem. For regularizing the ill-posed part a modified truncated singular value decomposition method is proposed that allows an easily performable a posteriori parameter choice. Then the whole problem is solved by combining the regularization method with a numerical procedure for the well-posed part. Finally, an error estimate is given to reveal the influence of the observation error on the approximation error of the numerical procedure. If the discretization parameter is specified as a known function of the noise level only, the optimal convergence order is achieved.