Determination of Discontinuity Points and the Jump Magnitude of the Original Based on Its Laplace Image

Abstract

The application of the integral Laplace transform to a wide class of problems leads to a simpler equation relative to the image of the desired original. At the next step, the inversion problem (i.e., the problem of finding the original based on its image) arises. As a rule, this step cannot be carried out analytically, and the problem arises of using approximate inversion methods. In this case, the approximate solution is represented in the form of a linear combination between the image and its derivatives at certain points of the complex half-plane, in which the image is regular. Unlike the image, however, the original may have even discontinuity points. Of undoubted interest is the task of developing methods for determining the possible discontinuity points of the original as well as the magnitudes of the original jump at these points. The suggested methods imply using values of high-order image derivatives in order to obtain a satisfactory accuracy of approximate solutions. The methods for accelerating the convergence of the obtained approximations are given. The results of numerical experiments which illustrate the efficiency of the suggested techniques are demonstrated.