Determination of breakpoints and the magnitude of the jump of the original according to its Laplace image

The application of the integral Laplace transform for a wide class of problems leads to a simpler equation for the image of the desired original. At the next step, the inversion problem arises, i. e., finding the original by its image. As a rule, it is not possible to carry out this step analytically. The problem arises of using approximate inversion methods. In this case, the approximate solution is represented as a linear combination of the image and its derivatives at a number of points of the complex half-plane in which the image is regular. However, the original, unlike the image, may even have break points. Of undoubted interest is the task of developing methods for determining the possible break points of the original and the magnitude of the original jump at these points. The proposed methods use the values of high-order image derivatives in order to obtain a satisfactory accuracy of approximate solutions. Methods for accelerating the convergence of the obtained approximations are indicated. The results of numerical experiments illustrating the effectiveness of the proposed methods are presented.