Regularization of Some Methods of Inversion of the Integral Laplace Transform

For a wide class of problems, the integral Laplace transform leads to a simpler equation with respect to the image of the sought original. The next step is the inversion problem: finding the original by its image. Often, this step cannot be implemented analytically, and approximate inversion methods are required. 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 where the image is analytic. However, unlike the image, the original may even have discontinuity points. Finding frameworks and approximate solutions is reduced to solving systems of linear algebraic equations (SLAEs), which, like the original problem, are ill-conditioned. SLAE matrices have different properties, which should be taken into account to reduce their condition number in comparison with existing regularization methods. The results of numerical experiments confirming the efficiency of the proposed inversion algorithms are presented.