Characteristics of Convergence and Stability of Some Methods for Inverting the Laplace Transform

The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations, in which the unknowns are either the coefficients of series expansion in special functions or approximate values of the sought original at a number of points. Various handling methods are considered, and their characteristics of accuracy and stability are indicated, which are required when choosing a handling method for solving applied problems. Quadrature inversion formulas adapted for inversion of long-term and slowly occurring processes of linear viscoelasticity were constructed. A method is proposed for deforming the integration contour in the Riemann-Mellin inversion formula, which leads the problem to the calculation of definite integrals and makes it possible to obtain estimates of the error.