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.