On the properties of some inversion methods of 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 the series expansion in terms of special functions, or the approximate values of the desired original at a number of points. Described method of inversion using special quadrature formulas of the highest degree of accuracy and the characteristics of the accuracy and stability of this method are indicated. Quadrature inversion formulas are constructed, which are adapted for the inversion of long-term and slow processes of linear viscoelasticity. A method of deformation of the integration contour in the Riemann-Mellin inversion formula is proposed, which leads the problem to the calculation of certain integrals and allows obtaining error estimates. A method is described for determining the possible breakpoints of the original and calculating the magnitude of the jump at these points.