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 (SLAEs), in which the unknowns are either the coefficients of the series expansion in terms of special functions or approximate values of the desired original at a number of points. A method of inversion by special quadrature formulas of the highest degree of accuracy (QFHDAs) is described, 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 linear viscoelastic processes. A method of deformation of the integration contour in the Riemann–Mellin inversion formula is proposed, which reduces the problem to the calculation of definite integrals and allows error estimates to be obtained. A method is described for determining the possible discontinuity points of the original and calculating the jump at these points.