Abstract

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.

中文翻译：

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关于拉普拉斯变换某些反演方法的性质

摘要

考虑了属于病态问题的积分拉普拉斯变换的反演问题。积分方程被简化为线性代数方程 (SLAE) 的病态系统，其中的未知数要么是根据特殊函数展开的级数系数，要么是所需原始值在多个点的近似值。描述了一种用最高精度特殊求积公式(QFHDAs)反演的方法，指出了该方法在精度和稳定性方面的特点。构造了正交反演公式，适用于反演长期和缓慢的线性粘弹性过程。提出了黎曼-梅林反演公式中积分等值线变形的方法，这将问题简化为定积分的计算，并允许获得误差估计。描述了一种用于确定原件的可能不连续点并计算这些点处的跳跃的方法。