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.
REFERENCES
M. A. Lavrent’ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable (Lan’, Moscow, 2002) [in Russian].
V. I. Krylov and N. S. Skoblya, Methods of the Approximate Fourier Transform and the Inversion of the Laplace Transform (Nauka, Moscow, 1974) [in Russian].
B. Davies and B. Martin, “Numerical inversion of the Laplace transform: A survey and comparison of methods,” J. Comput. Phys. 33 (1), 1–32 (1979).
V. A. Ditkin and A. P. Prudnikov, Operational Calculus (Vysshaya Shkola, Moscow, 1975) [in Russian].
D. V. Widder, The Laplace Transform (Princeton Univ. Press, Princeton, Calif., 1946).
G. Doetsch, Guide to the Applications of the Laplace and Z-Transforms (Van Nostrand, London, 1971; Nauka, Moscow, 1971).
A. I. Lur’e, Operational Calculus and Its Application to Problems in Mechanics (Gostekhizdat, Moscow, 1951) [in Russian].
L. I. Slepyan and Yu. S. Yakovlev, Integral Transformations in Unsteady Problems of Mechanics (Sudostroenie, Leningrad, 1980) [in Russian].
A. M. Cohen, Numerical Methods for Laplace Transform Inversion (Springer-Verlag, New York, 2007).
V. M. Ryabov, Numerical Inversion of the Laplace Transform (S.-Peterb. Gos. Univ., St. Petersburg, 2013) ) [in Russian].
P. K. Suetin, Classical Orthogonal Polynomials (Nauka, Moscow, 1976) [in Russian].
V. I. Krylov and N. S. Skoblya, Reference Book on the Numerical Inversion of the Laplace Transform (Nauka i Tekhnika, Minsk, 1968) [in Russian].
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1979; Halsted, New York, 1977).
S. I. Kabanikhin, Inverse and Ill-Posed Problem (Sib. Nauchn. Izd., Novosibirsk, 2009) [in Russian].
A. V. Lebedeva and V. M. Ryabov, “On regularization of the solution of integral equations of the first kind using quadrature formulas,” Vestn. St. Petersburg Univ.: Math. 54, 361–365 (2021). https://doi.org/10.1134/S1063454121040129
A. V. Lebedeva and V. M. Ryabov, “Method of moments in the problem of inversion of the Laplace transform and its regularization,” Vestn. St. Petersburg Univ.: Math. 55, 34–38 (2022). https://doi.org/10.1134/S1063454122010071
Yu. N. Rabotnov, Elements of Hereditary Solid Mechanics (Nauka, Moscow, 1977; Mir, Moscow, 1980).
V. S. Ekel’chik and V. M. Ryabov, “On the use of one class of hereditary kernels in linear equations of viscoelasticity,” Mekh. Kompoz. Mater. 3, 393–404 (1981) [in Russian].
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (Imperial College Press, London, 2010). https://doi.org/10.1142/p614
A. V. Lebedeva and V. M. Ryabov, “On integration contour deformation in a Laplace transform inversion formula,” Comput. Math. Math. Phys. 55, 1103–1109 (2015). https://doi.org/10.1134/S0965542515050139
T. A. Matveeva, Some Methods for Inverting the Laplace Transform and Their Applications, Candidate’s Dissertation in Mathematics and Physics (St. Petersburg State Univ., St. Petersburg, 2003).
V. M. Ryabov, “Calculation of original jumps from its image using quadrature formulas,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron., No. 1, 36–39 (1998) ) [in Russian].
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This study was supported by a grant of St. Petersburg State University Event 3 (Pure ID 75207094).
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Translated by I. Nikitin
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Lebedeva, A.V., Ryabov, V.M. On the Properties of Some Inversion Methods of the Laplace Transform. Vestnik St.Petersb. Univ.Math. 56, 27–34 (2023). https://doi.org/10.1134/S1063454123010089
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DOI: https://doi.org/10.1134/S1063454123010089