TY - JOUR

T1 - Methods for Laplace transform inversion

AU - Poroshina, N.I.

AU - Ryabov, V.M.

PY - 2011

Y1 - 2011

N2 - In solving problems by operational methods, the most difficult stage is inversion, i.e., the reconstruction of an original from its image. There is no universal inversion method which gives satisfactory results for any image F(p). Any particular inversion method must take into account the specific behavior of the image (or of the original function). The choice of an inversion method essentially depends on the representation of the available information about the image of the sought-for original. Typical situations are as follows: (i) the values of the image F(p) and its derivatives at some fixed point different from infinity are known (ii) the values of the image F(p) and its derivatives in some vicinity of the infinite point are known (iii) the values of the image F(p) on the real semiaxis p ≥ 0 are known (iv) the values of the image F(p) in a half-plane of the form Rep > λ are known. The purpose of the paper is to specify suitable inversion methods, give their detailed description or references, and develop new methods. Computational schemes of the methods and ways to accelerate their convergence are outlines. Methods for reconstructing originals in the form of Laguerre series, various quadrature inversion formulas (both real and complex), ways of deforming the contour in the Riemann-Mellin integral (which determines the inversion of the Laplace transform) and calculating it, and the Widder methods are also described.

AB - In solving problems by operational methods, the most difficult stage is inversion, i.e., the reconstruction of an original from its image. There is no universal inversion method which gives satisfactory results for any image F(p). Any particular inversion method must take into account the specific behavior of the image (or of the original function). The choice of an inversion method essentially depends on the representation of the available information about the image of the sought-for original. Typical situations are as follows: (i) the values of the image F(p) and its derivatives at some fixed point different from infinity are known (ii) the values of the image F(p) and its derivatives in some vicinity of the infinite point are known (iii) the values of the image F(p) on the real semiaxis p ≥ 0 are known (iv) the values of the image F(p) in a half-plane of the form Rep > λ are known. The purpose of the paper is to specify suitable inversion methods, give their detailed description or references, and develop new methods. Computational schemes of the methods and ways to accelerate their convergence are outlines. Methods for reconstructing originals in the form of Laguerre series, various quadrature inversion formulas (both real and complex), ways of deforming the contour in the Riemann-Mellin integral (which determines the inversion of the Laplace transform) and calculating it, and the Widder methods are also described.

U2 - 10.3103/S1063454111030071

DO - 10.3103/S1063454111030071

M3 - Article

VL - 44

SP - 214

EP - 222

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -