On the Generation of Rational Function Approximations for Laplace Transform Inversion with an Application to Viscoelasticity

1973 ◽  
Vol 24 (4) ◽  
pp. 429-440 ◽  
Author(s):  
I M. Longman
Keyword(s):  
Laplace Transform ◽  
Rational Function ◽  
Function Approximations ◽  
Laplace Transform Inversion

The solution of theoretical seismic problems by best rational function approximations for Laplace transform inversion

1975 ◽  
Vol 65 (4) ◽  
pp. 927-935
Author(s):  
I. M. Longman ◽  
T. Beer
Keyword(s):  
Laplace Transform ◽  
Rational Function ◽  
High Accuracy ◽  
Time Variable ◽  
Inversion Method ◽  
The Laplace Transform ◽  
Previously Treated ◽  
The Mean ◽  
Function Approximations ◽  
Laplace Transform Inversion

Abstract In a recent paper, the first author has developed a method of computation of “best” rational function approximations ḡn(p) to a given function f̄(p) of the Laplace transform operator p. These approximations are best in the sense that analytic inversion of ḡn(p) gives a function gn(t) of the time variable t, which approximates the (generally unknown) inverse f(t) of f̄(p in a minimum least-squares manner. Only f̄(p) but not f(t) is required to be known in order to carry out this process. n is the “order” of the approximation, and it can be shown that as n tends to infinity gn(t) tends to f(t) in the mean. Under suitable conditions on f(t) the convergence is extremely rapid, and quite low values of n (four or five, say) are sufficient to give high accuracy for all t ≧ 0. For seismological applications, we use geometrical optics to subtract out of f(t) its discontinuities, and bring it to a form in which the above inversion method is very rapidly convergent. This modification is of course carried out (suitably transformed) on f̄(p), and the discontinuities are restored to f(t) after the inversion. An application is given to an example previously treated by the first author by a different method, and it is a certain vindication of the present method that an error in the previously given solution is brought to light. The paper also presents a new analytical method for handling the Bessel function integrals that occur in theoretical seismic problems related to layered media.

Laplace transform inversion through the theta algorithm for power-system EMT analysis

2021 ◽  
Vol 197 ◽  
pp. 107342
Author(s):  
L.J. Castañón ◽  
J.L. Naredo ◽  
J.R. Zuluaga ◽  
E. Bañuelos-Cabral ◽  
Pablo Gómez
Keyword(s):  
Power System ◽  
Laplace Transform ◽  
Laplace Transform Inversion

scholarly journals The duality principle and new forms of the inverse Laplace transform for the analysis of signal propagation in inhomogeneous media with dispersion

2019 ◽  
Vol 489 (6) ◽  
pp. 558-563
Author(s):  
A. G. Pavelyev ◽  
A. A. Pavelyev
Keyword(s):  
Laplace Transform ◽  
Impulse Response ◽  
Angular Spectrum ◽  
Signal Propagation ◽  
Inhomogeneous Media ◽  
Inverse Laplace Transform ◽  
Duality Principle ◽  
Causality Principle ◽  
Speed Of Light ◽  
Laplace Transform Inversion

New equations for Laplace transform inversion are obtained. The equations satisfy the causality principle. The impulse response of a channel is determined in order to analyze dispersion distortions in inhomogeneous media. The impulse response excludes the possibility that the signal exceeds the speed of light in the medium. The transmission bandwidth, the angular spectrum, and the Doppler shift in the ionosphere are computed.

Sign in / Sign up

Export Citation Format

Share Document