The accurate numerical inversion of laplace transforms
ABSTRACT Inversion of almost arbitrary Laplace transforms is effected by trapezoidal integration along a special contour. The number n of points to be used is one of several parameters, in most cases yielding absolute errors of order 10-7 for n = 10, 10-11 for n = 20, 10-23 for n = 40 (with double precision working), and so on, for all values of the argument from 0+ up to some large maximum. The extreme accuracy of which the method is capable means that it has many possible applications of various kinds, and some of these are indicated.
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ABSTRACT:  A flowing partially penetrating well with infinitesimal well skin is a mixed boundary because a Cauchy condition is prescribed along the screen length and a Neumann condition of no flux is stipulated over the remaining unscreened part. An analytical approach based on the integral transform technique is developed to determine the Laplace domain solution for such a mixed boundary problem in a confined aquifer of finite thickness. First, the mixed boundary is changed into a homogeneous Neumann boundary by substituting the Cauchy condition with a Neumann condition in terms of well bore flux that varies along the screen length and is time dependent. Despite the well bore flux being unknown a priori, the modified model containing this homogeneous Neumann boundary can be solved with the Laplace and the finite Fourier cosine transforms. To determine well bore flux, screen length is discretized into a finite number of segments, to which the Cauchy condition is reinstated. This reinstatement also restores the relation between the original model and the solutions obtained. For a given time, the numerical inversion of the Laplace domain solution yields the drawdown distributions, well bore flux, and the well discharge. This analytical approach provides an alternative for dealing with the mixed boundary problems, especially when aquifer thickness is assumed to be finite.Water Resources Research. 06/2002; 38(6).
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ABSTRACT: A review has been made for the previous studies on safety of a geologic repository for high-level radioactive wastes (HLW) related to autocatalytic criticality phenomena with positive reactivity feedback. Neutronic studies on geometric and materials configuration consisting of rock, water and thermally fissile materials and the radionuclide migration and accumulation studies were performed previously for the Yucca Mountain Repository and a hypothetical water-saturated repository for vitrified HLW. In either case, it was concluded that it would be highly unlikely for an autocatalytic criticality event to happen at a geologic repository. Remaining scenarios can be avoided by careful selection of a repository site, engineered-barrier design and conditioning of solidified HLW. Thus, criticality safety should be properly addressed in regulations and site selection criteria. The models developed for radiological safety assessment to obtain conservatively overestimated exposure dose rates to the public may not be used directly for the criticality safety assessment, where accumulated fissile materials mass needs to be conservatively overestimated. The models for criticality safety also require more careful treatment of geometry and heterogeneity in transport paths because a minimum critical mass is sensitive to geometry of fissile materials accumulation.Nuclear Engineering and Technology 01/2006; 38(6). · 0.58 Impact Factor
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ABSTRACT: New methods are proposed for the numerical evaluation of f(𝐀) or f(𝐀)b, where f(𝐀) is a function such as 𝐀 1/2 or log(𝐀) with singularities in (-∞,0] and 𝐀 is a matrix with eigenvalues on or near (0,∞). The methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving Jacobi elliptic functions. The convergence is geometric, so that the computation of f(𝐀)b is typically reduced to one or two dozen linear system solves, which can be carried out in parallel.SIAM Journal on Numerical Analysis 01/2008; 46(5). · 1.48 Impact Factor
TR/61 April 1976
THE ACCURATE NUMERICAL INVERSION OF
The accurate numerical inversion of Laplace transforms
Mathematics Department, Brunei University, Uxbridge
Inversion of almost arbitrary Laplace transforms is
effected by trapezoidal integration along a special
contour. The number n of points to be used is one
of several parameters, in most cases yielding absolute
errors of order 10-7 for n = 10, 10-11 for n = 20,
10-23 for n = 40 (with double precision working), and
so on, for all values of the argument from 0+ up to
some large maximum.
The extreme accuracy of which the method is capable
means that it has many possible applications of
various kinds, and some of these are indicated.
- 1 -
The inversion of Laplace transforms is a topic of fundamental
importance in many areas of applied mathematics, as would be
evident by a glance at, for example, Carslaw and Jaeger (1948).
In the more standard applications the inversion can he accomplished
by the use of a dictionary of transforms, or in the case of
rational function transforms by partial fraction decomposition.
Where these methods are of no avail recourse may be had to the
inversion integral formula, which is likely to lead to an
intractable integral, or to an infinite series, often with
terms involving the roots of some transcendental function. It
is clear that in all but the simplest cases considerable effort
is needed to obtain an accurate numerical value of the inverse
for a specified value of the argument.
It is therefore natural that attention has been paid by
mathematicians, engineers, physicists and others to alternative
ways of evaluating the inverse. Early methods (e.g. Widder (1935),
Tricomi (1935), Shohat (1940)) involved expansion of the inverse
in series of Laguerre functions. Salzer (1955) evaluated the
inversion integral by Gaussian quadrature using an appropriate
system of orthogonal polynomials. Since 1955 et very large number
of methods for numerical inversion have been published: see for
example the partial bibliography in Piessens and Branders (1971)
or the fuller one in Piessens (1974) . A useful critical survey
of earlier work was given by Weeks (1966).
Many of the methods use either orthogonal series expansions, or
weighted sums of values of the transform at a set of points,
usually complex points. In either case considerable preliminary
work must be carried out. In the second type this may be done in
advance once and for all for each selected set of points, and the
points and weights stored in the computer. However, if more points
are desired for the sake of gaining increased accuracy,much further
computational effort must be expended first.
In general the methods hitherto published have been intended for
use with transforms of particular types, e.g. rational functions