This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this purpose, we also describe two methods for inverting Laplace transforms based on the Post-Widder inversion formula. The overall procedure is illustrated by several queueing examples.

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The Fourier-series method for inverting transforms of probability distributions

Fractional order theory of thermoelasticity

Theory of Fractional Order Generalized Thermoelasticity

A half-space problem in the theory of generalized thermoelastic diffusion

Faults often separate materials with different elastic properties. Nonuniform slip on such faults induces a change in normal stress. That suggests the possibility of self-sustained slip pulses [Weertman, 1980] propagating at the generalized Rayleigh wave speed even with a Coulomb constitutive law (i.e., with a constant coefficient of friction) and a remote driving shear stress that is arbitrarily less than the corresponding frictional strength. Following Andrews and Ben-Zion [1997] (ABZ), we study numerically, with a two-dimensional (2-D) plane strain geometry, the propagation of ruptures along such a dissimilar material interface. However, this problem has been shown to be ill-posed for a wide range of elastic material contrasts [Renardy, 1992; Martins and Simoes, 1995; Adams, 1995]. Ranjith and Rice [2000] (RR) showed that when the generalized Rayleigh speed exists, as is the case for the material contrast studied by ABZ, the problem is ill-posed for all values of the coefficient of friction, f, whereas when it does not exist, the problem is ill-posed only for f greater than a critical value. We illustrate the ill-posedness by showing that in the unstable range the numerical solutions do not converge through grid size reduction. By contrast, convergence is achieved in the stable range but, not unexpectedly, only dying pulses are then observed. RR showed that among other regularization procedures, use of an experimentally based law [Prakash and Clifton, 1993; Prakash, 1998], in which the shear strength in response to an abrupt change in normal stress evolves continuously with time or slip toward the corresponding Coulomb strength, provides a regularization. (Classical slip weakening or rate- and state-dependent constitutive laws having the same kind of abrupt response as Coulomb friction also do not regularize the problem.) Convergence through grid size reduction is then achieved in the otherwise ill-posed range. For sufficiently rapid shear strength evolution, self-sustained pulses are observed. When the generalized Ray-leigh wave speed exists, they propagate essentially at that velocity and, in consistence with Weertman's [1980] analysis, the propagation occurs only in one direction, which is that of slip in the more compliant medium. When the generalized Rayleigh wave speed does not exist, similar self-sustained pulses propagate at about the slower S wave speed and in the same direction. RR also suggested that for sufficiently high coefficient of friction, another kind of (less unstable) self-sustained pulses, propagating at a velocity close to the slower P wave speed and in the opposite direction, could also exist. We numerically verify that prediction.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JB900230

G. Honig

Papers

A method for the numerical inversion of Laplace transforms