scholarly journals Tricomi’s Method for the Laplace Transform and Orthogonal Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 589
Author(s):  
Paolo Emilio Ricci ◽  
Diego Caratelli ◽  
Francesco Mainardi
Keyword(s):  
Laplace Transform ◽  
Orthogonal Polynomials ◽  
Laguerre Polynomials ◽  
Laplace Transforms ◽  
Series Expansions ◽  
Connection Coefficients ◽  
General Series ◽  
The Laplace Transform

Tricomi’s method for computing a set of inverse Laplace transforms in terms of Laguerre polynomials is revisited. By using the more recent results about the inversion and the connection coefficients for the series of orthogonal polynomials, we find the possibility to extend the Tricomi method to more general series expansions. Some examples showing the effectiveness of the considered procedure are shown.

scholarly journals Laplace Transforms and Generalized Laguerre Polynomials

1958 ◽  
Vol 10 ◽  
pp. 177-182 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Laplace Transform ◽  
Laguerre Polynomials ◽  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Representation Theorems ◽  
Generalized Laguerre Polynomials ◽  
Inversion Operator ◽  
The Laplace Transform ◽  
Necessary And Sufficient

Various sets of necessary and sufficient conditions are known in order that a function ƒ(s), analytic for Re s > 0, be represented as the Laplace transform of a function in L p(0,∞), 1 < p ⩽ ∞ . Most of these theories are based on the properties of some inversion operator for the transformation— see, for example, (7, chap. 7). However in the case p = 2 a number of representation theorems of a much simpler type are available.

Analysis of the fluid weighted fair queueing system

2003 ◽  
Vol 40 (1) ◽  
pp. 180-199 ◽  
Author(s):  
Fabrice Guillemin ◽  
Ravi Mazumdar ◽  
Alain Dupuis ◽  
Jacqueline Boyer
Keyword(s):  
Laplace Transform ◽  
Performance Measures ◽  
Joint Distribution ◽  
Queueing System ◽  
Laplace Transforms ◽  
Fair Queueing ◽  
Mean Waiting Time ◽  
The Laplace Transform ◽  
The Mean

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

A result on the Laplace transform associated with the sticky Brownian motion on an interval

2020 ◽  
pp. 2150031
Author(s):  
Shiyu Song
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Laplace Transforms ◽  
Perturbation Approach ◽  
Integrated Process ◽  
Modified Bessel Function ◽  
Airy Functions ◽  
Occupation Times ◽  
The Laplace Transform

In this paper, we study the joint Laplace transform of the sticky Brownian motion on an interval, its occupation time at zero and its integrated process. The perturbation approach of Li and Zhou [The joint Laplace transforms for diffusion occupation times, Adv. Appl. Probab. 45 (2013) 1049–1067] is adopted to convert the problem into the computation of three Laplace transforms, which is essentially equivalent to solving the associated differential equations with boundary conditions. We obtain the explicit expression for the joint Laplace transform in terms of the modified Bessel function and Airy functions.

Analysis of the fluid weighted fair queueing system

2003 ◽  
Vol 40 (01) ◽  
pp. 180-199
Author(s):  
Fabrice Guillemin ◽  
Ravi Mazumdar ◽  
Alain Dupuis ◽  
Jacqueline Boyer
Keyword(s):  
Laplace Transform ◽  
Performance Measures ◽  
Joint Distribution ◽  
Queueing System ◽  
Laplace Transforms ◽  
Fair Queueing ◽  
Mean Waiting Time ◽  
The Laplace Transform ◽  
The Mean

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

scholarly journals Ultimate Ruin Probabilities for Generalized Gamma-Convolutions Claim Sizes

2001 ◽  
Vol 31 (1) ◽  
pp. 59-79 ◽  
Author(s):  
M. Usábel
Keyword(s):  
Laplace Transform ◽  
Laplace Transforms ◽  
Upper Bounds ◽  
Interesting Property ◽  
Ruin Probabilities ◽  
Lower And Upper Bounds ◽  
Recursive Methods ◽  
Stable Algorithm ◽  
The Laplace Transform ◽  
Generalized Gamma Convolutions

AbstractA method of inverting the Laplace transform based on the integration between zeros technique and a simple acceleration algorithm is presented. This approach was designed to approximate ultimate ruin probabilities for Γ-convolutions claim sizes, but it can be also used with other distributions. The stable algorithm obtained yields interval approximations (lower and upper bounds) to any desired degree of accuracy even for very large values of u (1,000,000), initial reserves, without increasing the number of computations. This last fact can be considered an interesting property compared with other recursive methods previously used in actuarial literature or other methods of inverting Laplace transforms.

A Student's Guide to Laplace Transforms

2022 ◽  
Author(s):  
Daniel Fleisch
Keyword(s):  
Laplace Transform ◽  
Physical Meaning ◽  
Applied Mathematics ◽  
Worked Examples ◽  
Laplace Transforms ◽  
Mathematical Tool ◽  
Plain Language ◽  
The Laplace Transform ◽  
Supporting Materials

The Laplace transform is a useful mathematical tool encountered by students of physics, engineering, and applied mathematics, within a wide variety of important applications in mechanics, electronics, thermodynamics and more. However, students often struggle with the rationale behind these transforms, and the physical meaning of the transform results. Using the same approach that has proven highly popular in his other Student's Guides, Professor Fleisch addresses the topics that his students have found most troublesome; providing a detailed and accessible description of Laplace transforms and how they relate to Fourier and Z-transforms. Written in plain language and including numerous, fully worked examples. The book is accompanied by a website containing a rich set of freely available supporting materials, including interactive solutions for every problem in the text, and a series of podcasts in which the author explains the important concepts, equations, and graphs of every section of the book.

NUMERICAL TRANSFORM INVERSION USING GAUSSIAN QUADRATURE

2005 ◽  
Vol 20 (1) ◽  
pp. 1-44 ◽  
Author(s):  
Peter den Iseger
Keyword(s):  
Laplace Transform ◽  
Stochastic Models ◽  
A Priori ◽  
Gaussian Quadrature ◽  
Laplace Transforms ◽  
Numerical Inversion ◽  
Inversion Algorithm ◽  
Machine Precision ◽  
The Laplace Transform ◽  
Laplace Transform Inversion

Numerical inversion of Laplace transforms is a powerful tool in computational probability. It greatly enhances the applicability of stochastic models in many fields. In this article we present a simple Laplace transform inversion algorithm that can compute the desired function values for a much larger class of Laplace transforms than the ones that can be inverted with the known methods in the literature. The algorithm can invert Laplace transforms of functions with discontinuities and singularities, even if we do not know the location of these discontinuities and singularities a priori. The algorithm only needs numerical values of the Laplace transform, is extremely fast, and the results are of almost machine precision. We also present a two-dimensional variant of the Laplace transform inversion algorithm. We illustrate the accuracy and robustness of the algorithms with various numerical examples.

Brownian excursions and Parisian barrier options: a note

2003 ◽  
Vol 40 (4) ◽  
pp. 855-864 ◽  
Author(s):  
Michael Schröder
Keyword(s):  
Laplace Transform ◽  
Laplace Transforms ◽  
Barrier Options ◽  
The Laplace Transform

This paper addresses Paris barrier options, as introduced by G. Kentwell and J. Cornwall at Bankers Trust Australia in the mid-1990s, and their valuation, as developed by Chesnay, Jeanblanc-Picqué and Yor using the Laplace-transform approach. The notion of Paris barrier options is extended so that their valuation becomes possible at any point during their lifespan, and the pertinent Laplace transforms of Chesnay, Jeanblanc-Picqué and Yor are modified when necessary.

scholarly journals The distribution of refracted Lévy processes with jumps having rational Laplace transforms

2017 ◽  
Vol 54 (4) ◽  
pp. 1167-1192
Author(s):  
Jiang Zhou ◽  
Lan Wu
Keyword(s):  
Laplace Transform ◽  
Diffusion Process ◽  
Jump Diffusion ◽  
Laplace Transforms ◽  
Variable Annuities ◽  
State Dependent ◽  
Approximating Procedure ◽  
Jump Diffusion Process ◽  
The Laplace Transform ◽  
Interesting Approach

Abstract We consider a refracted jump diffusion process having two-sided jumps with rational Laplace transforms. For such a process, by applying a straightforward but interesting approach, we derive formulae for the Laplace transform of its distribution. Our formulae are presented in an attractive form and the approach is novel. In particular, the idea in the application of an approximating procedure is remarkable. In addition, the results are used to price variable annuities with state-dependent fees.

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