Integrated Exponential Functions, the Weyl Fractional Calculus and the Laplace Transform

2016 ◽  
pp. 223-236
Author(s):  
Pedro J. Miana
Keyword(s):  
Fractional Calculus ◽  
Laplace Transform ◽  
Exponential Functions ◽  
The Laplace Transform ◽  
Weyl Fractional Calculus

The Error Bounds for Three-Dimensional Nodal LTSN Method

2008 ◽  
Author(s):  
Eliete Biasotto Hauser ◽  
Ruben Panta Pazos ◽  
Marco T. M. B. Vilhena ◽  
Ricardo C. Barros
Keyword(s):  
Spectral Analysis ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Error Bounds ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Exponential Functions ◽  
Decay Constants ◽  
The Laplace Transform ◽  
Cartesian Geometry

In this paper we present a proof about the convergence of the 3D Nodal-LTSN Method in order to solve the transport problem in a parallelepiped domain. For that, we define functions associated to the errors, one in the approximated flux, another in the quadrature formula and establish a relation between them. We present a Nodal-LTSN method to generate an analytical solution for discrete ordinates problems in three-dimensional cartesian geometry. We first transverse integrate the SN equations and then we apply the Laplace transform. The essence of this method is the diagonalization of the LTSN transport matrices and the spectral analysis garantees this. The transverse leakage terms that appear in the transverse integrated SN equations are represented by exponential functions with decay constants that depend on the characteristics of the material of the medium the particles leave behind. We present numerical results generated by the offered method applied to typical shielding model problems.

scholarly journals A Generalization of Integral Transform

2018 ◽  
Vol 11 (4) ◽  
pp. 1130-1142
Author(s):  
Benedict Barnes ◽  
C. Sebil ◽  
A. Quaye
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Integral Transform ◽  
Polynomial Function ◽  
Inverse Operator ◽  
Exponential Functions ◽  
Trans Form ◽  
The Laplace Transform ◽  
Func Tion

In this paper, the generalization of integral transform (GIT) of the func-tion G{f (t)} is introduced for solving both differential and interodif-ferential equations. This transform generalizes the integral transformswhich use exponential functions as their kernels and the integral trans-form with polynomial function as a kernel. The generalized integraltransform converts the differential equation in us domain (the trans-formed variables) and reconvert the result by its inverse operator. Inparticular, if u = 1, then the generalized integral transform coincideswith the Laplace transform and this result can be written in anotherform as the polynomial integral transform.

scholarly journals Parameter N Analysis on the Rational-Talbot Algorithm for Numerical Inversion of Laplace Transform

2021 ◽  
Vol 31 (2) ◽  
pp. 50-60
Author(s):  
Elisandra Freitas ◽  
George Ricardo Libardi Calixto ◽  
Juciara Alves Ferreira ◽  
Bárbara Denicol do Amaral Rodriguez ◽  
João Francisco Prolo Filho
Keyword(s):  
Laplace Transform ◽  
Free Parameter ◽  
Numerical Results ◽  
Numerical Inversion ◽  
Good Precision ◽  
Trigonometric Functions ◽  
Inversion Process ◽  
Elementary Functions ◽  
Exponential Functions ◽  
The Laplace Transform

This article investigates the numerical inversion of the Laplace Transform by the Rational-Talbot method and analyzes the influence on the variation of the free parameter N established by the technique when applied to certain functions. The set of elementary functions, for which the method is tested, has exponential and oscillatory characteristics. Based on the results obtained, it was concluded that the Rational-Talbot method is e cient for the inversion of decreasing exponential functions. At the same time, to perform the inversion process effectively for trigonometric forms, the algorithm requires a greater amount of terms in the sum. For higher values of N, the technique works well. In fact, this is observed in inverting the functions transform, that combine trigonometric and polynomial factors. The method numerical results have a good precision for the treatment of decreasing exponential functions when multiplied by trigonometric functions.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

scholarly journals Algebraic inversion of the Laplace transform

2005 ◽  
Vol 50 (1-2) ◽  
pp. 179-185 ◽  
Author(s):  
P.G. Massouros ◽  
G.M. Genin
Keyword(s):  
Laplace Transform ◽  
The Laplace Transform
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