Problems Involving Laplace Transforms with Fractional Powers

1994 ◽  
pp. 671-675
Keyword(s):  
Laplace Transforms ◽  
Fractional Powers

scholarly journals The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1273
Author(s):  
Alexander Apelblat ◽  
Armando Consiglio ◽  
Francesco Mainardi
Keyword(s):  
Hypergeometric Function ◽  
Laplace Transforms ◽  
Confluent Hypergeometric Function ◽  
Integral Function ◽  
Basic Properties ◽  
Differential Relations

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature.

scholarly journals Rational Krylov methods for fractional diffusion problems on graphs

2021 ◽  
Author(s):  
Michele Benzi ◽  
Igor Simunec
Keyword(s):  
Analytic Function ◽  
Diffusion Equation ◽  
Graph Laplacian ◽  
Fractional Diffusion ◽  
Singular Matrix ◽  
Krylov Methods ◽  
Fractional Powers ◽  
Directed Networks ◽  
Rank One ◽  
Diffusion Problems

AbstractIn this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product $$f(L^T) \varvec{b}$$ f ( L T ) b , where f is a non-analytic function involving fractional powers and $$\varvec{b}$$ b is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $$f(L^T) \varvec{b}$$ f ( L T ) b to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

A method for evaluating laplace transforms and other integrals

1997 ◽  
Vol 5 (3-4) ◽  
pp. 161-184
Author(s):  
M.L. Glasser ◽  
V. Kowalenko
Keyword(s):  
Laplace Transforms

scholarly journals Unitarity of theories containing fractional powers of the d’Alembertian operator

2014 ◽  
Vol 90 (10) ◽  
Author(s):  
E. C. Marino ◽  
Leandro O. Nascimento ◽  
Van Sérgio Alves ◽  
C. Morais Smith
Keyword(s):  
Fractional Powers

Closed-form representations of the laplace transforms of bessel funtions

2000 ◽  
Vol 9 (3) ◽  
pp. 217-228 ◽  
Author(s):  
Klaus Rottbrand ◽  
Christian Weddigen
Keyword(s):  
Closed Form ◽  
Laplace Transforms
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