Fractional integrals of generalized functions

1975 ◽  
pp. 151-170 ◽  
Author(s):  
Arthur Erdélyi
Keyword(s):  
Generalized Functions ◽  
Fractional Integrals

scholarly journals Two index laws for fractional integrals and derivatives

1972 ◽  
Vol 14 (4) ◽  
pp. 385-410 ◽  
Author(s):  
E. R. Love
Keyword(s):  
Integral Equations ◽  
Fractional Derivatives ◽  
Addition Theorem ◽  
Positive Real Part ◽  
Generalized Functions ◽  
Fractional Integrals ◽  
Positive Real ◽  
Integrable Functions ◽  
Hypergeometric Integral ◽  
Fractional Integrals And Derivatives

SummaryThe first index law, or addition theorem, is well known. The second is much less well known; but both have been found to be of importance in recent studies of hypergeometric integral equations. The first law has usually been considered only in the simple case of orders of integration which have positive real part, or in the context of generalized functions. Arising out of the need to manipulate expressions involving several fractional integrals and derivatives, our aim here is to establish both laws for all combinations of complex orders of integration and differentiation, and for nearly all functions for which the fractional derivatives involved exist as locally integrable functions.

scholarly journals On fractional integration of generalized functions on a half-line

1995 ◽  
Vol 38 (3) ◽  
pp. 387-396
Author(s):  
B. Rubin
Keyword(s):  
Large Class ◽  
Fractional Integration ◽  
Generalized Functions ◽  
Fractional Integrals ◽  
Half Line ◽  
New Approach

A new approach to fractional integrals of distributions on a half-line is suggested. The results admit an extension to a large class of Mellin convolutions.

scholarly journals Hilbert spaces of generalized functions extending L2, (I)

1960 ◽  
Vol 1 (3) ◽  
pp. 281-298 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Hilbert Spaces ◽  
Generalized Functions ◽  
Fractional Integrals ◽  
Invariance Properties ◽  
Fractional Integrals And Derivatives

By using certain fractional integrals and derivatives it is possible to construct a continuum of Hilbert spaces within the space L2 (0, ∞); these are the spaces gλ of functions f(x) for which 1xλf(λ)(x) є L2(0, ∞), and they exhibit invariance properties under generalized Fourier transformations. They are described in (6) and (7).

scholarly journals Eigenfunctions and Fundamental Solutions of the Fractional Two-Parameter Laplacian

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
Semyon Yakubovich
Keyword(s):  
Fractional Derivatives ◽  
Separation Of Variables ◽  
Fundamental Solutions ◽  
Generalized Functions ◽  
Fractional Integrals ◽  
Summable Function ◽  
Left Hand ◽  
Operational Technique ◽  
Two Parameter ◽  
Logarithmic Solution

We deal with the following fractional generalization of the Laplace equation for rectangular domains(x,y)∈(x0,X0)×(y0,Y0)⊂ℝ+×ℝ+, which is associated with the Riemann-Liouville fractional derivativesΔα,βu(x,y)=λu(x,y),Δα,β:=Dx0+1+α+Dy0+1+β, whereλ∈ℂ,(α,β)∈[0,1]×[0,1]. Reducing the left-hand side of this equation to the sum of fractional integrals byxandy, we then use the operational technique for the conventional right-sided Laplace transformation and its extension to generalized functions to describe a complete family of eigenfunctions and fundamental solutions of the operatorΔα,βin classes of functions represented by the left-sided fractional integral of a summable function or just admitting a summable fractional derivative. A symbolic operational form of the solutions in terms of the Mittag-Leffler functions is exhibited. The case of the separation of variables is also considered. An analog of the fractional logarithmic solution is presented. Classical particular cases of solutions are demonstrated.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

Seismic stability of the underground main pipeline

2011 ◽  
Vol 8 (1) ◽  
pp. 275-286
Author(s):  
R.G. Yakupov ◽  
D.M. Zaripov
Keyword(s):  
Laplace Transformation ◽  
Seismic Waves ◽  
Generalized Functions ◽  
Seismic Stability ◽  
Main Pipeline ◽  
Motion Equations ◽  
Numerical Example ◽  
Earthquake Force ◽  
Deformed State

The stress-deformed state of the underground main pipeline under the action of seismic waves of an earthquake is considered. The generalized functions of seismic impulses are constructed. The pipeline motion equations are solved with used Laplace transformation by the time. Tensions and deformations of the pipeline have been determined. A numerical example is reviewed. Diagrams of change of the tension depending on earthquake force are provided in earthquake-points.

Sign in / Sign up

Export Citation Format

Share Document