The distributional products of particular distributions

Several Results of Fractional Derivatives in Dʹ(R+)

“…How to define the products of distributions ( [7] and [8]) is a very difficult and not completely understood problem, and has been studied from several points of views since Schwartz established the theory of distributions by treating singular functions as linear and continuous functionals on D(R). Now, we turn our attention to the distribution x λ + given in [5] (λ is a complex number), which will play an important role in defining the fractional derivatives and integrals of distributions in D (R + ), the subspace of D (R) with the supports contained in R + .…”

On defining the distributions δk and (δ′Li1, Li2 2014Applied Mathematics and Computation10020View full textAdd to dashboardCite

“…In fact Fisher's [11] method in the computation of using the neutrix limit can be regarded as a particular application of the neutrix calculus. This is a general principle for the discarding of unwanted infinite quantities from asymptotic expansions and has been exploited in context of distribution by Fisher in connection with the problem of distributional powers, multiplication ( [12,13]), convolution and composition. In 2008, Aguirre [14] applied the Hankel transform to study d 2 ðxÞ as well as d ðmÞ ðxÞd ðlÞ ðxÞ under his definition of product of distributions.…”

The Convolution and Fractional Derivative of Distributions

“…One of the problems in distribution theory is the lack of definition for convolution and product of distributions in general ( [1], [2], [3] and [4]). It is well known that studying fractional calculus of distributions and many types of integral equations are based on distributional convolutions ( [5], [6], [7], [8] and [9]).…”

“…(1) (f * g) = f * g = f * g (2) where all the derivatives above are in the distributional sense. Definitions 1.1 and 1.2 are very restrictive and can only be used for a small class of distributions.…”