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|Issue Date: ||21-Feb-2018|
|Authors: ||Licciardi, Silvia|
|Title: ||Umbral Calculus A Different Mathematical Language|
|Abstract: ||The thesis is aimed at a thorough exposition of the Umbral Method, relevant in the
theory of special functions, for the solution of ordinary and partial differential
equations, including those of fractional nature. It will provide an account
of the theory and applications of Operational Methods allowing the "translation"
of the theory of special functions and polynomials into a "different"
mathematical language. The language we are referring to is that of symbolic
methods, largely based on a formalism of umbral type which provides a
tremendous simplification of the derivation of the associated properties, with
significant advantages from the computational point of view, either analytical
or to derive efficient numerical methods to handle integrals, ordinary and
partial differential equations, special functions and physical problems solutions.
The strategy we will follow is that of establishing the rules to replace
higher trascendental functions in terms of elementary functions, taking advantage
from such a recasting.
Albeit the point of view discussed here is not equivalent to that developed
by Rota and coworkers, we emphasize that it deepens its root into the Heaviside
operational calculus and into the methods introduced by the operationalists
(Sylvester, Boole, Glaisher, Crofton and Blizard) of the second half of the XIX century.
The method has opened new avenues to deal with rational, trascendental
and higher order trascendental functions, by the use of the same operational
forms. The technique had been formulated in general enough terms to be
readily extended to the fractional calculus. The starting point of our theory
is the use of the Borel transform methods to put the relevant mathematical
foundation on rigorous grounds.
Our target is the search for a common thread between special functions,
the relevant integral representation, the differential equations they satisfy
and their group theoretical interpretation, by embedding all the previously
quoted features within the same umbral formalism.
The procedure we envisage allows the straightforward derivation of (not
previously known) integrals involving e.g. the combination of special functions
or the Cauchy type partial differential equations (PDE) by means of
new forms of solution of evolution operator, which are extended to fractional
PDE. It is worth noting that our methods allow a new definition of fractional
forms of Poisson distributions different from those given in processes involving
A noticeable amount of work has been devoted to the rigorous definition
of the evolution operator and in particular the problem of its hermiticity
properties and more in general of its invertibility. Much effort is devoted to
the fractional ordering problem, namely the use of non-commuting operators
in fractional evolution equations and to time ordering.
We underscore the versatility and the usefulness of the proposed procedure
by presenting a large number of application of the method in different
fields of Mathematics and Physics .|
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