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Axioms 2019, 8(1), 35; https://doi.org/10.3390/axioms8010035

Operator Ordering and Solution of Pseudo-Evolutionary Equations

1
Institut de Recherche en Informatique Fondamentale (IRIF), Université Paris-Diderot, F-75013 Paris, France
2
ENEA—Frascati Research Center, Via Enrico Fermi 45, 00044 Rome, Italy
3
H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Science, 31-342 Kraków, Poland
*
Author to whom correspondence should be addressed.
Received: 1 February 2019 / Revised: 8 March 2019 / Accepted: 13 March 2019 / Published: 16 March 2019
(This article belongs to the Special Issue Fractional Differential Equations)
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Abstract

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance. View Full-Text
Keywords: pseudo-evolutionary differential equations; fractional differential equations; operational methods; umbral image techniques pseudo-evolutionary differential equations; fractional differential equations; operational methods; umbral image techniques
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Behr, N.; Dattoli, G.; Lattanzi, A. Operator Ordering and Solution of Pseudo-Evolutionary Equations. Axioms 2019, 8, 35.

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