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$$\psi $$ ψ -Hilfer pseudo-fractional operator: new results about fractional calculus
Computational and Applied Mathematics ( IF 2.5 ) Pub Date : 2020-08-27 , DOI: 10.1007/s40314-020-01304-6
J. Vanterler da C. Sousa , Gastão S. F. Frederico , E. Capelas de Oliveira

In this paper, we introduce the \(\psi \)-Hilfer pseudo-fractional operator, motivated by the \(\psi \)-Hilfer fractional derivative and the theory of pseudo-analysis. We investigate a wide class of important and essential results for pseudo-fractional calculus in a semiring \(([a, b], \oplus , \odot )\) and some particular cases are discussed. Specifically, we present a class of pseudo-fractional operators which are particular cases of the \(\psi \)-Hilfer pseudo-fractional operator. In addition, we present the pseudo-Leibniz-type rules I and II and pseudo-Leibniz rules and some particular cases of Leibniz-type rules I and II are discussed. Finally, we obtain formulas for the Hilfer pseudo-fractional derivative, for the pseudo-Laplace transform, and for the g-integration by parts of the \(\psi \)-Hilfer pseudo-fractional operator.

中文翻译:

$$ \ psi $$ψ-希尔弗(Hilfer)伪分数算子:关于分数演算的新结果

在本文中,我们介绍了\(\ psi \)- Hilfer伪分数算子,其受\(\ psi \)- Hilfer分数导数和伪分析理论的启发。我们研究半环\(([[a,b],\ oplus,\ odot)\)中伪分数阶微积分的一系列重要的重要结果,并对某些特殊情况进行了讨论。具体来说,我们提供了一类伪分数运算符,它们是\(\ psi \)的特殊情况-Hilfer伪分数运算符。另外,我们给出了伪莱布尼兹型规则I和II和伪莱布尼兹规则,并讨论了莱布尼兹型规则I和II的一些特殊情况。最后,我们得到公式的Hilfer的伪分数阶导数,对伪拉普拉斯变换,并为由的部分-整合\(\ PSI \) -Hilfer伪小数运算符。
更新日期:2020-08-27
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