$$\psi $$-Hilfer pseudo-fractional operator: new results about fractional calculus

Abstract

In this paper, we introduce the ψ-Hilfer pseudo-fractional operator, motivated by the ψ-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], ⊕,) and some particular cases are discussed. Specifically, we present a class of pseudo-fractional operators which are particular cases of the ψ-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 pseudofractional derivative, for the pseudo-Laplace transform, and for the g-integration by parts of the ψ-Hilfer pseudo-fractional operator.

Pseudo-fractional differential equations and generalized g-Laplace transform

“…( 9), respectively. Definition 12 [14] Let a generator g : J → [0, ∞] of the pseudo-addition ⊕ and the pseudo-multiplication be an increasing function. Also let ψ ∈ C n (J, R), a function such that ψ be an increasing and positive function on (a, b] having a continuous derivative ψ and ψ (x) = 0 for all x ∈ J.…”

“…When the first ideas about fractional calculus were discussed, there was no dimension of the importance and relevance that their theory and applications in physics, chemistry, biology, engineering, medicine, among other areas of knowledge, [1][2][3][4][5][6][7][8], and recently, fractional derivatives have been used to generalize and refine models that describe the epidemic disease of coronavirus [9,10]. Nowadays we have countless definitions of derivatives and fractional integrals [11][12][13][14][15][16].…”

“…Many works in the area come from Pap works. Recently, some researchers began to discuss more closely, the idea of unifying the g-calculus theory, pseudo-analysis with fractional calculation, since it is still a new field and because there are open questions [14,[26][27][28][29]. There are some works on the g-calculus theory with fractional calculus, in particular, involving inequalities, but it still needs more research and future contributions to the development and growth of the area [1,27,[30][31][32].…”

Analysis ofq‐fractional implicit boundary value problems having Stieltjes integral conditions

“…Fractional calculus has much applications in many engineering and scientific disciplines, such as the modeling of processes and systems in the fields of signal and image processing, 2 mathematical biology, 3 electronics, economics, control theory, 4 chemistry, biophysics, and blood flow phenomena. For more applications of fractional differential equations, we refer the reader to other studies 5‐13 . Since then, in order to describe and imitate the pragmatic phenomena with mathematical tools more accurately, the fractional q ‐integrals and q ‐derivatives make an appearance at the historic moment.…”

“…For more applications of fractional differential equations, we refer the reader to other studies. [5][6][7][8][9][10][11][12][13] Since then, in order to describe and imitate the pragmatic phenomena with mathematical tools more accurately, the fractional q-integrals and q-derivatives make an appearance at the historic moment.…”