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On right fractional calculus. (English) Zbl 1198.26006
Summary: Here are presented fractional Taylor type formulae with fractional integral remainder and fractional differential formulae, regarding the right Caputo fractional derivative, the right generalized fractional derivative of {\it J. A. Canavati} type [Nieuw Arch. Wiskd., IV. Ser. 5, 53--75 (1987; Zbl 0649.46026)] and their corresponding right fractional integrals.Then are given representation formulae of functions as fractional integrals of their above fractional derivatives, as well as of their right and left Weyl fractional derivatives.At the end, we mention some far reaching implications of our theory to mathematical analysis computational methods.Also we compare the right Caputo fractional derivative to right Riemann-Liouville fractional derivative. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:
26A33Fractional derivatives and integrals (real functions)
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Full Text: DOI
References:
[1] Anastassiou, G.: Quantitative approximations, (2001) · Zbl 0969.41001
[2] Canavati, J. A.: The Riemann -- Liouville integral, Nieuw archief voor wiskunde 5, No. 1, 53-75 (1987) · Zbl 0649.46026
[3] Kai Diethelm, Fractional differential equations. Available from: http://www.tu-bs.de/ diethelm/lehre/f-dgl02/fde-skript.ps.gz. · Zbl 1136.26302
[4] El-Sayed, A. M. A.; Gaber, M.: On the finite Caputo and finite Riesz derivatives, Electron J theor phys 3, No. 12, 81-95 (2006)
[5] Frederico, G. S.; Torres, D. F. M.: Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem, Int math forum 3, No. 10, 479-493 (2008) · Zbl 1154.49016
[6] Gorenflo R, Mainardi F, Essentials of fractional calculus. Maphysto Center; 2000. Available from: http://www.maphysto.dk/oldpages/events/LevyCAC2000/MainardiNotes/fm2k0a.ps. · Zbl 1030.26004
[7] Miller, K. S.: The Weyl fractional calculus, Lecture notes in mathematics 457, 80-89 (1975) · Zbl 0305.44004
[8] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives, theory and applications, (1993) · Zbl 0818.26003