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Generalized Taylor’s formula. (English) Zbl 1122.26006
The ordinary Taylor’s formula has been generalized by several authors [G. Hardy, J. Lond. Math. Soc. 20, 48–57 (1945; Zbl 0063.01925); J. J. Trujillo, M. Rivero and B. Bonilla, J. Math. Anal. 231, No. 1, 255–265 (1999; Zbl 0931.26004); Y. Watanabe, Tôhoku Math. J. 34, 28–41 (1931; JFM 57.0477.02)]. In this paper the authors obtain a generalized Taylor’s formula, using Caputo fractional derivative. Some applications involving approximation of functions and solutions of fractional differential equations are given.

MSC:
26A33 Fractional derivatives and integrals
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
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[1] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[2] Hardy, G., Riemann’s form of Taylor series, J. London math., 20, 48-57, (1945) · Zbl 0063.01925
[3] I. Podlubny, The Laplace transform method for linear differential equations of fractional order, Slovac Academy of Science, Slovak Republic, 1994.
[4] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego, CA · Zbl 0918.34010
[5] Truilljo, J.; Rivero, M.; Bonilla, B., On a riemann – liouville generalized taylor’s formula, J. math. anal., 231, 255-265, (1999) · Zbl 0931.26004
[6] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[7] Diethelm, K.; Ford, J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[8] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus, (1997), Carpinteri & Mainardi New York · Zbl 1030.26004
[9] Gorenflo, R.; Luchko, Yu.; Mainardi, F., Analytical properties and applications of the wrigth function, Fract. calc. appl. anal., 2, 4, 383-414, (1999) · Zbl 1027.33006
[10] Bagley, R.L., On the fractional order initial value problem and its engineering applications, (), 12-20 · Zbl 0751.73023
[11] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach London · Zbl 0818.26003
[12] Miller, S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley & Sons USA, p. 2 · Zbl 0789.26002
[13] Schneider, W.R., Completely monotone generalized mittag – leffler functions, Exposition. math., 14, 1, 3-16, (1996) · Zbl 0843.60024
[14] Luchko, Y.; Srivastava, H.M., The exact solution of certain differential equations of fractional order by using operational calculus, Comput. math. appl., 29, 73-85, (1995) · Zbl 0824.44011
[15] Y. Wantanable, Notes on the generalized derivatives of Riemann-Liouville and its application to Leibntz’s formula, Thoku Math. J. 34, 28-41.
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