Odibat, Zaid M.; Shawagfeh, Nabil T. Generalized Taylor’s formula. (English) Zbl 1122.26006 Appl. Math. Comput. 186, No. 1, 286-293 (2007). 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. Reviewer: S. L. Kalla (Kuwait) Cited in 4 ReviewsCited in 319 Documents MSC: 26A33 Fractional derivatives and integrals 33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) Keywords:fractional integral; Caputo fractional derivative Citations:Zbl 0063.01925; Zbl 0931.26004; JFM 57.0477.02 PDFBibTeX XMLCite \textit{Z. M. Odibat} and \textit{N. T. Shawagfeh}, Appl. Math. Comput. 186, No. 1, 286--293 (2007; Zbl 1122.26006) Full Text: DOI References: [1] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag Wien), 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.; 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: 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: 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: 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, (Nishimoto, K., Fractional Calculus and Its Applications (1990), Nihon University: Nihon University Tokyo), 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: 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: 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.; Y. Wantanable, Notes on the generalized derivatives of Riemann-Liouville and its application to Leibntz’s formula, Thoku Math. J. 34, 28-41. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.