Year: 2019
Author: Jinfa Cheng
Journal of Mathematical Study, Vol. 52 (2019), Iss. 1 : pp. 38–52
Abstract
In this paper, a generalized multivariate fractional Taylor's and Cauchy's mean value theorem of the kind
$$f(x,y) = \sum\limits_{j = 0}^n {\frac{{{D^{j\alpha }}f({x_{0,}}{y_0})}}{{\Gamma (j\alpha + 1)}}} + R_n^\alpha (\xi,\eta),\qquad\frac{{f(x,y) - \sum\limits_{j = 0}^n {\frac{{{D^{j\alpha }}f({x_{0,}}{y_0})}}{{\Gamma (j\alpha + 1)}}} }}{{g(x,y) - \sum\limits_{j = 0}^n {\frac{{{D^{j\alpha }}g({x_{0,}}{y_0})}}{{\Gamma (j\alpha + 1)}}} }} = \frac{{R_n^\alpha (\xi ,\eta )}}{{T_n^\alpha (\xi ,\eta )}},$$
where $0<\alpha \le 1$, is established. Such expression is precisely the classical Taylor's and Cauchy's mean value theorem in the particular case $\alpha=1$. In addition, detailed expressions for $R_n^\alpha (\xi,\eta)$ and $T_n^\alpha (\xi,\eta)$ involving the sequential Caputo fractional derivative are also given.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v52n1.19.04
Journal of Mathematical Study, Vol. 52 (2019), Iss. 1 : pp. 38–52
Published online: 2019-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
Keywords: Sequential Caputo fractional derivative generalized Taylor's mean value theorem generalized Taylor's formula generalized Cauchy' mean value theorem generalized Cauchy's formula.