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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
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.
Author Details
-
Fractional Bernstein Series Solution of Fractional Diffusion Equations with Error Estimate
Alshbool, Mohammed Hamed | Isik, Osman | Hashim, IshakAxioms, Vol. 10 (2021), Iss. 1 P.6
https://doi.org/10.3390/axioms10010006 [Citations: 10] -
Statistical Diagnostics of Electric Power Equipment
Information and Measuring Systems for Statistical Diagnostics of Electrical Equipment
Babak, Vitalii | Babak, Sergii | Zaporozhets, Artur2025
https://doi.org/10.1007/978-3-031-76253-6_7 [Citations: 0] -
Bernstein polynomials method for solving multi-order fractional neutral pantograph equations with error and stability analysis
Alshbool, M.H.T.
Results in Applied Mathematics, Vol. 22 (2024), Iss. P.100451
https://doi.org/10.1016/j.rinam.2024.100451 [Citations: 2] -
Infinite Series Representation of Functions in Fractional Calculus
Wei, Yiheng | Chen, YangQuan | Gao, Qing | Wang, Yong2019 Chinese Automation Congress (CAC), (2019), P.1697
https://doi.org/10.1109/CAC48633.2019.8997499 [Citations: 8] -
A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids
Vargas, A. M.
Communications on Applied Mathematics and Computation, Vol. (2023), Iss.
https://doi.org/10.1007/s42967-023-00323-4 [Citations: 3] -
Finite difference method for solving fractional differential equations at irregular meshes
Vargas, Antonio M.
Mathematics and Computers in Simulation, Vol. 193 (2022), Iss. P.204
https://doi.org/10.1016/j.matcom.2021.10.010 [Citations: 52]