Archive
Special Issues Volume 8, Issue 5, October 2019, Page: 83-87
Classical Properties on Conformable Fractional Calculus
Musraini M., Department of Mathematics, University of Riau, Pekanbaru, Indonesia
Rustam Efendi, Department of Mathematics, University of Riau, Pekanbaru, Indonesia
Endang Lily, Department of Mathematics, University of Riau, Pekanbaru, Indonesia
Ponco Hidayah, Department of Mathematics, University of Riau, Pekanbaru, Indonesia
Received: Sep. 3, 2019;       Accepted: Oct. 9, 2019;       Published: Oct. 23, 2019
Abstract
Recently, a definition of fractional which refers to classical calculus form called conformable fractional calculus has been introduced. The main idea of the concept of conformable fractional calculus is how to determine the derivative and integral with fractional order either rational numbers or real numbers. One of the most popular definitions of conformable fractional calculus is defined by Katugampola which is used in this study. This definition satisfies in some respects of classical calculus involved conformable fractional derivative and conformable fractional integral. In the branch of conformable fractional derivatives, some of the additional results such as analysis of fractional derivative in quotient property, product property and Rolle theorem are given. An application on classical calculus such as determining monotonicity of function is also given. Then, in the case of fractional integral, this definition showed that the fractional derivative and the fractional integral are inverses of each other. Some of the classical integral properties are also satisfied on conformable fractional integral. Additionally, this study also has shown that fractional integral acts as a limit of a sum. After that, comparison properties on fractional integral are provided. Finally, the mean value theorem and the second mean value theorem are also applicable for fractional integral.
Keywords
Fractional Calculus, Conformable Fractional Derivative, Conformable Fractional Integral
Musraini M., Rustam Efendi, Endang Lily, Ponco Hidayah, Classical Properties on Conformable Fractional Calculus, Pure and Applied Mathematics Journal. Vol. 8, No. 5, 2019, pp. 83-87. doi: 10.11648/j.pamj.20190805.11
Reference

T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics, (279), 57-66 (2015).

A. Akkurt, M. E. Yildirim dan H. Yildirim, Generalized new fractional derivative and integral, Konuralp Journal of Mathematics, (5), 248-259 (2017).

R. Almeida, M. Guzowska dan T. Odzijewicz, A remark on local fractional calculus and ordinary derivatives, Open Mathematics Research Article, (14), 1122-1124 (2016).

A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, arXiv: 1602.03408 [math. GM] 2016.

R. Anderson and D. J. Ulness, Properties of the Katugampola fractional derivative with potential applicationin quantum mechanics, Journal of Mathematical Physics, 56 (2015).

O. T. Birgani, S. Chandok, N. Dedović dan S. Radenović, A note on some recent results of theconformable derivative, Advances in the Theory of Nonlinear Analysis and its Applications, (3), 11-17 (2019).

C.-Q. Dai, Y. Wang, J. Liu, Spatiotemporal Hermite-Gaussian solitons of a (3+1)-dimensional partially nonlocal nonlinear Schrodinger equation. Nonlinear Dynamics, (84), 1157-1161 (2016).

J. Hadamard, Essai sur l’étude des fonctions donnés par leur développement de Taylor, Journalde Mathématiques Pures et Appliquées, (8), 101-186 (1892).

Hermann, Richard, Fractional Calculus: An Introduction for Physicists Analysis Second Edition, New Jersey, World Scientific Publishing, 2014.

U. Katumgapola, A new fractional derivative with classical properties, preprint, arXiv: 1410. 6535.

R. Khalil, M. Alhorani, A. Yousef dan M. Sababheh, A definition of fractional derivative, Journal of Computational Applied Mathematics, (264), 65-70 (2014).

B. Ross, The development of fractional calculus 1695-1900, Historia Mathematica, (4), 75-89 (1977).

N. A. Shah and A. F. I. Khan, Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo-Fabrizio derivatives, The European Physical Journal C. 76, (7), 1-11 (2016).

N. A. Sheikh, A. F. I. Khan, M. Saqib, A modern approach of Caputo-Fabrizio time-fractionalderivative to MHD free convection flow of generalized second-grade fluid in a porous medium, Neural Computing and Applications, 1-11 (2016).

M. H. Tavassoli, A. Tavassoli dan M. R. O. Rahimi, The geometric and physical interpretationof fractional order derivative of polynomial functions, Differential Geometry-Dynamical System, (15), 93-104 (2013).

F. Usta, A conformable calculus of radial basis functions and its applications, An International journal of optimization and control: Theories and Applications, (8), 176-182 (2018). 