New limit definition of fractional derivatives: Toward improved accuracy and generalization

Syamal K. Sen; J. Vasundhara Devi; R.V.G. Ravi Kumar

doi:10.26637/MJM0702/0009

Abstract

We computationally study 2 most recently defined fractional derivatives (FDs) with classical properties, both based on 1 st principles, also known as delta methods, involving limit approaches. Using the advantages of both the definitions we present a new limit definition of the FD that has always less computational error or, equivalently, more computational accuracy and at the same time satisfies all the classical properties that are observed by the foregoing 2 definitions. Such definitions are desirable so that these provide a smooth transition to/from the most extensively used and the best understood classical derivative (CD). Our study throws more light on the pros and
cons of these definitions and possibly encourage further innovative approach to improve the definitions for still better/complete compatibility/generalization, and possibly to understand and to write the physical significance of the FD readily.

Keywords:

Classical properties, Computational complexity, Fractional derivatives with classical properties, Improved accuracy, Limit definition of fractional derivatives

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Syamal K. Sen GVP-Prof. V. Lakshmikantham Institute for Advanced Studies, Gayatri Vidya Parishad College of Engineering Campus, Visakhaptnam 530048, India.
J. Vasundhara Devi GVP-Prof. V. Lakshmikantham Institute for Advanced Studies, Gayatri Vidya Parishad College of Engineering Campus, Visakhaptnam 530048, India. https://orcid.org/0000-0001-9616-5077
R.V.G. Ravi Kumar Department of Mathematics, Gayatri Vidya Parishad College of Engineering, Visakhaptnam 530048, India.

Pages: 182-191
Date Published: 01-04-2019
Vol. 7 No. 02 (2019): Malaya Journal of Matematik (MJM)

S.F. Lacroix, traite du calcul differential et du calcul integral, 2nd Edition, courcier, Paris, 1819.

S.G. Samko and B. Ross, Integration and differentiation to a variable fractional order Integral Transforms and Special Functions, 1(4)(2009), 277-300.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, California, U.S.A., 1999.

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, N.J., 2000.

A.A. Kilbas, H.M. Srivastava, and J.J.Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V., Amsterdam, 2006.

A.A. Kilbas and M. Saigo, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transform Spec Func., 15(1), 2004, $31-49$.

M. Klimek, Lagrangian fractional mechanics-a noncommutative approach, Czechoslovak Journal of Physics, $55(11) 2005,1447-1453$

J. Cresson, Fractional embedding of differential operators and Lagrangian systems, Journal of Mathematical Physics, 48(3) 2007, 12-23.

O. P. Agrawal, Generalized Variational Problems and Euler-Lagrange Equations, Computers and Mathematics with Applications, $59(5)(2010), 1852-1864$.

A.B. Malinowska, T. Odzijewicz, and D.F.M. Torres, Advanced Methods in the Fractional Calculus of Variations, Springer Briefs in Applied Sciences and Technology, 2015 , XII, 135 pages.

Z. Tomovski, R. Hilfer, H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag- Leffler type functions, Integral Transforms and special Functions, 21(11)(2010), 797814.

R. Khalil, M. A. Horani, A. Yousef, M. Sababheh. A new definition of fractional derivative, J. Comput. Appl. Math., $264(2014), 65-70$.

U.N. Katugampola, New approach to generalized fractional derivatives, B. Math. Anal. App., 6(4)(2014), 1-15.

R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific Publishing Co., Singapore, 2011.

Machado, J. A., And I say to myself: "What a fractional world!", Frac. Calc. Appl. Anal., 14(4)(2011), 635-654.

Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.

The authors thank the Science and Engineering Research Board (SERB) of the Department of Science and Technology (DST), Government of India for their support of the reported work under the project DST SERB EMR/2016/003572 dated February 06, 2017.