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

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Authors :

Syamal K. Sen 1,2 , J. Vasundhara Devi 1,3 *, and R.V.G. Ravi Kumar 3

Author Address :

1 GVP-Prof. V. Lakshmikantham Institute for Advanced Studies, Gayatri Vidya Parishad College of Engineering Campus, Visakhaptnam
530048, India.
2 Department of Computer Science and Engineering, Gayatri Vidya Parishad College of Engineering, Visakhaptnam 530048, India.
3 Department of Mathematics, Gayatri Vidya Parishad College of Engineering, Visakhaptnam 530048, India.

*Corresponding author.

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.



Article Info :

Received : February 21, 2019; Accepted : March 26, 2019.



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