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# New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator

by Sania Qureshi 1 , Norodin A. Rangaig 2 and Dumitru Baleanu 3,4,* 1
Department of Basic Sciences and Related Studies Mehran University of Engineering and Technology, Jamshoro 76062, Sindh, Pakistan
2
Department of Physics, Mindanao State University-Main Campus, Marawi City 9700, Philippines
3
Department of Mathematics, Cankaya University, Ankara 06530, Turkey
4
Institute of Atomic Physics, 077125 Magurele-Bucharest, Romania
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(4), 374; https://doi.org/10.3390/math7040374
Received: 3 March 2019 / Revised: 16 April 2019 / Accepted: 17 April 2019 / Published: 24 April 2019
In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f (t) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration. View Full-Text
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MDPI and ACS Style

Qureshi, S.; Rangaig, N.A.; Baleanu, D. New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator. Mathematics 2019, 7, 374.