High-order approximation for generalized fractional derivative and its application
International Journal of Numerical Methods for Heat & Fluid Flow
ISSN: 0961-5539
Article publication date: 19 July 2019
Issue publication date: 11 September 2019
Abstract
Purpose
This paper aims to present a high-order scheme to approximate generalized derivative of Caputo type for μ ∈ (0,1). The scheme is used to find the numerical solution of generalized fractional advection-diffusion equation define in terms of the generalized derivative.
Design/methodology/approach
The Taylor expansion and the finite difference method are used for achieving the high order of convergence which is numerically demonstrated. The stability of the scheme is proved with the help of Von Neumann analysis.
Findings
Generalization of fractional derivatives using scale function and weight function is useful in modeling of many complex phenomena occurring in particle transportation. The numerical scheme provided in this paper enlarges the possibility of solving such problems.
Originality/value
The Taylor expansion has not been used before for the approximation of generalized derivative. The order of convergence obtained in solving generalized fractional advection-diffusion equation using the proposed scheme is higher than that of the schemes introduced earlier.
Keywords
Acknowledgements
Authors sincerely thank the reviewers for providing constructive comments for improvement of the manuscript. The first author acknowledges the financial support from the Council of Scientific and Industrial Research (CSIR), New Delhi, India under the JRF schemes (File No: 09/1217(0009)/2016-EMR-I).
Citation
Yadav, S., Pandey, R.K., Shukla, A.K. and Kumar, K. (2019), "High-order approximation for generalized fractional derivative and its application", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29 No. 9, pp. 3515-3534. https://doi.org/10.1108/HFF-11-2018-0700
Publisher
:Emerald Publishing Limited
Copyright © 2019, Emerald Publishing Limited