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High-order approximation for generalized fractional derivative and its application

Swati Yadav (Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, India)
Rajesh K. Pandey (Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, India)
Anil K. Shukla (Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, India)
Kamlesh Kumar (Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, India)

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

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Emerald Publishing Limited

Copyright © 2019, Emerald Publishing Limited