Monte Carlo method for fractional-order differentiation of higher order

Leonenko, Nikolai ORCID: https://orcid.org/0000-0003-1932-4091 and Podlubny, Igor 2022. Monte Carlo method for fractional-order differentiation of higher order. Fractional Calculus and Applied Analysis 25 , pp. 841-857. 10.1007/s13540-022-00048-w

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Abstract

In this work the Monte Carlo method, introduced recently by the authors for orders of differentiation between zero and one, is further extended to differentiation of orders higher than one. Two approaches have been developed on this way. The first approach is based on interpreting the coefficients of the Grünwald–Letnikov fractional differences as so called signed probabilities, which in the case of orders higher than one can be negative or positive. We demonstrate how this situation can be processed and used for computations. The second approach uses the Monte Carlo method for orders between zero and one and the semi-group property of fractional-order differences. Both methods have been implemented in MATLAB and illustrated by several examples of fractional-order differentiation of several functions that typically appear in applications. Computational results of both methods were in mutual agreement and conform with the exact fractional-order derivatives of the functions used in the examples.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Additional Information:	This article is licensed under a Creative Commons Attribution 4.0 International License
Publisher:	Springer
ISSN:	1311-0454
Funders:	EPSRC
Date of First Compliant Deposit:	31 May 2022
Date of Acceptance:	30 April 2022
Last Modified:	24 May 2023 18:21
URI:	https://orca.cardiff.ac.uk/id/eprint/150155

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