Fast Evaluation of Grünwald-Letnikov Variable Fractional-Order Differentiation and Integration Based on the FFT Convolution

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1196)


The main topic of this research is the development of a new efficient method for solving fractional differential equations in the time domain using the Grünwald-Letnikov (GL) definition of the differintegral operator. The goal is to reduce or, in some cases, eliminate the necessity of introducing the maximum number of samples of the approximation, which is always a trade-off between accuracy, memory consumption, and computational speed. The algorithm involving Fast Fourier Transform and Fast Convolution operations has been proposed. The implementation in two different environments - the MATLAB/Simulink on a PC and on a hardware platform with the STM32H743 microcontroller - is described in this paper and the results of two iterations of experiments are presented. Fast Convolution algorithm is proven to be highly effective for processing block lengths \(N \ge 128\). In the most complex analyzed case (\(N=8192\) samples) on the Intel\(\textregistered \) Core\(^\mathrm{TM}\) i5-8250U CPU reduction of the computation time reached \(85\%\), compared to the implementation of the classic definition characterized by the \(\mathcal {O}(N^2)\) complexity.


Fractional-order calculus Fast Convolution Fast Fourier Transform Grünwald-Letnikov VFOD/I Arm® Cortex-M7 Microcontroller implementation 



This work was supported by Polish funds of the National Science Center under grant DEC-2016/23/B/ST7/03686.


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of Applied Computer ScienceLodz University of TechnologyLodzPoland

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