An Improved Approximation of Grunwald-Letnikov Fractional Integral

Abstract

Fractional calculus increases the flexibility of a system by studying the unexplored space between two integers. However, fractional calculus’s main challenge is its implementation due to its memory dependency, which appears in the amplitudes of the w coefficients in Grunwald–Letnikov(GL) definition. A modified GL approximation is proposed to control this dependency and decrease the error. The suggested approximation is based on the difference of the w binomial coefficients, which makes the new coefficients amplitudes decay faster. Three methods are discussed and compared for implementing the standard and the proposed GL approximation. The modified approximation shows an improvement, especially in the integration region of − 1 < α < −0.5. For example, the modified approximation results in an average absolute error of (0.1987) while the standard approximation results in an average absolute error of (0.8636) for sin(t) signal at α = −0.95, step size (h) of 0.01, window size of 64, and number of samples of 6283.