A new definition of fractional derivatives based on truncated left-handed grünwald-Letnikov formula with 0 < α < 1 and median correction

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Abstract

We propose a new definition of fractional derivatives based on truncated left-handed Grünwald-Letnikov formula with 0 < α < 1 and median correction. Analyzing the difficulties to choose the fractional orders and unsatisfied processing results in signal processing using fractional-order partial differential equations and related methods; we think that the nonzero values of the truncated fractional order derivatives in the smooth regions are major causes for these situations. In order to resolve the problem, the absolute values of truncated parts of the G-L formula are estimated by the median of signal values of the remainder parts, and then the truncated G-L formula is modified by replacing each of the original signal value to the differences of the signal value and the median. Since the sum of the coefficients of the G-L formula is zero, the median correction can reduce the truncated errors greatly to proximate G-L formula better. We also present some simulation results and experiments to support our theory analysis. © 2014 Zhiwu Liao.

Figures

  • Figure 1: Truncated errors of truncated G-L formula with support lengths 10, 50, 100, and 500 and fractional orders 𝛼 from 0.1 to 0.9.
  • Table 1: Truncated errors of truncated left-handed G-L formula for 𝑠(𝑥) = 1with support lengths from 10 to 500 (rows) and fractional-orders 𝛼 from 0.1 to 0.9 (columns).
  • Figure 2:The original signal (blocks with 1024 samples, blue lines), its fractional derivatives of truncated G-L with different fractional orders ((a) and (c)), and its fractional derivatives of the new method with different fractional orders ((b) and (d)), where the support length of (a) and (b) is 30 and the support length of (c) and (d) is 500.
  • Figure 3:The original signal (bumps with 1024 samples, blue line), its fractional derivatives of truncated G-L with different fractional orders ((a) and (c)), and its fractional derivatives of the new method with different fractional orders ((b) and (d)), where the support length of (a) and (b) is 30 and the support length of (c) and (d) is 500.

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APA

Liao, Z. (2014). A new definition of fractional derivatives based on truncated left-handed grünwald-Letnikov formula with 0 < α < 1 and median correction. Abstract and Applied Analysis, 2014. https://doi.org/10.1155/2014/914386

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