A New Definition of Fractional Derivatives Based on Truncated Left-Handed Grünwald-Letnikov Formula with 0<𝛼<1 and Median Correction

Table 1

Truncated errors of truncated left-handed G-L formula for 𝑠(𝑥)=1 with support lengths from 10 to 500 (rows) and fractional-orders 𝛼 from 0.1 to 0.9 (columns).

Length of support	𝛼=0.1	𝛼=0.2	𝛼=0.3	𝛼=0.4	𝛼=0.5	𝛼=0.6	𝛼=0.7	𝛼=0.8	𝛼=0.9
10	−0.7475	−0.5487	−0.3939	−0.2752	−0.1855	−0.1190	−0.0710	−0.0372	−0.0145
30	−0.6672	−0.4368	−0.2795	−0.1739	−0.1043	−0.0595	−0.0315	−0.0147	−0.0051
50	−0.6335	−0.3937	−0.2392	−0.1412	−0.0804	−0.0435	−0.0219	−0.0097	−0.0032
100	−0.5908	−0.3432	−0.1939	−0.1067	−0.0566	−0.0286	−0.0134	−0.0055	−0.0017
200	−0.5510	−0.2979	−0.1573	−0.0808	−0.0400	−0.0188	−0.0082	−0.0032	−0.0009
500	−0.5027	−0.2479	−0.1194	−0.0559	−0.0253	−0.0108	−0.0043	−0.0015	−0.0004