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Comparison of Fractional Order Derivatives Computational Accuracy - Right Hand vs Left Hand Definition


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Fig. 1

Graphical Representation of Left-hand definition of fractional derivative for order ν = 2.3.
Graphical Representation of Left-hand definition of fractional derivative for order ν = 2.3.

Fig. 2

Graphical Representation of Right-hand definition of fractional derivative of order ν = 2.3.
Graphical Representation of Right-hand definition of fractional derivative of order ν = 2.3.

Fig. 3

Two fractional orders ν1 and ν2 of different values concatenated to obtain an integer order n, i.e. ν1 + ν2 = n.
Two fractional orders ν1 and ν2 of different values concatenated to obtain an integer order n, i.e. ν1 + ν2 = n.

Fig. 4

Plot of the kernel of integrand in RL and C formulas.
Plot of the kernel of integrand in RL and C formulas.

C||er||L∞,h=t−t0100 points$\begin{array}{} \displaystyle C||{e.r}|{|.{L\infty }},h = {{t - {t.0}} \over {100{\rm{ }}points}} \end{array}$d - differentiation step, n = 1; n−v = 0:7; v = 0:3

Function (18)Function (19)Function (20)Function (21)
N1.0e−061.0e− 361.0e 061.0e 361.0e 061.0e 361.0e 061.0e 36←d
41.24e-141.76e-212.33e-062.33e-061.22e-027.99e-022.81e-092.81e-09
81.24e-142.07e-451.62e-134.28e-114.16e-024.16e-021.67e-133.52e-22
161.24e-145.52e-671.62e-133.43e-402.13e-022.13e-021.67e-131.13e-52
321.24e-148.05e-681.62e-134.29e-651.07e-021.07e-021.67e-134.45e-67
641.24e-141.38e-671.62e-133.65e-655.43e-035.43e-031.67e-134.37e-66

RL||er||L∞,h=t−t0100points$\begin{array}{} \displaystyle {\rm{RL||}}{e.r}{\rm{|}}{{\rm{|}}.{L\infty }},h = {{t - {t.0}} \over {100points}} \end{array}$ ,d - differentiation step, n − ν = 0.7, n = 1,ν = 0.3

Function (18)Function (19)Function (20)Function (21)
N1.0e−061.0e− 361.0e− 061.0e− 361.0e− 061.0e− 361.0e− 061.0e− 36←d
42.92e-113.05e-198.04e-068.04e-061.22e-031.22e-031.60e-081.06e-08
82.92e-116.89e-431.64e-132.66e-151.07e-041.07e-042.01e-133.60e-21
162.92e-115.21e-661.66e-133.91e-392.26e-052.26e-052.01e-132.19e-51
322.92e-115.21e-661.66e-131.39e-652.91e-062.91e-062.01e-131.03e-64
642.92e-111.16e-661.66e-131.96e-643.07e-073.07e-072.01e-131.54e-64

RL||er||L∞,h=t−t0100 points$\begin{array}{} \displaystyle {\rm{RL||}}{e.r}{\rm{|}}{{\rm{|}}.{L\infty }},h = {{t - {t.0}} \over {100{\rm{ }}points}} \end{array}$d - differentiation step, n− v = 0:7; n = 3; v = 2:3

Function (18)Function (19)Function (20)Function (21)
N1.0e − 061.0e − 161.0e − 061.0e − 161.0e − 061.0e − 161.0e − 061.0e − 16←d
47.38e-103.56e-172.13e-096.78e-191.22e-031.22e-035.41e-055.41e-05
87.38e-107.38e-302.13e-092.13e-291.07e-041.07e-041.62e-111.59e-11
167.38e-107.38e-302.13e-092.13e-292.26e-052.26e-052.51e-134.60e-32
327.38e-107.38e-302.13e-092.13e-292.91e-062.91e-062.51e-132.51e-33
647.38e-107.38e-302.13e-092.13e-293.07e-073.07e-072.51e-132.51e-33

C||er||L∞,h=t−t0100 points$\begin{array}{} \displaystyle C||{e.r}||{L.\infty },h = \frac{{t - {t.0}}}{{100points}} \end{array}$d - differentiation step, n = 3; n− v = 0:7; v = 2:3

Function (18)Function (19)Function (20)Function (21)
N1.0e−061.0e− 161.0e −061.0e − 161.0e − 061.0e − 161.0e − 061.0e − 16←d
46.54e-156.14e-153.90e-153.93e-15nannan1.08e-041.08e-04
86.54e-156.14e-153.90e-153.93e-15nannan6.16e-133.57e-13
166.54e-156.14e-153.90e-153.93e-15nannan2.53e-135.86e-15
326.54e-156.14e-153.90e-153.93e-15nan nan2.53e-135.86e-15
646.54e-156.14e-153.90e-153.93e-15nannan2.53e-135.86e-15
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