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Fig. 1
Plot of the kernel of the integrand in the formulas (3), (5) and (6).
Fig. 2
(a) Graph of the original integrand g(τ), (b) graph of transformed variable τ into u and (c) transformed integrand g(τ) into G(u).
Fig. 3
Computational accuracy of fractional integral of order ν = −0.5 for the function (11)(a) and (12)(b) applying unmodified(RL ...) and modified(mRL ...) formula (6).
Fig. 4
Computational accuracy of fractional derivative of order ν = 0.5 for the function (11)(a) and (12)(b) applying unmodified(RL ...) and modified(mRL ...) formula (5).
Fig. 5
The Double Exponential Transformation: (a) graph of the original integrand,(b) graph of transforming expression and (c) transformed integrand.
Fig. 6
Accuracy of the DE Transformation for orders ν = 0.9(a), ν = 0.5(b) and N=1000 in context of applied precision during computations.
Fig. 7
Accuracy of the DE Transformation for orders ν = −0.0001(a), ν = −0.1(b) and N=1000 in context of applied precision during computations.
Fig. 8
Time complexity for increasing precision (digits), N=1000, fractional integral ν = −0.0001 for the function (26) and ν = 0.9 for the function (27).
Fig. 9
Accuracy of GJ with N = 32 for calculating fractional integrals of orders ν = −0.0001(a) and ν = −0.1(b); 16 vs 100 digits precision.
Fig. 10
Accuracy of GJ with N = 32 for calculating fractional derivatives of orders ν = 0.5(a) and ν = 0.9(b); 16 vs 100 digits precision.
Fig. 11
Accuracy of GJ for calculating fractional integrals of orders ν = −0.0001(a) and ν = 0.9(b). N = 4,8,16,32 with 100 digits precision.
Fig. 12
Accuracy of GJ for calculating fractional derivatives of orders ν = 0.5(a) and ν = 0.9(b). N = 1,2,4,8,16,32 with 100 digits precision.
Fig. 13
Accuracy of GJ for calculating fractional derivatives of orders (0.1(0.1)0.9) (a) and of order ν = 0.9999 for intervals (1(1)6) (b), N = 32 with 100 digits precision.
Fig. 14
Programs running time for the function (23), (a) programmed with double precision and (b) with 100-digits precision.
D(1/2)f(t), f(t) = t, t ∈ (0,1), relative error in %
N
GL
NCm
Diet
Odiba
8
8.11
10.68
0.0004
0.0024
15
4.25
7.81
0.0004
0.0023
21
3.02
6.6
0.0004
0.0023
61
1.03
3.88
0.0004
0.0022
300
0.21
1.75
0.0002
0.0021
600
0.11
1.24
0.0002
0.0021
1000
0.06
0.96
0.0002
0.0021
D(1/2)f(t), f(t) = e−t, t ∈ (0,5), relative error in %
N
GL
NCm
8
61.87
13.52
15
29.32
5.88
21
20.17
4.12
61
6.54
1.81
300
1.3
0.76
600
0.65
0.53
1000
0.39
0.41
D(1/2)f(t), f(t) = sin(t), t ∈ (0,2π), relative error in %