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Accuracy Problems of Numerical Calculation of Fractional Order Derivatives and Integrals Applying the Riemann-Liouville/Caputo Formulas

Dariusz W. Brzeziński
Dariusz W. Brzeziński
   | Jan 01, 2016
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Published Online: Jan 01, 2016
Page range: 23 - 44
Received: Jun 01, 2015
Accepted: Nov 04, 2015
DOI: https://doi.org/10.21042/AMNS.2016.1.00003
© 2016 Dariusz W. Brzeziński, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Plot of the kernel of the integrand in the formulas (3), (5) and (6).
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).
(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).
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).
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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 %

NGLNCmDietOdiba
88.1110.680.00040.0024
154.257.810.00040.0023
213.026.60.00040.0023
611.033.880.00040.0022
3000.211.750.00020.0021
6000.111.240.00020.0021
10000.060.960.00020.0021

D(1/2)f(t), f(t) = e−t, t ∈ (0,5), relative error in %

NGLNCm
861.8713.52
1529.325.88
2120.174.12
616.541.81
3001.30.76
6000.650.53
10000.390.41

D(1/2)f(t), f(t) = sin(t), t ∈ (0,2π), relative error in %

NGLNCmDietOdiba
8130.0345.4217.675.78
1566.1433.647.953.46
2146.0928.775.182.94
6116.117.841.532.94
3005.029.350.622.43
6003.677.290.562.35
10003.146.180.552.35
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