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An Improvement of a Nonclassical Numerical Method for the Computation of Fractional Derivatives

[+] Author and Article Information
Kai Diethelm

 GNS Gesellschaft für Numerische Simulation mbH, Am Gaußberg 2, 38114 Braunschweig, Germanydiethelm@gns-mbh.com

J. Vib. Acoust 131(1), 014502 (Jan 08, 2009) (4 pages) doi:10.1115/1.2981167 History: Received January 25, 2008; Revised May 23, 2008; Published January 08, 2009

Standard methods for the numerical calculation of fractional derivatives can be slow and memory consuming due to the nonlocality of the differential operators. Yuan and Agrawal (2002, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives  ,” ASME J. Vibr. Acoust., 124, pp. 321–324) have proposed a more efficient approach for operators whose order is between 0 and 1 that differs substantially from the traditional concepts. It seems, however, that the accuracy of the results can be poor. We modify the approach, adapting it better to the properties of the problem, and show that this leads to a significantly improved quality. Our idea also works for operators of order greater than 1.

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Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Log-log plot of transformed nodes and weights for eight-point Gauss–Laguerre rule (original method; triangles) and eight-point Gauss–Jacobi rule (modified method; circles)

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Figure 2

Exact solution (solid line) of Eq. 6 with data given in Eq. 7 and numerical solution with modified method, using trapezoidal ODE solver and 2 (dash-dotted), 7 (dotted), and 15 (dashed) quadrature points, respectively

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Figure 3

Absolute errors of the original Yuan–Agrawal method for the approximation of D*αy(x) with y(x)=x3 and α=0.6 for x∊[0,1] with 8, 16, and 32 Gauss–Laguerre nodes and trapezoidal method with h=1∕10 (dashed) and h=1∕100 (solid lines)

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Figure 4

Absolute errors of the modified Yuan–Agrawal method for the approximation of D*αy(x) with y(x)=x3 and α=0.6 for x∊[0,1] with 8 (dotted) and 16 (solid lines) Gauss–Jacobi nodes and trapezoidal method with h=1∕10 and h=1∕100

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Figure 5

Absolute errors of the original Yuan–Agrawal method for the approximation of D*αy(x) with y(x)=exp(−x) and α=1.3 for x∊[0,1] with 8, 16, and 32 Gauss–Laguerre nodes and trapezoidal method with h=1∕10 (dashed) and h=1∕100 (solid lines)

Grahic Jump Location
Figure 6

Absolute errors of the modified Yuan–Agrawal method for the approximation of D*αy(x) with y(x)=exp(−x) and α=1.3 for x∊[0,1] with 8 (dotted) and 16 (solid lines) Gauss–Jacobi nodes and trapezoidal method with h=1∕10 and h=1∕100

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Figure 7

Absolute errors of the modified Yuan–Agrawal method for the solution of Eq. 11 with initial condition 12 and α=1.25 for x∊[0,1] with five and ten Gauss–Jacobi nodes and trapezoidal method with h=1∕40 (dashed) and h=1∕80 (solid lines)

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