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A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations

  • * Corresponding author: S. Wang

    * Corresponding author: S. Wang 
This work is supported by the AOARD Project # 15IOA095 from the US Air Force.
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  • In this paper we propose an efficient and easy-to-implement numerical method for an $α$-th order Ordinary Differential Equation (ODE) when $α∈ (0, 1)$, based on a one-point quadrature rule. The quadrature point in each sub-interval of a given partition with mesh size $h$ is chosen judiciously so that the degree of accuracy of the quadrature rule is 2 in the presence of the singular integral kernel. The resulting time-stepping method can be regarded as the counterpart for fractional ODEs of the well-known mid-point method for 1st-order ODEs. We show that the global error in a numerical solution generated by this method is of the order $\mathcal{O}(h^{2})$, independently of $α$. Numerical results are presented to demonstrate that the computed rates of convergence match the theoretical one very well and that our method is much more accurate than a well-known one-step method when $α$ is small.

    Mathematics Subject Classification: Primary: 65L05, 65L20; Secondary: 49M25.

    Citation:

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  • Table 1.  Maximum Errors and Convergence Rates for Example 1

    h Our results Results from [9] Our results Results from [9] Our results Results from [9] Our results Results from [9]
    α=0.1 Order α=0.1 Order α=0.3 Order α=0.3 Order α=0.5 Order α=0.5 Order α=0.9 Order α=0.9 Order
    1/10 4.06e-3 - - - 7.67e-3 - - - 8.49e-3 - 0.0355 - 7.88e-3 - 0.0107 -
    1/20 1.11e-3 1.86 1.86 1.10 2.00e-3 1.94 - - 2.16e-3 1.98 0.00879 2.01 1.97e-3 2.00 0.00231 2.21
    1/40 3.02e-4 1.89 1.89 1.26 5.17e-4 1.95 - - 5.43e-4 1.99 2.16e-3 2.03 4.93e-4 2.00 5.21e-4 2.15
    1/80 8.08e-5 1.90 1.90 1.31 1.32e-4 1.97 - - 1.37e-4 1.99 5.31e-4 2.02 1.23e-4 2.00 1.22e-4 2.09
    1/160 2.14e-5 1.92 1.92 1.32 3.37e-5 1.97 - - 3.43e-5 2.00 1.31e-4 2.02 3.08e-5 2.00 2.94e-5 2.06
    1/320 5.65e-6 1.93 1.93 1.31 8.55e-6 1.98 - - 8.58e-6 2.00 3.24e-5 2.02 7.70e-6 2.00 7.18e-6 2.03
    1/640 1.47e-6 1.93 1.93 1.30 2.16e-6 1.98 - - 2.15e-6 2.00 8.03e-6 2.01 1.92e-6 2.00 1.77e-6 2.01
    1/1280 3.84e-7 1.94 1.94 - 5.46e-7 1.99 - - 5.38e-7 2.00 - - 4.81e-7 2.00 - -
     | Show Table
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    Table 2.  Maximum Errors and Convergence Rates for Example 2

    h α=0.1 Order α=0.3 Order α=0.5 Order α=0.9 Order
    1/10 2.37e-3 - 5.08e-3 - 6.40e-3 - 7.50e-3 -
    1/20 6.45e-4 1.88 1.31e-3 1.95 1.62e-3 1.98 1.88e-3 2.00
    1/40 1.73e-4 1.90 3.39e-4 1.96 4.08e-4 1.99 4.69e-4 2.00
    1/80 4.60e-5 1.91 8.65e-5 1.97 1.03e-4 1.99 1.17e-4 2.00
    1/160 1.21e-5 1.92 2.20e-5 1.98 2.57e-5 2.00 2.93e-5 2.00
    1/320 3.18e-6 1.93 5.58e-6 1.98 6.44e-6 2.00 7.32e-6 2.00
    1/640 8.30e-7 1.94 1.41e-6 1.98 1.61e-6 2.00 1.83e-6 2.00
    1/1280 2.15e-7 1.95 3.55e-7 1.99 4.04e-7 2.00 4.57e-7 2.00
     | Show Table
    DownLoad: CSV

    Table 3.  Maximum Errors and Convergence Rates for Example 3

    h α=0.1 Order α=0.3 Order α=0.5 Order α=0.9 Order
    1/10 6.19e-2 - 4.27e-2 - 2.56e-2 - 3.70e-3 -
    1/20 1.69e-2 1.87 1.03e-2 2.05 5.70e-3 2.17 6.90e-4 2.42
    1/40 4.42e-3 1.93 2.368e-3 2.13 1.10e-3 2.37 1.88e-4 1.88
    1/80 1.10e-3 2.00 5.055e-4 2.23 1.85e-4 2.57 5.92e-5 1.67
    1/160 2.65e-4 2.05 1.025e-4 2.30 2.63e-5 2.82 2.11e-5 1.49
    1/320 6.26e-5 2.08 2.00e-5 2.36 2.63e-6 3.32 6.19e-6 1.77
    1/640 1.46e-5 2.10 3.77e-6 2.41 5.30e-7 2.31 1.68e-6 1.88
    1/1280 3.41e-6 2.10 6.86e-7 2.46 1.55e-7 1.77 4.37e-7 1.94
     | Show Table
    DownLoad: CSV
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