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Two-step collocation methods for fractional differential equations

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The work is supported by GNCS-Indam project

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  • We propose two-step collocation methods for the numerical solution of fractional differential equations. These methods increase the order of convergence of one-step collocation methods, with the same number of collocation points. Moreover, they are continuous methods, i.e. they furnish an approximation of the solution at each point of the time interval. We describe the derivation of two-step collocation methods and analyse convergence. Some numerical experiments confirm theoretical expectations.

    Mathematics Subject Classification: Primary: 65L05, 65R20; Secondary: 34A08, 65D07.

    Citation:

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  • Figure 1.  One-step collocation method with $m = 2$ and two-step collocation methods with $m = 2$ and with $m = 4$

    Figure 2.  One-step collocation method with $m = 3$ and two-step collocation methods with $m = 3$ and with $m = 6$

    Table 1.  Two-step collocation method with $m = 2$, $\eta = \left[\frac{1}{3}, \frac{2}{3}\right]$

    Problem 1Problem 2Problem 3
    N $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$
    8 $0.02$ $4.49e-03$ $0.31$
    16 $1.20e-03$ $3.78$ $3.08e-04$ $3.87$ $0.10$ $1.62$
    32 $8.02e-05$ $3.90$ $1.93e-05$ $3.99$ $0.01$ $2.78$
    64 $5.19e-06$ $3.95$ $1.20e-06$ $4.01$$1.26e-03$ $3.52$
    128 $3.33e-07$ $3.96$ $7.42e-08$ $4.01$$8.32e-05$ $3.92$
    256 $2.12e-08$ $3.97$ $4.62e-09$ $4.01$$4.82e-06$ $4.11$
    512 $1.35e-09$$3.98$ $2.88e-10$ $4.00$$2.68e-07$ $4.17$
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    Table 2.  Two-step collocation method with $m = 3$, $\eta = \left[\frac{1}{4}, \frac{1}{2}, \frac{3}{4}\right]$

    Problem 1 Problem 2 Problem 3
    N $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$ $e_{N}$ $p_{\mbox{eff}}$
    8 $0.13$ $1.60e-02$ $0.11$
    16 $1.44e-03$ $6.54$ $1.73e-04$ $6.53$ $1.93e-02$ $ 2.56$
    32 $1.78e-05$ $6.34$ $ 2.12e-06$ $6.34$ $1.06e-03$ $ 4.18$
    64 $2.40e-07$ $6.21$ $2.91e-08$ $6.19$$2.78e-05$ $5.26$
    128 $3.43e-09$ $6.13$ $4.25e-10$ $6.10$$5.00e-07$ $5.80$
    256 $5.09e-11$ $6.08$ $6.43e-12$ $6.05$$ 7.53e-09$ $6.05$
    512 $7.66e-13$ $6.05$ $9.90e-14$ $ 6.02$$1.05e-10$ $6.17$
     | Show Table
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