Article Contents
Article Contents

# Approximation methods for the distributed order calculus using the convolution quadrature

• * Corresponding author: mathliuyang@imu.edu.cn (Y. Liu)

The second author is supported in part by the NSFC grant 11661058; The third author is supported in part by the NSFC grant 11761053, the NSF of Inner Mongolia 2017MS0107, and the program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region NJYT-17-A07; The fourth author was supported in part by grants NSFC 11871092 and NSAF U1930402

• In this article we generalize the convolution quadrature (CQ) method, which aims at approximating the fractional calculus, to the case for the distributed order calculus. Our method is a natural expansion that the approximation formulas, convergence results and correction technique reduce to the cases for the CQ method if the weight function $\mu(\alpha)$ is defined by $\delta(\alpha-\alpha_0)$. Further, we explore a new structure of the solution of an ODE with the distributed order fractional derivative, which differs from those of the solutions of traditional fractional ODEs, and propose a new correction technique for this new structure to restore the optimal convergence rate. Numerical tests with smooth and nonsmooth solutions confirm our theoretical results and the efficiency of our correction technique.

Mathematics Subject Classification: Primary: 26A33, 65D25; Secondary: 65D30.

 Citation:

• Figure 1.  Figures of $f_n(x)$ and the numerical solution

Figure 2.  Error plot at each node with correction terms, $h = \frac{1}{40}$

Figure 3.  Error plot at each node without correction terms, $h = \frac{1}{40}$

Table 2.  Comparison of rates for methods with and without correction terms, $h = \frac{1}{2000}$

 Methods $\tau$ $E_U$ Rate CPU(s) $E_C$ Rate CPU(s) D-Euler ($\ell$=0.1) 1/10 2.66E-01 – 0.441 5.14E-02 – 0.435 1/20 2.90E-01 – 0.426 2.55E-02 1.01 0.461 1/40 3.05E-01 – 0.480 1.28E-02 1.00 0.496 1/80 3.10E-01 – 0.611 6.40E-03 1.00 0.611 D-BDF2 ($\ell$=0.5) 1/10 5.54E-02 – 0.442 4.20E-03 – 0.435 1/20 4.97E-02 0.16 0.443 1.11E-03 1.92 0.447 1/40 4.23E-02 0.23 0.461 2.85E-04 1.96 0.481 1/80 3.43E-02 0.30 0.501 7.15E-05 1.99 0.810 D-BT-$\theta$($\theta$=0.45, $\ell$=0.3) 1/10 1.55E-01 – 0.450 1.87E-03 – 0.428 1/20 1.50E-01 0.05 0.457 4.70E-04 2.00 0.461 1/40 1.38E-01 0.11 0.476 1.17E-04 2.00 0.498 1/80 1.23E-01 0.17 0.589 2.88E-05 2.02 0.605 D-BN-$\theta$($\theta$=1, $\ell$=0.8) 1/10 1.59E-02 – 0.448 6.76E-03 – 0.444 1/20 1.27E-02 0.33 0.467 1.82E-03 1.90 0.414 1/40 9.47E-03 0.42 0.478 4.70E-04 1.95 0.491 1/80 6.69E-03 0.50 0.619 1.19E-04 1.98 0.570 D-BDF3($\ell$=0.9) 1/10 1.48E-02 – 0.437 3.98E-04 – 0.428 1/20 1.03E-02 0.53 0.437 5.08E-05 2.97 0.443 1/40 6.74E-03 0.60 0.494 5.76E-06 3.14 0.479 1/80 4.21E-03 0.68 0.582 5.20E-07 3.47 0.585

Table 3.  Temporal convergence rates for methods with and without correction terms

 Methods $h$ $E_U$ Rate CPU(s) $E_C$ Rate CPU(s) D-Euler 1/20 1.82E-02 – 1.516 5.18E-03 – 1.579 1/40 9.85E-03 0.89 3.526 2.64E-03 0.97 3.266 1/80 5.23E-03 0.91 7.992 1.33E-03 0.99 8.236 1/160 2.74E-03 0.93 33.652 6.65E-04 1.00 34.729 D-BDF2 1/20 5.32E-02 – 1.525 2.79E-04 – 1.507 1/40 3.28E-02 0.70 3.285 7.09E-05 1.97 3.203 1/80 1.96E-02 0.75 7.897 1.79E-05 1.99 8.142 1/160 1.19E-02 0.72 34.870 4.49E-06 1.99 32.841 D-BT-$\theta$($\theta$=0.2) 1/20 5.94E-02 – 1.468 1.95E-04 – 1.523 1/40 3.68E-02 0.69 3.254 4.97E-05 1.97 3.308 1/80 2.20E-02 0.74 7.876 1.25E-05 1.99 8.160 1/160 1.28E-02 0.78 34.141 3.14E-06 1.99 34.439 D-BN-$\theta$($\theta$=0.5) 1/20 5.44E-02 – 1.509 2.46E-04 – 1.517 1/40 3.35E-02 0.70 3.281 6.24E-05 1.98 3.479 1/80 1.99E-02 0.75 8.090 1.57E-05 1.99 7.998 1/160 1.20E-02 0.73 35.025 3.95E-06 1.99 34.751

Table 1.  Temporal convergence rates for smooth solutions, $h = \frac{1}{2000}$

 Methods $\tau$ $E_T$ Rate CPU(s) $E_S$ Rate CPU(s) D-Euler 1/10 9.03E-02 – 0.401 9.10E-02 – 0.441 1/20 4.59E-02 0.98 0.513 4.60E-02 0.98 0.453 1/40 2.31E-02 0.99 0.554 2.31E-02 0.99 0.492 1/80 1.16E-02 1.00 0.688 1.16E-02 1.00 0.617 D-BDF2 1/10 1.48E-02 – 0.441 1.49E-02 – 0.426 1/20 3.98E-03 1.89 0.453 3.98E-03 1.90 0.435 1/40 1.03E-03 1.95 0.474 1.03E-03 1.95 0.468 1/80 2.62E-04 1.98 0.579 2.62E-04 1.98 0.607 D-BT-$\theta$($\theta$=0.45) 1/10 5.36E-03 – 0.405 5.41E-03 – 0.452 1/20 1.36E-03 1.97 0.454 1.37E-03 1.98 0.449 1/40 3.44E-04 1.99 0.516 3.44E-04 1.99 0.469 1/80 8.61E-05 2.00 0.604 8.61E-05 2.00 0.598 D-BN-$\theta$($\theta$=1) 1/10 2.63E-02 – 0.413 2.65E-02 – 0.438 1/20 7.17E-03 1.88 0.453 7.18E-03 1.88 0.437 1/40 1.87E-03 1.94 0.490 1.87E-03 1.94 0.494 1/80 4.76E-04 1.97 0.623 4.76E-04 1.97 0.583 D-BDF3 1/10 2.00E-03 – 0.473 2.03E-03 – 0.407 1/20 2.65E-04 2.92 0.475 2.66E-04 2.93 0.440 1/40 3.39E-05 2.97 0.496 3.39E-05 2.97 0.496 1/80 4.20E-06 3.01 0.611 4.20E-06 3.01 0.614

Table 4.  Temporal convergence rates for methods with and without correction terms

 Methods $h$ $E_U$ CPU(s) $E_C$ CPU(s) D-Euler 1/20 1.8231582E-02 1.010 1.40274270E-02 1.054 1/40 9.8461370E-03 2.195 2.16534296E-03 2.278 1/80 5.2308982E-03 5.646 1.18484214E-03 5.556 1/160 2.7366136E-03 22.973 6.24937949E-04 23.143 D-BDF2 1/20 5.3244469E-02 0.982 1.40274270E-02 0.998 1/40 3.2848868E-02 2.202 2.16534296E-03 2.233 1/80 1.9587693E-02 5.452 2.69407811E-04 5.554 1/160 1.1890696E-02 23.290 4.62329665E-06 23.297 D-BT-$\theta$($\theta$=0.2) 1/20 5.9360151E-02 0.971 1.40274270E-02 0.998 1/40 3.6795394E-02 2.226 2.16534295E-03 2.187 1/80 2.1969727E-02 5.443 2.69407811E-04 5.579 1/160 1.2752778E-02 23.064 4.62329717E-06 23.388 D-BN-$\theta$($\theta$=0.5) 1/20 5.4412292E-02 0.959 1.40274270E-02 1.013 1/40 3.3527144E-02 2.210 2.16534296E-03 2.199 1/80 1.9922404E-02 5.520 2.69407811E-04 5.530 1/160 1.2015071E-02 23.077 4.62329657E-06 23.571
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