Approximation methods for the distributed order calculus using the convolution quadrature

Baoli Yin; Yang Liu; Hong Li; Zhimin Zhang

doi:10.3934/dcdsb.2020168

Article Contents

2021, Volume 26, Issue 3: 1447-1468. Doi: 10.3934/dcdsb.2020168

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Approximation methods for the distributed order calculus using the convolution quadrature

1.
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
2.
Beijing Computational Science Research Center, Beijing 100193, China
3.
Department of Mathematics, Wayne State University Detroit, MI 48202, USA

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

Received: August 2019

Revised: March 2020

Early access: May 2020

Published: March 2021

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.

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Abstract

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.

Keywords:

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

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

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

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

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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

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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

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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

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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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