Article Contents

2019, Volume 24, Issue 4: 1989-2015. Doi: 10.3934/dcdsb.2019026

This issue Previous Article Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness Next Article

Well-posedness and numerical algorithm for the tempered fractional differential equations

1.
Department of Applied Mathematics, Xi'an University of Technology, Xi'an, Shaanxi 710054, China
2.
Beijing Computational Science Research Center, Beijing 10084, China
3.
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China
4.
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710054, China

Received: November 2015

Revised: January 2019

Early access: January 2019

Published: January 2019

Abstract / Introduction Full Text(HTML) Figure(0) / Table(6) Related Papers Cited by

Abstract

Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cytoplasm. The famous continuous time random walk (CTRW) model with power law waiting time distribution (having diverging first moment) describes this phenomenon. Because of the finite lifetime of biological particles, sometimes it is necessary to temper the power law measure such that the waiting time measure has convergent first moment. Then the time operator of the Fokker-Planck equation corresponding to the CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. This paper focus on discussing the properties of the time tempered fractional derivative, and studying the well-posedness and the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation. By adjusting the parameter of the proposed algorithm, high convergence order can be obtained and the computational cost linearly increases with time. The numerical results shows that our algorithm converges with order $ N_I $, where $ N_I $ is the number of used interpolating points.

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Mathematics Subject Classification: Primary: 34A08, 74S25; Secondary: 26A33.

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Table 1. Maximum errors and convergence orders of Example 1 solved by the scheme (56)-(57) with $ T = 1,N = 20,N_I = 7 $, and $ \alpha = 0.5 $

	$ \lambda=0 $		$ \lambda=2 $		$ \lambda=6 $
$ \tau $	error	order	error	order	error	order
1/10	1.5207e-004		2.3516e-005		1.4300e-006
1/20	4.6202e-007	8.3626	1.4040e-007	7.3879	3.3507e-008	5.4154
1/40	1.6877e-009	8.0967	6.3106e-010	7.7976	2.7846e-010	6.9109
1/80	8.1135e-012	7.7005	2.5491e-012	7.9517	1.4371e-012	7.5982
1/160	3.5305e-014	7.8443	1.2794e-014	7.6383	7.0913e-015	7.6629

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Table 2. Maximum errors and convergence orders of Example 1 solved by the scheme (56)-(57) with $T = 1,N = 20,N_I = 6$, and $\alpha = 1.0$

	$\lambda=0$		$\lambda=2$		$\lambda=6$
$\tau$	error	order	error	order	error	order
1/10	8.1108e-005		1.2528e-005		1.1365e-006
1/20	7.8788e-007	6.6857	1.5673e-007	6.3207	2.3299e-008	5.6082
1/40	1.2817e-008	5.9418	2.1909e-009	6.1606	3.2657e-010	6.1567
1/80	2.2418e-010	5.8373	3.4124e-011	6.0046	4.4768e-012	6.1888
1/160	3.6193e-012	5.9528	5.3461e-013	5.9962	6.7955e-014	6.0417

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Table 3. Maximum errors and convergence orders of Example 1 solved by the scheme (56)-(57) with $ T = 1,N = 20,N_I = 6 $, and $ \alpha = 1.5 $

	$ \lambda=0 $		$ \lambda=2 $		$ \lambda=6 $
$ \tau $	error	order	error	order	error	order
1/10	6.6386e-005		9.6009e-006		8.5068e-007
1/20	9.2847e-007	6.1599	1.4297e-007	6.0694	1.9943e-008	5.4147
1/40	1.5767e-008	5.8799	2.1338e-009	6.0661	3.0437e-010	6.0339
1/80	2.3505e-010	6.0678	3.5138e-011	5.9242	3.8203e-012	6.3159
1/160	3.8498e-012	5.9320	5.3434e-013	6.0391	6.7433e-014	5.8241

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Table 4. Maximum errors and convergence orders of Example 2 solved by the scheme (66) with $ T = 1.1, N = 26, \tilde{N} = 40, N_I = 2, T_0 = 0.1,\mu = 1 $, and $ \lambda = 5 $

	$ \alpha=0.2 $		$ \alpha=0.9 $		$ \alpha=1.8 $
$ \tau $	error	order	error	order	error	order
1/20	5.4805e-004		1.9043e-005		2.1461e-006
1/40	1.8749e-004	1.5475	4.3478e-006	2.1309	5.6685e-007	1.9207
1/80	5.0838e-005	1.8828	1.0851e-006	2.0025	1.5416e-007	1.8786
1/160	1.3492e-005	1.9138	3.1549e-007	1.7821	4.0386e-008	1.9324

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Table 5. Maximum errors and convergence orders of Example 2 solved by the scheme (66) with $ T = 1.1, N = 26, \tilde{N} = 40, N_I = 2, T_0 = 0.1,\mu = 1 $, and $ \lambda = 10 $

	$ \alpha=0.2 $		$ \alpha=0.9 $		$ \alpha=1.8 $
$ \tau $	error	order	error	order	error	order
1/20	2.0162e-004		7.0054e-006		4.0825e-007
1/40	8.8563e-005	1.1868	1.6897e-006	2.0517	1.0286e-007	1.9887
1/80	2.7211e-005	1.7025	4.1757e-007	2.0167	2.7730e-008	1.8912
1/160	7.4508e-006	1.8688	1.1169e-007	1.9025	7.3208e-009	1.9214

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Table 6. Maximum errors and convergence orders of Example 3 solved by the scheme (56)-(57) with $ T = 1,N = 20,N_I = 5 $, and $ \alpha = 0.4 $

	$ \lambda=0 $		$ \lambda=3 $		$ \lambda=5 $
$ \tau $	error	order	error	order	error	order
1/10	2.4208e-004		7.4482e-007		2.1996e-007
1/20	4.9371e-006	5.6157	6.8759e-008	3.4373	3.2736e-008	2.7483
1/40	1.0390e-007	5.5704	3.1715e-009	4.4383	2.1801e-009	3.9084
1/80	2.2895e-009	5.5041	1.1961e-010	4.7288	9.9159e-011	4.4585
1/160	5.3248e-011	5.4261	4.1052e-012	4.8647	3.7378e-012	4.7295

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