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

Wen Li; Song Wang; Volker Rehbock

doi:10.3934/naco.2017018

Article Contents

2017, Volume 7, Issue 3: 273-287. Doi: 10.3934/naco.2017018

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

Department of Mathematics & Statistics, Curtin University, GPO Box U1987, Perth WA 6845, Australia

* Corresponding author: S. Wang
* Corresponding author: S. Wang

Received: December 31, 2016

Revised: May 31, 2017

Published: October 2017

This work is supported by the AOARD Project # 15IOA095 from the US Air Force.

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Abstract

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.

Keywords:

Fractional ordinary differential equation,
optimal control,
one-step time-stepping scheme,
one-point quadrature rule,
super-convergence,
Caputo's fractional derivative.

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]
h	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	-	-

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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

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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

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