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September  2017, 7(3): 273-287. doi: 10.3934/naco.2017018

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

Received  January 2017 Revised  June 2017 Published  July 2017

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

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.

Citation: Wen Li, Song Wang, Volker Rehbock. A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 273-287. doi: 10.3934/naco.2017018
References:
[1]

J. Cao and C. Xu, A high order scheme for the numerical solution of the fractional ordinary differential equations, Journal of Computational Physics, 238 (2013), 154-168. doi: 10.1016/j.jcp.2012.12.013. Google Scholar

[2]

S. Campbell and P. Kunkel, Solving higher index DAE optimal control problems, Numerical Algebra, Control & Optimization, 6 (2016), 447-472. doi: 10.3934/naco.2016020. Google Scholar

[3]

A. Cartea and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A: Statistical Mechanics and its Applications, 374 (2007), 749-763. doi: 10.1063/1.2336114. Google Scholar

[4]

W. Chen and S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Comp. Math. With Appl., 67 (2014), 77-90. doi: 10.1016/j.camwa.2013.10.007. Google Scholar

[5]

W. Chen and S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, Journal of Industrial & Management Optimization, 11 (2015), 241-264. doi: 10.3934/jimo.2015.11.241. Google Scholar

[6]

W. Chen and S. Wang, A 2nd-Order FDM for a 2D fractional Black-Scholes equation, in Numerical Analysis and Its Applications. NAA 2016 (eds. Dimov I., Farag I., Vulkov L.), Lecture Notes in Computer Science, Springer, 10187 (2017), 46–57.Google Scholar

[7]

W. Chen and S. Wang, A power penalty method for a 2D fractional partial differential linear complementarity problem governing two-asset American option pricing, Applied Mathematics and Computation, 305 (2017), 174-187. doi: 10.1016/j.amc.2017.01.069. Google Scholar

[8]

C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phis. (Leipzig), 12 (2003), 692-703. doi: 10.1002/andp.200310032. Google Scholar

[9]

W. Deng and C. Li, Numerical schemes for fractional ordinary differential equations, Numerical Modeling, Dr. Peep Miidla (Ed.), InTech, 2012.Google Scholar

[10]

K. Diethelm and N. J. Ford, Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265 (2002), 229-248. doi: 10.1006/jmaa.2000.7194. Google Scholar

[11]

K. DiethelmN. J. Ford and A. D. Freed, A predictor corrector approach for the numerical solution of fractional differential equation, Nonlinear Dynam, 29 (2002), 2-22. doi: 10.1023/A:1016592219341. Google Scholar

[12]

K. DiethelmN. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52. doi: 10.1023/B:NUMA.0000027736.85078.be. Google Scholar

[13]

K. DiethelmN. J. FordA. D. Freed and Yu. Luchko, Algorithms for the fractional calculus: a selection of numerical methods, Comput. Method appl. Mech. Engrg., 194 (2005), 743-773. doi: 10.1016/j.cma.2004.06.006. Google Scholar

[14]

B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, World Scientific, 2015. doi: 10.1142/9543. Google Scholar

[15]

H. HuangY. Tang and L. Vazquez, Convergence analysis of a block-by block method for fractional differential equation, Numer. Math. Theor. Methods Appl., 5 (2012), 229-241. doi: 10.4208/nmtma.2012.m1038. Google Scholar

[16]

A. A. Kilbas and S. A. Marzan, Cauchy problem for differential equation with Caputo derivative, Fractional Calculus and Applied Analysis, 7 (2014), 297-321. Google Scholar

[17]

K. Kumar and O. P. Agrawal, An approximate method for numerical solution of fractional differential equations, Signal Process, 86 (2006), 2602-2610. Google Scholar

[18]

A. Laforgia and P. Natalini, Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities, J. Math. Anal. Appl., 407 (2013), 459-504. doi: 10.1016/j.jmaa.2013.05.045. Google Scholar

[19]

C. Li and C. Tao, On the fractional Adams method, Comput. Math. Appl., 58 (2009), 1573-1588. doi: 10.1016/j.camwa.2009.07.050. Google Scholar

[20]

C. Li and F. Zeng, The finite difference methods for fractional ordinary differential equations, Numerical Functional Analysis and Optimization, 34 (2013), 149-179. doi: 10.1080/01630563.2012.706673. Google Scholar

[21]

R. Lin and F. Liu, Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear Analysis, 66 (2007), 856-869. doi: 10.1016/j.na.2005.12.027. Google Scholar

[22] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010. Google Scholar
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F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010. doi: 10.1142/9781848163300. Google Scholar

[24]

S. Mohd MahaliS. Wang and X. Lou, Determination of effective diffusion coefficients of drug delivery devices by a state observer approach, Discrete and Continuous Dynamical Systems Series B, 17 (2011), 1119-1136. Google Scholar

[25]

S. Mohd MahaliS. Wang and X. Lou, Estimation of effective diffusion coefficients of drug delivery devices in a flow-through system, Journal of Engineering Mathematics, 87 (2014), 139-152. doi: 10.1007/s10665-013-9669-y. Google Scholar

[26]

M. D. Ortigueria and J. A. T. Machodo, Special section: Fractional calculus applications in signals and systems, Signal Processing, 86 (2006), 2503-3094. Google Scholar

[27]

B. ShenX. Wang and C. Liu, Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control, Numerical Algebra, Control & Optimization, 5 (2015), 369-380. doi: 10.3934/naco.2015.5.369. Google Scholar

[28]

S. Sorokin and M. Staritsyn, Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control, Numerical Algebra, Control & Optimization, 7 (2017), 201-210. doi: 10.3934/naco.2017014. Google Scholar

[29]

M. Y. TanL. S. Jennings and S. Wang, Analysing human periodic walking at different speeds using parametrization enhancing transform in dynamic optimization, Pacific Journal of Optimization, 12 (2016), 557-586. Google Scholar

[30]

Y. WangC. Yu and K. L. Teo, A new computational strategy for optimal control problem with a cost on changing control, Numerical Algebra, Control & Optimization, 6 (2016), 339-364. doi: 10.3934/naco.2016016. Google Scholar

show all references

References:
[1]

J. Cao and C. Xu, A high order scheme for the numerical solution of the fractional ordinary differential equations, Journal of Computational Physics, 238 (2013), 154-168. doi: 10.1016/j.jcp.2012.12.013. Google Scholar

[2]

S. Campbell and P. Kunkel, Solving higher index DAE optimal control problems, Numerical Algebra, Control & Optimization, 6 (2016), 447-472. doi: 10.3934/naco.2016020. Google Scholar

[3]

A. Cartea and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A: Statistical Mechanics and its Applications, 374 (2007), 749-763. doi: 10.1063/1.2336114. Google Scholar

[4]

W. Chen and S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Comp. Math. With Appl., 67 (2014), 77-90. doi: 10.1016/j.camwa.2013.10.007. Google Scholar

[5]

W. Chen and S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, Journal of Industrial & Management Optimization, 11 (2015), 241-264. doi: 10.3934/jimo.2015.11.241. Google Scholar

[6]

W. Chen and S. Wang, A 2nd-Order FDM for a 2D fractional Black-Scholes equation, in Numerical Analysis and Its Applications. NAA 2016 (eds. Dimov I., Farag I., Vulkov L.), Lecture Notes in Computer Science, Springer, 10187 (2017), 46–57.Google Scholar

[7]

W. Chen and S. Wang, A power penalty method for a 2D fractional partial differential linear complementarity problem governing two-asset American option pricing, Applied Mathematics and Computation, 305 (2017), 174-187. doi: 10.1016/j.amc.2017.01.069. Google Scholar

[8]

C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phis. (Leipzig), 12 (2003), 692-703. doi: 10.1002/andp.200310032. Google Scholar

[9]

W. Deng and C. Li, Numerical schemes for fractional ordinary differential equations, Numerical Modeling, Dr. Peep Miidla (Ed.), InTech, 2012.Google Scholar

[10]

K. Diethelm and N. J. Ford, Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265 (2002), 229-248. doi: 10.1006/jmaa.2000.7194. Google Scholar

[11]

K. DiethelmN. J. Ford and A. D. Freed, A predictor corrector approach for the numerical solution of fractional differential equation, Nonlinear Dynam, 29 (2002), 2-22. doi: 10.1023/A:1016592219341. Google Scholar

[12]

K. DiethelmN. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31-52. doi: 10.1023/B:NUMA.0000027736.85078.be. Google Scholar

[13]

K. DiethelmN. J. FordA. D. Freed and Yu. Luchko, Algorithms for the fractional calculus: a selection of numerical methods, Comput. Method appl. Mech. Engrg., 194 (2005), 743-773. doi: 10.1016/j.cma.2004.06.006. Google Scholar

[14]

B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and Their Numerical Solutions, World Scientific, 2015. doi: 10.1142/9543. Google Scholar

[15]

H. HuangY. Tang and L. Vazquez, Convergence analysis of a block-by block method for fractional differential equation, Numer. Math. Theor. Methods Appl., 5 (2012), 229-241. doi: 10.4208/nmtma.2012.m1038. Google Scholar

[16]

A. A. Kilbas and S. A. Marzan, Cauchy problem for differential equation with Caputo derivative, Fractional Calculus and Applied Analysis, 7 (2014), 297-321. Google Scholar

[17]

K. Kumar and O. P. Agrawal, An approximate method for numerical solution of fractional differential equations, Signal Process, 86 (2006), 2602-2610. Google Scholar

[18]

A. Laforgia and P. Natalini, Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities, J. Math. Anal. Appl., 407 (2013), 459-504. doi: 10.1016/j.jmaa.2013.05.045. Google Scholar

[19]

C. Li and C. Tao, On the fractional Adams method, Comput. Math. Appl., 58 (2009), 1573-1588. doi: 10.1016/j.camwa.2009.07.050. Google Scholar

[20]

C. Li and F. Zeng, The finite difference methods for fractional ordinary differential equations, Numerical Functional Analysis and Optimization, 34 (2013), 149-179. doi: 10.1080/01630563.2012.706673. Google Scholar

[21]

R. Lin and F. Liu, Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear Analysis, 66 (2007), 856-869. doi: 10.1016/j.na.2005.12.027. Google Scholar

[22] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010. Google Scholar
[23]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010. doi: 10.1142/9781848163300. Google Scholar

[24]

S. Mohd MahaliS. Wang and X. Lou, Determination of effective diffusion coefficients of drug delivery devices by a state observer approach, Discrete and Continuous Dynamical Systems Series B, 17 (2011), 1119-1136. Google Scholar

[25]

S. Mohd MahaliS. Wang and X. Lou, Estimation of effective diffusion coefficients of drug delivery devices in a flow-through system, Journal of Engineering Mathematics, 87 (2014), 139-152. doi: 10.1007/s10665-013-9669-y. Google Scholar

[26]

M. D. Ortigueria and J. A. T. Machodo, Special section: Fractional calculus applications in signals and systems, Signal Processing, 86 (2006), 2503-3094. Google Scholar

[27]

B. ShenX. Wang and C. Liu, Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control, Numerical Algebra, Control & Optimization, 5 (2015), 369-380. doi: 10.3934/naco.2015.5.369. Google Scholar

[28]

S. Sorokin and M. Staritsyn, Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control, Numerical Algebra, Control & Optimization, 7 (2017), 201-210. doi: 10.3934/naco.2017014. Google Scholar

[29]

M. Y. TanL. S. Jennings and S. Wang, Analysing human periodic walking at different speeds using parametrization enhancing transform in dynamic optimization, Pacific Journal of Optimization, 12 (2016), 557-586. Google Scholar

[30]

Y. WangC. Yu and K. L. Teo, A new computational strategy for optimal control problem with a cost on changing control, Numerical Algebra, Control & Optimization, 6 (2016), 339-364. doi: 10.3934/naco.2016016. Google Scholar

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]
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 - -
h Our results Results from [9] Our results Results from [9] Our results Results from [9] Our results Results from [9]
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 - -
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
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
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
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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