Analysis of a finite difference scheme for a nonlinear Caputo fractional differential equation on an adaptive grid

Yong Zhang; Xiaobing Bao; Li-Bin Liu; Zhifang Liang; Yong Zhang; Xiaobing Bao; Li-Bin Liu; Zhifang Liang

doi:10.3934/math.2021500

AIMS Mathematics

2021, Volume 6, Issue 8: 8611-8624. doi: 10.3934/math.2021500

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Analysis of a finite difference scheme for a nonlinear Caputo fractional differential equation on an adaptive grid

1.
School of Big Data and Artificial Intelligence, Chizhou University, Chizhou, Anhui 247000, China
2.
School of Mathematics and Statistics, Nanning Normal University, Nanning 530029, China

Received: 25 April 2021 Accepted: 02 June 2021 Published: 07 June 2021
MSC : 65L05, 65L12, 65L20

A nonlinear initial value problem whose differential operator is a Caputo derivative of order $ \alpha $ with $ 0 < \alpha < 1 $ is studied. By using the Riemann-Liouville fractional integral transformation, this problem is reformulated as a Volterra integral equation, which is discretized by using the right rectangle formula. Both a priori and an a posteriori error analysis are conducted. Based on the a priori error bound and mesh equidistribution principle, we show that there exists a nonuniform grid that gives first-order convergent result, which is robust with respect to $ \alpha $. Then an a posteriori error estimation is derived and used to design an adaptive grid generation algorithm. Numerical results complement the theoretical findings.
- Caputo fractional derivative,
- initial value problem,
- adaptive grid,
- mesh equidistribution
Citation: Yong Zhang, Xiaobing Bao, Li-Bin Liu, Zhifang Liang. Analysis of a finite difference scheme for a nonlinear Caputo fractional differential equation on an adaptive grid[J]. AIMS Mathematics, 2021, 6(8): 8611-8624. doi: 10.3934/math.2021500

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Abstract

A nonlinear initial value problem whose differential operator is a Caputo derivative of order $ \alpha $ with $ 0 < \alpha < 1 $ is studied. By using the Riemann-Liouville fractional integral transformation, this problem is reformulated as a Volterra integral equation, which is discretized by using the right rectangle formula. Both a priori and an a posteriori error analysis are conducted. Based on the a priori error bound and mesh equidistribution principle, we show that there exists a nonuniform grid that gives first-order convergent result, which is robust with respect to $ \alpha $. Then an a posteriori error estimation is derived and used to design an adaptive grid generation algorithm. Numerical results complement the theoretical findings.

References

[1]	R. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Redding, 2006.
[2]	A. Kaur, P. S. Takhar, D. M. Smith, J. E. Mann, M. M. Brashears, Fractional differential equations based modeling of microbial survival and growth curves: Model development and experimental validation, Food Eng. Phy. Properties, 73 (2008), 403-414.
[3]	D. Lokenath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci., 54 (2003), 3413-3442.
[4]	P. A. Naik, J. Zu, K. M. Owolabi, Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order, Physica A., 545 (2020), 123816. doi: 10.1016/j.physa.2019.123816
[5]	P. A. Naik, J. Zu, K. M. Owolabi, Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control, Chaos. Solition. Fract., 138 (2020), 109826. doi: 10.1016/j.chaos.2020.109826
[6]	P. A. Naik, K. M. Owolabi, M. Yavuz, Z. Jian, Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells, Chaos. Solition. Fract., 140 (2020), 110272 doi: 10.1016/j.chaos.2020.110272
[7]	K. M. Owolabi, B. Karaagac, Dynamics of multi-puls splitting process in one-dimensional Gray-Scott system with fractional order operator, Chaos. Soliton. Fract., 136 (2020), 109835. doi: 10.1016/j.chaos.2020.109835
[8]	C. Li, F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim., 34 (2013), 149-179. doi: 10.1080/01630563.2012.706673
[9]	M. Stynes, J. L. Gracia, A finite difference method for a two-point boundary value problem with a Caputo fractional derivative, IMA J. Numer. Anal., 35 (2015), 698-721. doi: 10.1093/imanum/dru011
[10]	S. Vong, P. Lyu, X. Chen, S.-L. Lei, High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives, Numer. Algorithm., 72 (2016), 195-210. doi: 10.1007/s11075-015-0041-3
[11]	K. M. Owolabi, Numerical simulation of fractional-order reaction-diffusion equations with the Riesz and Caputo derivatives, Neural. Comput. Appl., 2019. Available from: https://doi.org/10.1007/s00521-019-04350-2.
[12]	K. Nedaiasl, R. Dehbozorgi, Galerkin finite element method for nonlinear fractional differential equations, Numer. Algorithms., 2021. Available from: https://doi.org/10.1007/s11075-020-01032-2.
[13]	H. Liang, M. Stynes, Collocation methods for general Caputo two-point boundary value problems, J. Sci. Comput., 76 (2018), 390-425. doi: 10.1007/s10915-017-0622-5
[14]	H. Liang, M. Stynes, Collocation methods for general Riemann-Liouville two-point boundary value problems, Adv. Comput. Math., 45 (2019), 897-928. doi: 10.1007/s10444-018-9645-1
[15]	N. Kopteva, M. Stynes, An efficient collocation method for a Caputo two-point boundary value problem, BIT Numer. Math., 55 (2015), 1105-1123. doi: 10.1007/s10543-014-0539-4
[16]	C. Wang, Z. Wang, L. Wang, A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative, J. Sci. Comput., 76 (2018), 166-188. doi: 10.1007/s10915-017-0616-3
[17]	Z. Gu, Spectral collocation method for nonlinear Caputo fractional differential system, Adv. Comput. Math., 46 (2020), 66. doi: 10.1007/s10444-020-09808-9
[18]	M. Stynes, J. L. Gracia, A finite difference method for a two-point boundary value problem with a Caputo fractional derivative, IMA J. Numer. Anal., 35 (2015), 698-721. doi: 10.1093/imanum/dru011
[19]	J. Gracia, M. Stynes, Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems, J. Comput. Appl. Math., 273 (2015), 103-115. doi: 10.1016/j.cam.2014.05.025
[20]	M. Stynes, E. O'Riordan, J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079. doi: 10.1137/16M1082329
[21]	H. Liao, W. Mclean, J. Zhang, A discrete Grönwall inequality with applications to numerical scheme for sudiffusion problems, SIAM J. Numer. Anal., 57 (2019), 218-237. doi: 10.1137/16M1175742
[22]	J. L. Gracia, E. O'Riordan, M. Stynes, A fitted scheme for a Caputo initial-boundary value problem, J. Sci. Comput., 76 (2018), 583-609. doi: 10.1007/s10915-017-0631-4
[23]	N. Kopteva, X. Meng, Error analysis for a fractional-derivative parabolic problem on quasi-graded meshes using barrier function, SIAM J. Numer. Anal., 2020, 58 (2020), 1217-1238.
[24]	Z. Cen, L. B. Liu, J. Huang, A posteriori error estimation in maximum norm for a two-point boundary value problem with a Riemann-Liouville fractional derivative, Appl. Math. Lett., 102 (2020), 106086. doi: 10.1016/j.aml.2019.106086
[25]	L. B. Liu, Z. Liang, G. Long, Y. Liang, Convergence analysis of a finite difference scheme for a Riemann-Liouville fractional derivative two-point boundary value problem on an adaptive grid, J. Comput. Appl. Math., 375 (2020), 112809. doi: 10.1016/j.cam.2020.112809
[26]	J. Huang, Z. Cen, L. B. Liu, J. Zhao, An efficient numerical method for a Riemann-Liouville two-point boundary value problem, Appl. Math. Lett., 103 (2020), 106201. doi: 10.1016/j.aml.2019.106201
[27]	Z. Cen, A. Le, A. Xu, A posteriori error analysis for a fractional differential equation, Int. J. Comput. Math., 94 (2017), 1182-1195.
[28]	L. B. Liu, Y. Chen, A posteriori error estimation and adaptive strategy for a nonlinear fractional differential equation, Int. J. Comput. Math., 2021. Available from: https://doi.org/10.1080/00207160.2021.1906420.
[29]	K. Diethelm, The analysis of fractional differential equations. Springer, Berlin, 2010.
[30]	C. Li, Q. Yi, A. Chen, Finite difference methods with non-uniform meshes for nonlinear differential equations, J. Comput. Phys., 316 (2016), 614-631. doi: 10.1016/j.jcp.2016.04.039
[31]	H. Ye, J. Gao, Y. Ding, A generalized Grönwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081. doi: 10.1016/j.jmaa.2006.05.061

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