The Cubic B-Spline Method for a Class of Caputo-Fabrizio Fractional Differential Equations

HU Xinghua; CAI Junying

doi:10.21656/1000-0887.430195

Volume 44 Issue 6

Jun. 2023

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2023 > 44(6): 744-756

HU Xinghua, CAI Junying. The Cubic B-Spline Method for a Class of Caputo-Fabrizio Fractional Differential Equations[J]. Applied Mathematics and Mechanics, 2023, 44(6): 744-756. doi: 10.21656/1000-0887.430195

Citation:

PDF( 3123 KB)

The Cubic B-Spline Method for a Class of Caputo-Fabrizio Fractional Differential Equations

doi: 10.21656/1000-0887.430195

HU Xinghua,
CAI Junying^,

School of Sciences, Liaoning Technical University, Fuxin, Liaoning 123000, P.R.China

Received Date: 2022-06-07
Rev Recd Date: 2022-08-24
Publish Date: 2023-06-01

Abstract

Abstract

Based on the basic theorem of fractional calculus and the cubic B-spline theory, the cubic B-spline method for numerical solution of linear Caputo-Fabrizio fractional differential equations was proposed. The basic theorem of fractional calculus was used to transform the initial value problem into an expression about the solution function, and the cubic B-spline function was used to approximate the integrand function in the expression. Then the numerical solutions of the Caputo-Fabrizio fractional differential equations were calculated. The error estimation, convergence and stability of the constructed cubic B-spline method were given theoretically. Numerical experiments show that, the presented numerical method is feasible and effective in solving a class of Caputo-Fabrizio fractional differential equations, and the computation accuracy and efficiency are better than the 2 existing numerical methods.
- Caputo-Fabrizio fractional derivative,
- cubic B-spline method,
- error estimation,
- convergence,
- stability

FullText(HTML)

References(25)

References

[1]	张嫚, 曹艳华, 杨晓忠. 一类分数阶Langevin方程block-by-block算法的数值分析[J]. 应用数学和力学, 2021, 42(6): 562-574. doi: 10.21656/1000-0887.410337 ZHANG Man, CAO Yanhua, YANG Xiaozhong. Numerical analysis of a class of fractional Langevin equations with the block-by-block method[J]. Applied Mathematics and Mechanics, 2021, 42(6): 562-574. (in Chinese) doi: 10.21656/1000-0887.410337
[2]	鲍四元, 沈峰. 分数阶常微分方程的改进精细积分法[J]. 应用数学和力学, 2019, 40(12): 1309-1320. doi: 10.21656/1000-0887.390355 BAO Siyuan, SHEN Feng. An improved precise integration method for fractional ordinary differential equations[J]. Applied Mathematics and Mechanics, 2019, 40(12): 1309-1320. (in Chinese) doi: 10.21656/1000-0887.390355
[3]	CAPUTO M, FABRIZIO M. A new definition of fractional derivative without singular kernel[J]. Progress in Fractional Differentiation and Applications, 2015, 1: 73-85.
[4]	CAPUTO M, FABRIZIO M. Applications of new time and spatial fractional derivatives with exponential kernels[J]. Progress in Fractional Differentiation and Applications, 2016, 2(1): 1-11. doi: 10.18576/pfda/020101
[5]	LOSADA J, NIETO J J. Properties of a new fractional derivative without singular kernel[J]. Progress in Fractional Differentiation and Applications, 2015, 1(2): 87-92.
[6]	OWOLABI K M, ATANGANA A. Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative[J]. Chaos, Solitons and Fractals, 2017, 105: 111-119. doi: 10.1016/j.chaos.2017.10.020
[7]	ATANGANA A, ALQAHTANI R T. Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation[J]. Advances in Difference Equations, 2016, 2016(1): 156. doi: 10.1186/s13662-016-0871-x
[8]	BALEANU D, JAJARMI A, MOHAMMADI H, et al. A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative[J]. Chaos, Solitons and Fractals, 2020, 134: 109705. doi: 10.1016/j.chaos.2020.109705
[9]	BOUDAOUI A, MOUSSA Y, HAMMOUCH Z, et al. A fractional-order model describing the dynamics of the novel coronavirus (COVID-19) with nonsingular kernel[J]. Chaos, Solitons and Fractals, 2021, 146: 110859. doi: 10.1016/j.chaos.2021.110859
[10]	GONG X, FATMAWATI, KHAN M A. A new numerical solution of the competition model among bank data in Caputo-Fabrizio derivative[J]. Alexandria Engineering Journal, 2020, 59(4): 2251-2259. doi: 10.1016/j.aej.2020.02.008
[11]	LIAO X, LIN D, DONG L, et al. Analog implementation of fractional-order electric elements using Caputo-Fabrizio and Atangana-Baleanu definitions[J]. Fractals, 2021, 29(7): 2150235. doi: 10.1142/S0218348X21502352
[12]	ARIF M, ALI F, SHEIKH N A, et al. Fractional model of couple stress fluid for generalized Couette flow: a comparative analysis of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives[J]. IEEE Access, 2019, 7: 88643-88655. doi: 10.1109/ACCESS.2019.2925699
[13]	YOUSSEF H M, EL-BARY A A. Generalized fractional viscothermoelastic nanobeam under the classical Caputo and the new Caputo-Fabrizio definitions of fractional derivatives[J]. Waves in Random and Complex Media, 2023, 33(3): 545-566. doi: 10.1080/17455030.2021.1883767
[14]	CAO J, WANG Z, XU C. A high-order scheme for fractional ordinary differential equations with the Caputo-Fabrizio derivative[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 179-199. doi: 10.1007/s42967-019-00043-8
[15]	GUO X, LI Y T, ZENG T Y. A finite difference scheme for Caputo-Fabrizio fractional differential equations[J]. International Journal of Numerical Analysis and Modeling, 2020, 17(2): 195-211.
[16]	AL-SMADI M, DJEDDI N, MOMANI S, et al. An attractive numerical algorithm for solving nonlinear Caputo-Fabrizio fractional Abel differential equation in a Hilbert space[J]. Advances in Difference Equations, 2021, 271: 2-18.
[17]	DOUAIFIA R, BENDOUKHA S, ABDELMALEK S. A Newton interpolation based predictor-corrector numerical method for fractional differential equations with an activator-inhibitor case study[J]. Mathematics and Computers in Simulation, 2021, 187: 391-413.
[18]	何广婷. 带有Caputo-Fabrizio导数的分数阶微分方程的快速高阶算法的研究[D]. 硕士学位论文. 哈尔滨: 哈尔滨师范大学, 2021. HE Guangting. Fast higher order algorithms research for fractional differential equations with Caputo-Fabrizio derivatives[D]. Master Thesis. Harbin: Harbin Normal University, 2021. (in Chinese)
[19]	李鹏柱, 李风军, 李星, 等. 基于三次样条插值函数的非线性动力系统数值求解[J]. 应用数学和力学, 2015, 36(8): 887-896. doi: 10.3879/j.issn.1000-0887.2015.08.010 LI Pengzhu, LI Fengjun, LI Xing, et al. A numerical method for the solutions to nonlinear dynamic systems based on cubic spline interpolation functions[J]. Applied Mathematics and Mechanics, 2015, 36(8): 887-896. (in Chinese) doi: 10.3879/j.issn.1000-0887.2015.08.010
[20]	PODLUBNY I. Fractional Differential Equations[M]. New York: Academic Press, 1999.
[21]	RAMEZANI M, MOKHTARI R, HAASE G. Some high order formulae for approximating Caputo fractional derivatives[J]. Applied Numerical Mathematics, 2020, 153: 300-318.
[22]	李庆扬, 关治, 白峰杉. 数值计算原理[M]. 北京: 清华大学出版社, 2000. LI Qingyang, GUAN Zhi, BAI Fengshan. Theory of Numerical Calculation[M]. Beijing: Tsinghua University Press, 2000. (in Chinese)
[23]	黎天送. 基于粒子群的全局双三次B样条曲面插值方法及实验研究[D]. 硕士学位论文. 昆明: 云南大学, 2014. LI Tiansong. Global bicubic B-spline surface interpolation method based on particle swarm optimization and experimental research[D]. Master Thesis. Kunming: Yunnan University, 2014. (in Chinese)
[24]	KINCAID D R, CHENEY W. Numerical Analysis: Mathematics of Scientific Computing[M]. Pacific Grove, CA, USA: Cole Publishing Company, 2002.
[25]	HALL C A, MEYER W W. Optimal error bounds for cubic spline interpolation[J]. Journal of Approximation Theory, 1976, 16(2): 105-122.