A Piecewise Linear Interpolation Polynomial Method for Nonlinear Fractional Ordinary Differential Equations

GAO Xinghua; LI Hong; LIU Yang

doi:10.21656/1000-0887.410149

Volume 42 Issue 5

May 2021

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2021 > 42(5): 531-540

GAO Xinghua, LI Hong, LIU Yang. A Piecewise Linear Interpolation Polynomial Method for Nonlinear Fractional Ordinary Differential Equations[J]. Applied Mathematics and Mechanics, 2021, 42(5): 531-540. doi: 10.21656/1000-0887.410149

Citation:

PDF( 1138 KB)

A Piecewise Linear Interpolation Polynomial Method for Nonlinear Fractional Ordinary Differential Equations

doi: 10.21656/1000-0887.410149

School of Mathematical Sciences, Inner Mongolia University, Huhhot 010021, P.R.China

Funds: The National Natural Science Foundation of China（11761053;11661058）

Received Date: 2020-05-26
Rev Recd Date: 2020-11-16
Publish Date: 2021-05-01

Abstract

Abstract

A numerical scheme with the piecewise linear interpolation polynomial method was established to solve a class of nonlinear fractional ordinary differential equations including the Hadamard finite part integral. In the time direction, the fractional derivative was approximated with the piecewise linear interpolation polynomial method, and the integer order time derivative was discretized by means of the 2ndorder backward difference scheme. Through detailed proof, the error estimates with an accuracy of OO(τ^{min{1+α,1+β}}) were obtained. The comparison between the numerical results and the theoretical solution shows the correctness of the theoretical analysis.

FullText(HTML)

References(20)

References

[1]	LIU Y, DU Y W, LI H, et al. Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem[J]. Computers and Mathematics With Applications,2015,70(4): 573-591.
[2]	CHEN H Z, WANG H. Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation[J]. Journal of Computational and Applied Mathematics,2016,296: 480-498.
[3]	SHI D Y, YANG H J. Superconvergence analysis of a new low order nonconforming MFEM for time-fractional diffusion equation[J]. Applied Numerical Mathematics,2018,131: 109-122.
[4]	LIU Y, DU Y W, LI H, et al. A two-grid finite element approximation for a nonlinear time-fractional Cable equation[J]. Nonlinear Dynamics,2016,85: 2535-2548.
[5]	HENRY B I, LANGLANDS T A M, WEARNE S L. Fractional cable models for spiny neuronal dendrites[J]. Physical Review Letters,2008,100(12): 128103.
[6]	GOUFO E F D. Stability and convergence analysis of a variable order replicator-mutator process in a moving medium[J]. Journal of Theoretical Biology,2016,403: 178-187.
[7]	杨旭, 梁英杰, 孙洪广, 等. 空间分数阶非Newton流体本构及圆管流动规律研究[J]. 应用数学和力学, 2018,39(11): 1213-1226.(YANG Xu, LIANG Yingjie, SUN Hongguang, et al. A study on the constitutive relation and the flow of spatial fractional non-Newtonian fluid in circular pipes[J]. Applied Mathematics and Mechanics,2018,39(11): 1213-1226.(in Chinese))
[8]	KOH C G, KELLY J M. Application of fractional derivatives to seismic analysis of base-isolated models[J]. Earthquake Engineering and Structural Dynamics,1990,19(2): 229-241.
[9]	KLAS A, MIKAEL E, STIG L. Adaptive discretization of fractional order viscoelasticity using sparse time history[J]. Computer Methods in Applied Mechanics and Engineering,2004,193(42/44): 4567-4590.
[10]	LI C, ZHAO S. Efficient numerical schemes for fractional water wave models[J]. Computers and Mathematics With Applications,2016,71(1): 238-254.
[11]	LIU Y, YU Z D, LI H, et al. Time two-mesh algorithm combined with finite element method for time fractional water wave model[J]. International Journal of Heat and Mass Transfer,2018,120: 1132-1145.
[12]	ZHANG J, XU C J. Finite difference/spectral approximations to a water wave model with a nonlocal viscous term[J]. Applied Mathematical Modelling,2014,38(19): 4912-4925.
[13]	孙志忠, 高广花. 分数阶微分方程的有限差分方法[M]. 北京: 科学出版社, 2015.(SUN Zhizhong, GAO Guanghua. Finite Difference Method for Fractional Differential Equations [M]. Beijing: Science Press, 2015.(in Chinese))
[14]	DIETHELM K. An algorithm for the numerical solution of differential equations of fractional order[J]. Electronic Transactions on Numerical Analysis,1997,5: 1-6.
[15]	DIETHELM K, FORD N J, FREED A D, et al. Algorithms for the fractional calculus: a selection of numerical methods[J]. Computer Methods in Applied Mechanics and Engineering,2005,194(6/8): 743-773.
[16]	LI Z Q, YAN Y B, FORD N J. Error estimates of a high order numerical method for solving linear fractional differential equations[J].Applied Numerical Mathematics,2017,114: 201-220.
[17]	HADAMARD J, MORSE P M. Lectures on Cauchy’s Problem in Linear Partial Differential Equations [M]. New York: Dover Publications, 1953.
[18]	MONEGATO G. Definitions, properties and applications of finite-part integrals[J]. Journal of Computational and Applied Mathematics,2008,229(2): 425-439.
[19]	MA L, LI C P. On finite part integrals and Hadamard-type fractional derivatives[J]. Journal of Computational and Nonlinear Dynamics,2018,13(9): 090905.
[20]	GALAPON E A. The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals[J]. Journal of Mathematical Physics,2016,57(3): 033502.