Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-31T16:46:43.369Z Has data issue: false hasContentIssue false
  • English
  • Français

A Finite Difference Method for Boundary Value Problems of a Caputo Fractional Differential Equation

Part of: Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  31 January 2018

Pin Lyu ,
Seakweng Vong  and
Zhibo Wang
Pin Lyu*
Affiliation:
Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China
Seakweng Vong*
Affiliation:
Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China
Zhibo Wang*
Affiliation:
School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, Guangdong, China
*
*Corresponding author. Email addresses:lyupin1991@163.com (P. Lyu), swvong@umac.mo (S. Vong), wzbmath@gdut.edu.cn (Z. Wang)
*Corresponding author. Email addresses:lyupin1991@163.com (P. Lyu), swvong@umac.mo (S. Vong), wzbmath@gdut.edu.cn (Z. Wang)
*Corresponding author. Email addresses:lyupin1991@163.com (P. Lyu), swvong@umac.mo (S. Vong), wzbmath@gdut.edu.cn (Z. Wang)
Get access

Abstract

In this paper, we consider a two-point boundary value problem with Caputo fractional derivative, where the second order derivative of the exact solution is unbounded. Based on the equivalent form of the main equation, a finite difference scheme is derived. The L∞ convergence of the difference system is discussed rigorously. The convergence rate in general improves previous results. Numerical examples are provided to demonstrate the theoretical results.

Keywords

Boundary value problemCaputo fractional derivativeFinite difference methodConvergenceDerivative bound

MSC classification

Secondary: 65L10: Boundary value problems 65L12: Finite difference methods 65L20: Stability and convergence of numerical methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 4 , November 2017 , pp. 752 - 766
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alikhanov, A. A., A new difference scheme for the fractional diffusion equation, J. Comput. Phys. 280, 424–438 (2015).Google Scholar
[2] Alrefai, M., Basic results on nonlinear eigenvalue problems of fractional order, Electron. J. Differ Equ. 191, 1–12 (2012).Google Scholar
[3] Chen, M., Deng, W., Fourth order accuracy scheme for the space fractional diffusion equations, SIAM. J. Numer. Anal. 52, 1418–1438 (2014).Google Scholar
[4] Chen, C., Liu, F., Anh, V., Turner, I., Numerical schemes with high spatial accuracy for a variableorder anomalous subdiffusion equation, SIAM J. Sci. Comput. 32, 1740–1760 (2010).Google Scholar
[5] Cui, M., Compact exponential scheme for the time fractional convection-diffusion reaction equation with variable coefficients, J. Comput. Phys. 280, 143–163 (2015).Google Scholar
[6] Gao, G., Sun, Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys. 230, 586–595 (2011).Google Scholar
[7] Gracia, J. L., Stynes, M., Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems, J. Comput. Appl. Math. 273, 103–115 (2015).Google Scholar
[8] Gracia, J. L., Stynes, M., Formal consistency versus actual convergence rates of difference schemes for fractional-derivative boundary value problems, Fract. Calc. Appl. Anal. 18, 419–436 (2015).Google Scholar
[9] Guo, B., Xu, Q., Zhu, A., A second-order finite difference method for two-dimensional fractional percolation equations, Commun. Comput. Phys. 19, 733–757 (2016).Google Scholar
[10] Hao, Z., Sun, Z., Cao, W., A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281, 787–805 (2015).Google Scholar
[11] Ji, C., Sun, Z., An unconditionally stable and high-order convergent difference scheme for Stokes’ first problemfor a heated generalized second grade fluid with fractional derivative, Numer.Math.- Theory Methods Appl. 10, 597–613 (2017).Google Scholar
[12] Lei, S., Huang, Y., Fast algorithms for high-order numerical methods for space-fractional diffusion equations, Int. J. Comput. Math. 94, 1062–1078 (2017).Google Scholar
[13] Li, C., Ding, H., Higher order finite difference method for the reaction and anomalous-diffusion equation, Appl. Math. Model. 38, 3802–3821 (2014).Google Scholar
[14] Li, G., Sun, C., Jia, X., Du, D., Numerical solution to the multi-term time fractional diffusion equation in a finite domain, Numer. Math.-Theory Methods Appl. 9, 337–357 (2016).Google Scholar
[15] Liao, H., Zhang, Y., Zhao, Y. and Shi, H., Stability and convergence of modified Du Fort-Frankel schemes for solving time-fractional subdiffusion equations, J. Sci. Comput. 61, 629–648 (2014).Google Scholar
[16] Liao, H., Zhao, Y., Teng, X., A weighted ADI scheme for subdiffusion equations, J. Sci. Comput. 69, 1144–1164 (2016).Google Scholar
[17] Lin, Y., Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225, 1533–1552 (2007).Google Scholar
[18] Meerschaert, M. M., Tadjeran, C., Finite difference approximations for fractional advectiondispersion flow equations, J. Comput. Appl. Math. 172, 65–77 (2004).Google Scholar
[19] Pedas, A., Tamme, E., Piecewise polynomial collocation for linear boundary value problems of fractional differential equations, J. Comput. Appl. Math. 236, 3349–3359 (2012).Google Scholar
[20] Ren, J., Sun, Z., Efficient and stable numerical methods for the multi-term time fractional subdiffusion equations, East Asian J. Appl. Math. 3, 242–266 (2014).Google Scholar
[21] Sousa, E., Finite difference approximations for a fractional advection diffusion problem, J. Comput. Phys. 228, 4038–4054 (2009).Google Scholar
[22] Sousa, E., How to approximate the fractional derivative of order 1 < α ≤ 2, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22, 1250075, 13.Google Scholar
[23] Stynes, M., Gracia, J. L., A finite difference method for a two-point boundary value problem with a Caputo fractional derivative, IMA J. Numer. Anal. 35, 698–721 (2015).Google Scholar
[24] Sun, Z., Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56, 193–209 (2006).Google Scholar
[25] Tian, W., Zhou, H., Deng, W., A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput. 84, 1703–1727 (2015).Google Scholar
[26] Vong, S., Lyu, P., Wang, Z., A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under Neumann boundary conditions, J. Sci. Comput. 66, 725–739 (2016).Google Scholar
[27] Wang, Z., Vong, S., A compact difference scheme for a two dimensional nonlinear fractional Klein-Gordon equation in polar coordinates, Comput. Math. Appl. 71, 2524–2540 (2016).Google Scholar
[28] Wang, Z., Vong, S., Compact difference schemes for the modified anomalous fractional subdiffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277, 1–15 (2014).Google Scholar
[29] Yuste, S. B.,Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216, 264–274 (2006).Google Scholar