Stability and Convergence Analysis of a Class of Continuous Piecewise Polynomial Approximations for Time-Fractional Differential Equations

Han Zhou; Paul Andries Zegeling

doi:10.1007/s10915-018-0704-z

Stability and Convergence Analysis of a Class of Continuous Piecewise Polynomial Approximations for Time-Fractional Differential Equations

Han Zhou^*, Paul Andries Zegeling

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We propose and study a class of numerical schemes to approximate time-fractional differential equations. The methods are based on the approximations of the Caputo fractional derivative of order α∈ (0 , 1) by using continuous piecewise polynomials, which are strongly related to the backward differentiation formulae. We investigate their theoretical properties, such as the local truncation error and global error estimates with respect to sufficiently smooth solutions, and the numerical stability in terms of stability region and A(π2)-stability. Numerical experiments are given to verify our theoretical investigations.

Original language	English
Pages (from-to)	225-262
Number of pages	38
Journal	Journal of Scientific Computing
Volume	77
Issue number	1
DOIs	https://doi.org/10.1007/s10915-018-0704-z
Publication status	Published - 1 Oct 2018

Funding

Acknowledgements The authors would like to thank Jason Frank and WenYi Tian for helpful discussions. Han Zhou also thanks the support from China Scholarship Council (CSC).

Keywords

Caputo fractional derivative
Continuous piecewise polynomial
Convergence analysis
Stability
Time-fractional differential equations

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Stability and Convergence Analysis of a Class of Continuous Piecewise Polynomial Approximations for Time-Fractional Differential Equations

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