TY - JOUR
T1 - Spectral Deferred Correction Methods for Fractional Differential Equations
AU - Chunwan Lv, Mejdi Azaiez & Chuanju Xu
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 729
EP - 751
PY - 2018
DA - 2018/06
SN - 11
DO - http://doi.org/10.4208/nmtma.2018.s03
UR - https://global-sci.org/intro/article_detail/nmtma/12469.html
KW - 
AB - <p style="text-align: justify;">In this paper, we propose and analyze a spectral deferred correction method
for the fractional differential equation of order α. The proposed method is based on a
well-known finite difference method of $(2−α)$-order, see [Sun and Wu, Appl. Numer.
Math., 56(2), 2006] and [Lin and Xu, J. Comput. Phys., 225(2), 2007], for prediction
of the numerical solution, which is then corrected through a spectral deferred correction
method. In order to derive the convergence rate of the prediction-correction iteration,
we first derive an error estimate for the $(2−α)$-order finite difference method
on some non-uniform meshes. Then the convergence rate of orders $\mathcal{O}(τ^{(2−α)(p+1)})$ and&nbsp; $\mathcal{O}(τ^{(2−α)+p})$ of the overall scheme is demonstrated numerically for the uniform mesh
and the Gauss-Lobatto mesh respectively, where $τ$ is the maximal time step size and $p$ is the number of correction steps. The performed numerical test confirms the efficiency
of the proposed method.</p>