TY - JOUR
T1 - Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy
AU - Cao , Junying
AU - Cai , Zhenning
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 71
EP - 112
PY - 2020
DA - 2020/10
SN - 14
DO - http://doi.org/10.4208/nmtma.OA-2020-0039
UR - https://global-sci.org/intro/article_detail/nmtma/18328.html
KW - Caputo derivative, fractional ordinary differential equations, high-order numerical scheme, stability and convergence analysis.
AB - <p style="text-align: justify;">We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing
the domain into a number of subintervals, and applying the quadratic interpolation
on each subinterval. The method is shown to be unconditionally stable, and for general nonlinear equations, the uniform sharp numerical order 3 − $ν$ can be rigorously
proven for sufficiently smooth solutions at all time steps. The proof provides a general guide for proving the sharp order for higher-order schemes in the nonlinear
case. Some numerical examples are given to validate our theoretical results.</p>