非线性分数阶微分方程的一致精度高阶格式的数值分析
Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy
摘要：
介绍一种求解含Caputo导数的分数阶常微分方程的高阶数值格式。该方法通过将区域划分为若干子区间，并在每个子区间上应用二次插值来实现。该方法被证明是无条件稳定的，并为一般的非线性方程，一致尖锐的数值阶$3-ν$可以严格证明在所有时间步长足够光滑的解决方案。该证明为证明高阶格式在非线性情况下的锐阶性提供了一般指导。最后给出了一些数值例子来验证我们的理论结果。
英文摘要：
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.
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