构造了α∈(0,1)阶Caputo导数的一种数值逼近公式.考虑到Caputo导数在初始时刻的弱奇异性,在非均匀网格的第一个小区间上使用线性插值,其后的每个小区间上使用二次插值,从而推导出非均匀L1-2公式.证明了其截断误差可以达到(3-α)阶精度,并讨论了相应的系数性质.将得到的公式应用到时间分数阶扩散方程的数值求解中,数值实验验证了该公式的有效性和正确性.
This paper constructs a numerical approximation formula for the Caputo derivative of order α∈(0，1). Considering the weak regularity of the Caputo derivative at the initial time, linear interpolation is employed over the first subinterval of the non-uniform mesh, while quadratic interpolation is utilized for each subsequent subinterval, leading to the derivation of a non-uniform L1-2 formula. It is proven that the truncation error can achieve (3-fi)-order accuracy, and the corresponding coefficient properties are discussed. The derived formula is applied to the numerical solution of the time-fractional diffusion equation, and numerical experiments have verified the effectiveness and correctness of the formula.
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