用于Caputo分数阶微分的带块卷积的精确矩局部勒让德框架方法
An Exact-Moment Local Legendre Frame Method with Block Convolution for Caputo Fractional Differentiation
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中文总结 AI 辅助
该研究提出带块卷积的精确矩局部勒让德框架方法，用于精确计算0<α<1的Caputo分数阶导数，通过解析处理奇异核实现指数型收敛，数值实验验证了其精度与效率。
中文摘要 AI 辅助
我们提出一种局部勒让德框架方法，用于精确计算阶数为0<α<1的Caputo分数阶导数。在每个局部子区间上，函数由从扩展区间上的缩放勒让德多项式得到的受限勒让德框架表示。通过指数加权的GTSVD正则化，从等距采样点计算局部系数。随后，通过将弱奇异分数阶积分应用于局部框架基函数的导数，来计算Caputo导数。由于这些导数是多项式，对应的Caputo权重可表示为有限加权矩的形式，因此奇异核可通过解析方式处理，而非采用低阶求积规则。对于均匀剖分，历史权重具有依赖于块的结构，可被高效复用。误差分析将局部框架重构与Caputo积分分离，具体而言，Caputo误差受导数重构误差约束，而导数重构误差通过插值论证，由L²重构误差和GTSVD近似的加权光滑性界得到。对于具有指数系数衰减的解析局部函数，这会导出导数的指数型收敛，进而得到Caputo近似的指数型收敛。数值实验验证了精确矩权重的精度、局部加权重构的有效性以及块实现的效率。
英文摘要
We propose a local Legendre frame method for the accurate computation of Caputo fractional derivatives of order \(0<α<1\). On each local subinterval, the function is represented by a restricted Legendre frame obtained from scaled Legendre polynomials on an extended interval. The local coefficients are computed from equispaced samples by an exponentially weighted GTSVD regularization. The Caputo derivative is then evaluated by applying the weakly singular fractional integral to the derivatives of the local frame basis functions. Since these derivatives are polynomials, the corresponding Caputo weights can be written in terms of finite weighted moments, so that the singular kernel is treated analytically rather than by a low-order quadrature rule. For uniform partitions, the history weights have a block-dependent structure and can be reused efficiently. The error analysis separates the local frame reconstruction from the Caputo integration. In particular, the Caputo error is bounded by the derivative reconstruction error, while the latter is obtained from the \(L^2\) reconstruction error and a weighted smoothness bound of the GTSVD approximation through an interpolation argument. For analytic local functions with exponential coefficient decay, this leads to exponential-type convergence of the derivative and hence of the Caputo approximation. Numerical experiments confirm the accuracy of the exact moment weights, the effectiveness of the local weighted reconstruction, and the efficiency of the block implementation.
