The convolution quadrature method originally developed for the Riemann-Liouville fractional calculus is extended in this work to the Hadamard fractional calculus by using the exponential type meshes. Local truncation error analysis is presented for singular solutions. By adopting the fractional BDF-$p(1\leq p \leq 6)$ for the Caputo-Hadamard fractional derivative in solving subdiffusion problem with singular source terms, and using the finite element method to discretize the space variable, we carry out the sharp error analysis rigorously and obtain the optimal accuracy by the novel correction technique. Our correction method is a natural generalization of the one developed for subdiffusion problems with smooth source terms. Numerical tests confirm the correctness of our theoretical results.

翻译：本文将通过使用指数型网格，将最初为黎曼-刘维尔分数阶微积分开发的卷积求积法推广至Hadamard分数阶微积分。针对奇异解，给出了局部截断误差分析。在求解具有奇异源项的子扩散问题时，采用分数阶BDF-$p(1\leq p \leq 6)$格式离散Caputo-Hadamard分数阶导数，并利用有限元方法离散空间变量，我们严格地进行了尖锐误差分析，通过新颖的校正技术获得了最优精度。我们的校正方法是对已有针对光滑源项子扩散问题校正方法的自然推广。数值实验证实了理论结果的正确性。