To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time‐fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time‐fractional Benjamin–Bona–Mahony‐type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H¹‐norm. Numerical examples are provided to justify the accuracy.

中文翻译：

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Caputo导数的非均匀L2公式及其在具有非光滑解的分数阶Benjamin–Bona–Mahony型方程中的应用

为了恢复离散分数导数的全部精度，非均匀网格技术是一种自然且简单的方法，可以有效地解决时间分数线性和非线性微分方程解中总是出现的初始奇点。我们首先为1 < α  <2阶的Caputo分数阶导数构造一个非均匀L2逼近， 并在一些合理的正则性假设下给出了全局一致性分析。然后，使用时间非均匀L2公式为时间分数阶本杰明-波纳-马洪尼型方程建立线性差分方案。关于离散H ¹证明了我们的方案在均匀和非均匀（渐变）时间网格上的无条件收敛-规范。提供了数值示例以证明准确性。