Abstract A class of two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order α ∈ (1, 2] with a convection term is considered. Its boundary conditions are of Robin type including the Dirichlet boundary conditions as a special case. An explicit formula for the associated Green function is obtained in terms of two-parameter Mittag-Leffler functions. This work improves Meng and Stynes’ work [9] in three aspects. Firstly, the Green function is constructed by the use of the Laplace transform. This method may be more straightforward and easy to be generalized to solving other problems. Secondly, the monotonicity of a function is proven using monotone definition rather than using its derivative. Thirdly, a direct proof of the necessity of the positive property of Green function is given so that we can avoid the difficulty of construction any counter-example.

中文翻译：

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一类具有对流项的 Caputo 分数阶微分方程的格林函数

摘要 考虑了一类最高阶项为具有对流项的α ∈ (1, 2] 阶Caputo 分数阶导数的两点边值问题，其边界条件为Robin 型，其中Dirichlet 边界条件为特殊情况。根据二参数 Mittag-Leffler 函数获得相关格林函数的显式公式。这项工作在三个方面改进了 Meng 和 Stynes 的工作 [9]。首先，格林函数是通过使用拉普拉斯变换。这种方法可能更直接，更容易推广到解决其他问题。其次，使用单调定义而不是使用其导数来证明函数的单调性。第三，给出了格林函数正性质的必要性的直接证明，这样我们就可以避免构造任何反例的困难。