In this paper, we study the stability and logarithmic decay of the solutions to fractional differential equations (FDEs). Both linear and nonlinear cases are included. And the fractional derivative is in the sense of Hadamard or Caputo–Hadamard with order \(\alpha \,(0<\alpha <1)\). The solutions can be expressed by Mittag–Leffler functions through applying the modified Laplace transform. In view of the asymptotic expansions of Mittag–Leffler function, we discuss the stability and logarithmic decay of the solution to FDEs in great detail.

中文翻译：

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Hadamard型分数阶微分方程解的稳定性和对数衰减

在本文中，我们研究分数阶微分方程（FDE）解的稳定性和对数衰减。线性和非线性情况都包括在内。分数导数在Hadamard或Caputo–Hadamard的意义上为\（\ alpha \，（0 <\ alpha <1）\）。解决方案可以通过应用改进的Laplace变换由Mittag-Leffler函数表示。鉴于Mittag-Leffler函数的渐近展开，我们将详细讨论FDE的解的稳定性和对数衰减。