26、常分数阶Volterra方程的分析与数值方案

常分数阶Volterra方程的分析与数值方案

1. 特殊极限情况分析

1.1 当α趋近于1时的情况

当α趋近于1时，有如下推导：
[
\begin{align }
\lim_{\alpha \to 1} {} {0}^{FFE}D {t}^{\alpha,\beta(t)}f(t) &= \lim_{\alpha \to 1} \frac{M(\alpha)}{1 - \alpha} \frac{d}{dt^{\beta(t)}} \int_{0}^{t} \exp\left(-\frac{\alpha (t - \tau)}{1 - \alpha}\right) f(\tau) d\tau\
&= \lim_{\alpha \to 1} \frac{M(\alpha)}{1 - \alpha} \frac{d}{dt} \int_{0}^{t} \exp\left(-\frac{\alpha (t - \tau)}{1 - \alpha}\right) f(\tau) d\tau \frac{t^{-\beta(t)}}{\beta’(t) \ln t + \frac{\beta(t)}{t}}\
&= \frac{d}{dt} \int_{0}^{t} \lim_{\alpha \to 1} \frac{M(\alpha)}{1 - \alpha} \exp\left(-\frac{\alpha (t - \tau)}{1 - \alpha}\right) f(\tau) d\tau \frac{t^{-\beta(t)}}{\beta’(t) \ln t + \frac{\beta(t