Hartley Tom T, Lorenzo Carl F
Department of Electrical and Computer Engineering, University of Akron, Akron, Ohio, USA.
Crit Rev Biomed Eng. 2008;36(1):23-38. doi: 10.1615/critrevbiomedeng.v36.i1.30.
This paper provides a solution to the fundamental linear fractional order differential equation, namely, cdtqx(t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi-infinite lossy line, is used to demonstrate the theory.
本文给出了基本线性分数阶微分方程的一个解，即(cd_{t}^{q}x(t)+ax(t)=bu(t))。脉冲响应解被证明是一个级数，称为F函数，它推广了普通指数函数。F函数为q阶“分数极点”提供了基础。阐明了复平面行为，并通过一个简单的例子——终端接有电感的半无限有损线——来演示该理论。
