Zhang Zhi-Yong
College of Science, Minzu University of China, Beijing 100081, People's Republic of China.
Proc Math Phys Eng Sci. 2020 Jan;476(2233):20190564. doi: 10.1098/rspa.2019.0564. Epub 2020 Jan 29.
We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.
我们首先表明，时间分数阶偏微分方程（PDE）的李对称性无穷小生成元具有统一且简单的形式，然后将李对称性条件分为两个不同的部分，其中一个是线性时间分数阶偏微分方程，另一个是占据主导地位的整数阶偏微分方程，甚至能完全确定特定类型时间分数阶偏微分方程的对称性。此外，我们表明线性时间分数阶偏微分方程总是允许无穷维的无穷小生成元李代数，就如同线性偏微分方程的情况一样，而非线性时间分数阶偏微分方程至多允许有限维李代数。因此，不存在将非线性时间分数阶偏微分方程转换为线性方程的可逆映射。我们通过考虑两个例子来说明这些结果。
