Lie group investigation of fractional partial differential equation using symmetry

Abstract

In this paper, we make use of Lie group investigation of the following non-linear of fractional partial differential equations using symmetry.
$$
\begin{aligned}
& D^{\alpha_1} u_1=\frac{1}{2} u_1 \\
& D^{\alpha_2} u_2=u_2+u_1{ }^2,
\end{aligned}
$$
where \(\alpha_1\) and \(\alpha_2\) are real constant, \(0<\alpha_1 \alpha_2 \leq 1\) and \(u_1\) and \(u_2\) are functions of independent variable \(\mathrm{x}\) and \(D^\alpha u\) is a fractional derivative of \(u\) w.r.t. \(x\) which can be the following \(R-L\) type.
$$
\frac{\partial^\alpha u}{\partial x^\alpha}=\left\{\begin{array}{cc}
\frac{\partial^m u}{\partial x^m} & \alpha=m \in N \\
\frac{1}{(m-\alpha) !} \frac{\partial^m}{d x^m} \int_0^t(t-\tau)^{m-\alpha-1} u(\tau, x) d \tau & m-1<\alpha<m, m \in N
\end{array}\right.
$$