In this paper, we further extend the theories of Lie symmetry group and conservation law to study the time-fractional $b$-family peakon equations. The main distinction of the equations with the usual time-fractional partial differential equations is the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order $x$-derivative. Thus we first give a prolongation formula of the infinitesimal generator for the case of mixed derivative, then after finding the Lie symmetries, we use them to transform the equations into fractional and integer-order ordinary differential equations respectively. Some exact solutions and power series solutions are constructed. Finally, a general conservation law formula is given based on the idea of nonlinear self-adjointness and some nontrivial conservation laws of the equations are presented.

在本文中，我们进一步扩展了李对称群和守恒定律的理论来研究时间分形 $乙$-家庭峰值方程。该方程与通常的时间分数阶偏微分方程的主要区别是黎曼-刘维尔时间分数阶导数和整数阶的混合导数$X$-衍生物。因此，我们首先给出混合导数情况下无穷小发生器的扩展公式，然后在找到Lie对称性后，我们将它们分别转化为分数阶和整数阶常微分方程。构造了一些精确解和幂级数解。最后，基于非线性自伴随的思想，给出了一个通用的守恒定律公式，并给出了方程的一些非平凡守恒定律。