Mild solution for the time fractional magneto-hydrodynamics system

In this paper, by using the Mittag–Leffler operators \(\{\mathcal {L}_{\alpha }(-t^{\alpha }\mathbb {I}):t\ge 0\}\) $\{{\mathcal{L}}_{\alpha }(-t^{\alpha }\mathbb{I}):t\ge 0\}$ and \(\{\mathcal {L}_{\alpha ,\alpha }(-t^{\alpha }\mathbb {I}):t\ge 0\}\) $\{{\mathcal{L}}_{\alpha ,\alpha }(-t^{\alpha }\mathbb{I}):t\ge 0\}$ we will prove the mild soltion of the time fractional magneto-hydrodynamics system with a fractional derivative of Caputo. Furthermore, by Itô integral, we will establish the mild solution of stochastic time fractional magneto-hydrodynamics system in \(\mathcal{E}\mathcal{N}_{p}^{\lambda } \cap \textrm{N}_{p,\lambda }^{2\alpha }\) $\mathcal{E}{\mathcal{N}}_p^{\lambda }\cap {\text{N}}_{p,\lambda }^{2\alpha }$ .