Mild Solution for the Time Fractional Hall-Magneto-Hydrodynamics Stochastic Equations

Abstract

In this paper, we establish the existence and uniqueness of mild solutions for the time fractional hall-magneto-hydrodynamics stochastic equations with a fractional derivative of Caputo. Initially, we focus on the existence and uniqueness in the deterministe case. Using the Mittag-Leffler operators $\{\mathcal{Q}_α(−t^α \mathbb{J}): t ≥ 0\}$ and $\{\mathcal{Q}_{α,α}(−t^α \mathbb{J}) : t ≥ 0\}$ and applying the bilinear fixed-point theorem, we will prove the frist result. Next, by Itô integral, and by similair analogy we will establish the existence and uniqueness in the stochastic case in $\mathcal{EN}^{\mu}_a ∩ N^{2α}_{a,\mu}.$