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| Mild Solution for the Time Fractional Hall-Magneto-Hydrodynamics Stochastic Equations |
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| KeyWord:Time fractional hall-magneto-hydrodynamics equations, It\^o integral, derivative of Caputo, stochastic |
| Author Name | Affiliation | | Hassan Khaider | Laboratory LMACS, FST of Beni-Mellal, Sultan Moulay Slimane University, Morocco | | Achraf Azanzal | Laboratory of Education, Sciences and Techniques - LEST, Higher School of Education and Training Berrechid (ESEFB), Hassan First University, Avenue de l'Universit\'{e}, B.P :218, 26100, Berrechid, Morocco | | Abderrahmane Raji | Laboratory LMACS, FST of Beni-Mellal, Sultan Moulay Slimane University, Morocco |
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| 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 $\{\mathcal{Q}_{\alpha}(-t^{\alpha}\mathbb{J}):t\geq 0\}$ and $\{\mathcal{Q}_{\alpha,\alpha}(-t^{\alpha}\mathbb{J}):t\geq 0\}$ and applying the bilinear fixed-point theorem, we will prove the frist result. Next, by It\^o integral, and by similair analogy we will establish the existence and uniqueness in the stochastic case in $\mathcal{EN}_{a}^{\mu} \cap \mathrm{N}_{a,\mu}^{2\alpha}$. |
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