In　this　paper,　we　study　the　magnetohydrodynamic　(MHD)　flow　and　heat　transfer　of　the　fractional　Jeffrey　fluid　in　a　straight　channel,　of　which　the　cross　section　is　a　two-dimensional　irregular　convex　domain.　We　consider　a　spatial　fractional　operator　to　modify　the　classical　Fourier＇s　law　of　thermal　conduction,　and　obtain　the　time-space　fractional　coupled　model.　The　fractional　coupled　model　is　solved　numerically　by　the　L2−1σ　method　in　the　temporal　direction　and　the　unstructured　mesh　finite　element　method　in　the　spatial　direction.　Besides,　we　prove　the　stability　and　convergence　of　the　numerical　scheme.　In　order　to　reduce　the　computational　time,　a　fast　method　is　proposed.　Finally,　a　numerical　example　is　given　to　verify　the　theoretical　analysis　and　the　efficiency　of　the　fast　method.　Furthermore,　an　example　is　considered　to　discuss　the　effects　of　the　fractional　parameters　on　the　MHD　flow　and　heat　transfer　of　the　fractional　Jeffrey　fluid.　©　2023　The　Author(s)