This　paper　introduces　a　new　model　for　the　Fourier　law　of　heat　conduction　with　the　time-fractional　order　to　the　generalized　Maxwell　fluid.　The　flow　is　influenced　by　magnetic　field,　radiation　heat,　and　heat　source.　A　fractional　calculus　approach　is　used　to　establish　the　constitutive　relationship　coupling　model　of　a　viscoelastic　fluid.　We　use　the　Laplace　transform　and　solve　ordinary　differential　equations　with　a　matrix　form　to　obtain　the　velocity　and　temperature　in　the　Laplace　domain.　To　obtain　solutions　from　the　Laplace　space　back　to　the　original　space,　the　numerical　inversion　of　the　Laplace　transform　is　used.　According　to　the　results　and　graphs,　a　new　theory　can　be　constructed.　Comparisons　of　the　associated　parameters　and　the　corresponding　flow　and　heat　transfer　characteristics　are　presented　and　analyzed　in　detail.