[en] In this work, we construct a mathematical model of two-temperature Green–Naghdi (III) magneto-thermo viscoelasticity theory based on the fractional order of heat transfer. Some essential theorems of coupled and generalized thermo-viscoelasticity can be easily obtained. The Laplace transform and state-space techniques are used to obtain the general solution for any set of boundary conditions. The resulting formulation is applied to the specific problem of a perfect electrically conducting viscoelastic half-space subjected to a moving heat source with constant velocity and ramp-type heating. The inverse transforms are obtained by using a numerical method based on Fourier expansion techniques. The effects of the heat source speed, the two-temperature parameter, Alfven velocity and the ramping time parameter on a polymethyl methacrylate (Plexiglas) material are discussed. (author)