Magneto-thermoelasticity with two fractional order heat transfer

A new mathematical model of the equations of two-temperature magneto-thermoelasticity theory for a perfect conducting solid has been constructed in the context of a new consideration of heat conduction with a time-fractional derivative of order α (0 < α ⩽ 1) and a time-fractional integral of order υ (0 < υ ⩽ 2). This model is applied to one-dimensional problem for a perfect conducting half-space of elastic solid with heat source distribution in the presence of a constant magnetic field. Laplace transforms and state-space techniques will be used to obtain the general solution for any set of boundary conditions. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and their graphs, conclusions about the new theory are given. Some comparisons are shown in figures to estimate the effects of the fractional order parameters and the temperature discrepancy on all the studied fields.