The model of equations of thermo-viscoelasticity with fractional order heat transfer is constructed. Some fundamental theorems on the linear coupled and generalized theories of thermo-viscoelasticity can be easily obtained as special cases. The medium is assumed initially quiescent. Laplace and Fourier integral transforms are utilized. The method of the matrix exponential which constitutes the basis of the state–space approach of modern control theory is applied to the system of two-dimensional equations. The resulting formulation is applied to a thermal shock half-space problem. The inversion process for Fourier and Laplace transforms is carried out using numerical method based on Fourier series expansions. Numerical results are given and illustrated graphically for the problem considered. Comparisons are made with the results predicted by the coupled theory and generalized theory. The effect of the fractional order parameter on all the considered fields is examined.

A mathematical model of electro-thermoelasticity has been constructed in the context of a new consideration of heat conduction with phase-lag Green-Naghdi theory without energy dissipation (GN-II). The equations for one-dimensional problems including heat sources are cast into matrix form using the state space and Laplace transform techniques. The resulting formulation is applied to a problem for the whole space with a plane distribution of heat sources. It is also applied to a thermoelectric semi-space problem with a traction-free surface and plane distribution of heat sources located inside the medium. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for the temperature, displacement, and stress distributions are given and illustrated graphically for the whole space with a plane distribution of heat sources problem.

In this article, the problem of combined heat and mass transfer stability of an electrically conducting fluid in MHD free convection with the diffusion of chemically reactive species adjacent to a vertical surface is analyzed. The effects of Ohmic heating and viscous dissipation, radiation and chemical reaction are taken into account. The governing fundamental equations are approximated by a system of non-linear ordinary differential equations and are solved numerically by using a Chebyshev spectral collocation method. The solutions are found to be dependent on the governing parameters including the magnetic field parameter, the buoyancy ratio between species and thermal diffusion, the Eckert number, the Prandtl number, the radiation parameter, chemical reaction parameter and the Schmidt number. The study of magnetohydrodynamics flow fluid over a stretching surface has become the basis of several industrial, scientific and engineering applications such as in chemical production, car cooling systems, cooling of electronic chips, cooling of heat sinks, power generation in power plant and in nuclear systems to obtain high rates of heat extraction from reactors. The effects of these parameters on the velocity, temperature and concentration are investigated and represented graphically.