A new mathematical model of magnetohydrodynamic (MHD) theory has been constructed in the context of a new consideration of heat conduction with a time-fractional derivative of order 0 < α ≤ 1 and a time-fractional integral of order 0 < γ ≤ 2. This model is applied to one-dimensional problems for a thermoelectric viscoelastic fluid flow in the absence or presence of heat sources. Laplace transforms and state-space techniques will be used to obtain the general solution for any set of boundary conditions. According to the numerical results and its graphs, conclusion about the new theory has been constructed. Some comparisons have been shown in figures to estimate the effects of the fractional order parameters on all the studied fields.

An exact analysis of deformation and stress field in a laminated elastic tube under extension and uniform pressuring is presented. The problem of finite laminated tube is considered with emphasis on the end effect. The problem is formulated on the basis of the state space formalism for axisymmetric deformation of transversely isotropic layer. The transfer matrix transmits the state vector in radial direction from inner surface to outer surface and takes into account the interfacial continuity and lateral boundary conditions in a rigorous manner. Upon delineating the symplectic orthogonality relations of the eigenvectors and by using eigenfunction expansion, an exact solution which satisfies the end conditions is determined. The results show that the end effect is significant but confined to a local region near the base where the displacement and stress distributions are remarkably different from those according to the simplified solutions that only satisfy lateral BC.