Group classification of nonlinear time-fractional dual-phase-lag heat equation with a small parameter

In this paper, we solve the group classification problem for a nonlinear one-dimensional time-fractional heat conduction equation with full memory and dual-phase-lag, including thermal relaxation and thermal damping. The characteristic times of relaxation processes are assumed to be small enough and therefore a small parameter for fractional differential relaxation terms is introduced. All thermal properties of a medium are considered as functions of temperature. Group classification is performed with respect to groups of approximate point transformations (groups of approximate symmetries) admitted by the equation up to equivalence transformations. We prove that generally the equation admits five-parameter group of approximate transformations, and the cases of its extension to seven- and nine-parameters groups are found. Also, it is shown that the considered nonlinear equation has an infinite approximate symmetry group if the corresponding unperturbed equation is linear. We find that the equation in question always exactly inherits the symmetries of the unperturbed equation. The obtained results make it possible to construct approximately invariant solutions of equation under consideration. In particular, it follows from the classification found that the equation always has a traveling wave solution. The self-similar solutions can be constructed only if the medium thermal properties have power-law dependences on temperature. Ansatzes of these types of solutions are obtained and symmetry reductions of the equation under consideration to the corresponding ordinary fractional differential equations are performed.