Implicit scheme of the finite difference method for the second-order dual phase lag equation

The second-order dual phase lag equation (DPLE) as a mathematical model of the microscale
heat transfer is considered. It is known that the starting point determining the final form of
this equation is the generalized Fourier law in which two positive constants (the relaxation
and thermalization times) appear. Depending on the order of the generalized Fourier law
expansion into the Taylor series, different forms of the DPLE can be obtained. As an example
of the problem described by the second-order DPLE equation, thermal processes proceeding
in the domain of a thin metal film subjected to a laser pulse are considered. The numerical
algorithm is based on an implicit scheme of the finite difference method. At the stage of
numerical modeling, the first, second and mixed order of the dual phase lag equation are
considered. In the final part of the paper, examples of different solutions are presented and
conclusions are formulated.