  Working with a magnetic field periodic along $Oz$ and decaying in time, we
deal with the Dirac-type equation characterizing the fermions evolving in
magnetar's crust. For ultra-relativistic particles, one can employ the
perturbative approach, to compute the conserved current density components. If
the magnetic field is frozen and the magnetar is treated as a stationary
object, the fermion's wave function is expressed in terms of the Heun's
Confluent functions. Finally, we are extending some previous investigations on
the linearly independent fermionic modes solutions to the Mathieu's equation
and we discuss the energy spectrum and the Mathieu Characteristic Exponent.
