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

Approximative Analytic Study of Fermions in Magnetar's Crust; Ultra-relativistic Plane Waves, Heun and Mathieu Solutions and Beyond