  We present a new approach to study a class of non-Hermitian (1+1)-dimensional
Dirac Hamiltonian in the presence of local Fermi velocity. We apply the well
known Nikiforov-Uvarov method to solve such a system. We discuss applications
and explore the solvability of both $\mathcal{PT}$-symmetric and
non-$\mathcal{PT}$ symmetric classes of potentials. In the former case we
obtain the solution of a harmonic oscillator in the presence of a linear vector
potential while in the latter case we solve the shifted harmonic oscillator
problem.
