In many given physical problems and in the course of dispersion curve through a spectral line under the influence of the Doppler-effect and in collision damping, the Voigt functions have been widely utilized. By taking advantage of the fractional calculus in spectral theory and the Sturm-Liouville problems, in this paper, we obtain the Voigt functions via the quadrature formulae of one dimensional fractional in time evolution diffusion and wave problems consisting of different initial and inhomogeneous boundary conditions.