Complex eigenwdue problems are encountered in the solution of mung engineering problems; but no work appears to be available ,f?w solving such problems directly in the comp1e.u domuin. In this study, u methodology is presented@ solving such problems directly in the complex domuin by utilizing CI shooting upprouch along with the Runge-Kuttu method. To illustrate the application, the problem of' transient,forced convection to an incompressible W. S. Kim et al. y+ dimensionless distance from wall Greek s~whols thermal diffusivity of fluid dimensionless time = at/h' dimensionless temperature profile = (TC,,V,,I -T,)/ATO separated dimensionless temperature distribution dimensionless transverse variable = p/h dimensionless axial variable = (16.x/Dc/Re Pr) dimensionless total diffusivity = 1 + (E,,/x) eddy diffusivity for heat eddy diffusivity for momentum cigcnvalues of eigenvalue problem (4) = ~1: eigenfunctions of eigenvalue problem (4) frequency of inlet oscillations dimensionless frequency of inlet oscillations = wb'/a kinematic viscosity of fluid order of eigenvalue Superscripts I.