This study is devoted to investigate the unsteady fully developed time-dependent Couette flow in a composite channel partially filled with porous material. The Brinkman-extended Darcy model is used to simulate momentum transfer in the porous medium. The fluid and porous regions are interlinked by equating the velocity and by considering shear stress jump conditions in the interface. The solutions of the governing equations are obtained using a Laplace transform technique. However, the Riemann-sum approximation method is used to invert from Laplace domain to time domain. The solution obtained is validated by presenting comparisons with closed form solution obtained for steady flow which has been derived separately and also by implicit finite difference method. During the course of numerical comparison, an excellent agreement was found between steady-state solution obtained exactly and unsteady solution obtained by implicit finite difference method or Riemann-sum approximation method at large values of time. The effect of various flow parameters entering into the problem is discussed with the aid of line graphs. The results obtained here may be further used to verify the validity of obtained numerical solutions for more complicated time-dependent Couette flow in composite channel.