In this talk, we discuss a new class of numerical methods for the accurate and efficient integration of time dependent partial differential equations. Unlike traditional method of lines $(MoL)$, the new Krylov deferred correction $(KDC)$ accelerated method of lines transpose $(Mol^T)$ first discretizes the temporal direction using Gaussian type nodes and spectral integration, and the resulting coupled elliptic system is solved iteratively using Newton-Krylov techniques such as Newton-GMRES method, in which each function evaluation is simply one low order time stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers. Preliminary numerical experiments show that the KDC accelerated $MoL^T$ technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time step sizes in long-time simulations. Numerical experiments for parabolic type equations including the Schrodinger equation will be discussed.