Detonation initiation in a tube with parabolic contraction and conical expansion was investigated numerically and experimentally. The optimized geometry of conical expansion with sine-shaped wall is proposed. The generalized diagram in the form of detonation curves at the contraction slope angle versus incident shock Mach number plane is presented. For solving the governing Euler equations, the numerical method based on finite volume approach with Godunov flux approximation adapted for multiprocessor systems is used. It has been shown experimentally that the parabolic contraction and conical expansion ensure shock-to-detonation transition in a stoichiometric propane-air mixture under normal conditions at a very low minimal incident shock wave velocity of 680 +/- 20m/s, which approximately corresponds to a Mach number of 2. This result is important for novel jet propulsion systems with detonative burning of fuel-pulse detonation engines.