Hyperbolic conservation law equations and convection-dominated convection-diffusion equations can be solved numerically using high-order weighted essentially non-oscillatory (WENO) techniques in computational fluid mechanics. In this paper, we extend the modified WENO scheme to mixed-type equations. Numerical examples for the van der Waals equations and the corresponding viscosity-capillarity equations are presented to illustrate the efficiency and structure of our algorithm. The emphasis is on the Riemann problem of mixed equations under different pressure states.