A numerical model is developed to examine steady, laminar flame spread and extinction over a thin solid fuel in low-speed concurrent flow. The model incorporates an elliptic treatment of the upstream flame stabilization zone near the fuel burnout point, and a parabolic treatment of the downstream flame, which has a higher flow Reynolds number. This provides a more precise fluid-mechanical description of the flame than using parabolic equations throughout, and is the first time such an approach has been used in concurrent flame spread modeling. The parabolic and elliptic regions are coupled smoothly by matching boundary conditions. The solid phase consists of an energy equation with surface radiative loss and a surface pyrolysis relation. Calculations (with the flame spread rate being an eigenvalue) are performed for forced flow without gravitational influences in a range of velocities which are lower than those induced in a normal gravity buoyant environment. Steady spread with constant flame and pyrolysis lengths is found possible for thin fuels and this facilitates the adoption of a moving coordinate system attached to the flame.