The Shock Location for a Class of Sensitive Boundary Value Problems

The sensitive internal layer behaviour in an autonomous nonlinear singularly perturbed boundary value problem is investigated. For this problem we show that the internal layers solutions exhibit either an exponential or an algebraic sensitivity in reponse to small changes in the boundary conditions as well as in the coefficients of the equation and we derive a geometric method to determine the shock location as a function of the perturbations. The results are then applied to study the behavior of both the viscous shock location for the two-point problem for the stationary Burgers equation and the supersonic-subsonic shock that arises in modelling compressible flows as a result of perturbations of the boundary conditions of order 0(e-1/ε). For the corresponding time-dependent partial differential equation we also show how the exponentially small perturbations in the ordinary differential equation are associated with the metastable viscous shock layer motion. Some associated boundary layer resonance problems with turning points are also considered