Second and third order perturbation solutions of a generalized Burgers' equation

The differential equation u_τ-uu_x=k(u_xx+cu_xτ) with initial values on τ=0 is considered. When c≠0 this represents a hyperbolic generalization of Burgers' equation. For k≪1 perturbation solutions are obtained, the outer solution being given completely up to third order, the inner solution (i.e. close to the shock) being given to second. The determination of the unknown functions in the second order inner solution is completed using an integral conservation technique. While the third order inner solution is not explicitly determined, it is shown that matching of the inner and outer solutions at third order is satisfied.