Shock-wave structure in the presence of low viscosity

The initial value problem for a first-order nonlinear equation is examined in the case when the solution has one smooth line of discontinuity. The solution of the problem is smooth everywhere if the dissipative term with a small parameter is taken into account in the equation. The asymptotic-expansion matching method is used to obtain a uniform approximation to this solution with an accuracy up to an arbitrary degree of the small parameter.