Asymptotic approximation of solutions of weakly linear differential systems (Q1072702)

An asymptotic method is obtained to solve the system \[ \partial u_ j/\partial t+\lambda_ j(\tau,\epsilon)\partial u_ j/\partial x=\epsilon f_ j(\tau,u,\partial u/\partial x,\epsilon),\quad u=(u_ 1,...,u_ n),\quad j=1,...,n,\quad 0<\epsilon \leq 1,\quad \tau =\epsilon t \] with periodic initial conditions \(u_ j(0,x,\epsilon)=u_{0j}(x,\epsilon)\). The asymptotic solution of the system, uniformly valid when \(t\in [0;O(\epsilon^{-1})]\), is received. The restrictions on the properties of the coefficients \(\lambda_ j\) are less stringent compared to other results. For example, this method can be used for all systems with constant \(\lambda_ j\). The application to ordinary differential equations is considered, too.