Burgers方程的奇摄动初值问题的激波解

Shock solutions of singularly perturbed initial value problems for Burgers equations

利用合成渐近展开法研究具有初值间断的Burgers方程的奇摄动问题.初始条件的突变使得问题的解在过渡层产生激波.首先,在激波位置两侧分别寻求具有边界层性质的近似式;再使用衔接法将对应的曲面光滑地衔接,构成激波解的形式近似;最后,运用渐近展开理论分析解的渐近性质,得到一致有效的渐近展开式.

Singularly perturbed problems for Burgers equations with initial value discontinuity are studied by using composite asymptotic expansion methods.The sharp changes of initial condition cause the solution of the problem to produce a shock wave in the transition layer.First,the formal approximation of shock solutions is constructed by seeking approximations which exhibit boundary layer behavior at both sides of shock location respectively.Then,the corresponding surfaces are connected smoothly by the joint method to form a formal approximation of the shock solution.Finally,the asymptotic properties of the solution are analyzed by using the asymptotic expansion theory,and a uniformly valid asymptotic expansion is obtained.