Asymptotic Analysis for Hyperbolic Equation with Neumann Condition

Abstract

The second-order hyperbolic equation with small parameter and Neumann con- dition is considered. This kind of problem loses the boundary conditions both in x = 0 and x = 1, while it also loses two initial boundary conditions in t = 0. The solution changes rapidly near two boundary layers and one initial layer. Firstly, the asymptotic solution was studied. The analytical solution was approximated by the degenerate solution and two boundary layer functions and one initial layer function. Secondly, three transition points were presented ac- cording to Shishkin’s idea. Non-equidistant mesh partitions both in x direction and t direction were introduced. An effective computational method is given according to non-equidistant mesh partitions. Finally, numerical experiment was given.