An Asymptotic Formula for the Solution of a Singular Perturbation Problem

一個奇界攝動問題解的漸近形式

盧性良(Shing-Liang Lu)

《東海科學》 1期 (1999/07) Pp. 49-63

https://doi.org/10.29723/TS.199907.0003

摘要

We study, in the rectangle Ω=(0, α)×(0, b), the Dirichlet boundary value problem for the elliptic partial differential equation Lu=-εΔu+pu(subscript x)+gu(subscript y)+qu=f, where 0＜ε≪1, Δ is the Laplacian operator, and the functions p, g, q and f satisfy certain hypotheses; in particular, p＞0, q≥0. We construct a formal asymptotic expansion of the solution u of this problem for small ε. This expansion contains the solution of the reduced equation and boundary layer functions. The parabolic boundary layer functions satisfy a parabolic equation with an unbounded coefficient. We transform the parabolic equation into a heat equation to develop properties of the parabolic boundary layer. Estimates for the remainder in the expansion are established that are of the order of magnitude of powers of ε.

關鍵字

奇界攝動 ； 邊界層

並列摘要

We study, in the rectangle Ω=(0, α)×(0, b), the Dirichlet boundary value problem for the elliptic partial differential equation Lu=-εΔu+pu(subscript x)+gu(subscript y)+qu=f, where 0＜ε≪1, Δ is the Laplacian operator, and the functions p, g, q and f satisfy certain hypotheses; in particular, p＞0, q≥0. We construct a formal asymptotic expansion of the solution u of this problem for small ε. This expansion contains the solution of the reduced equation and boundary layer functions. The parabolic boundary layer functions satisfy a parabolic equation with an unbounded coefficient. We transform the parabolic equation into a heat equation to develop properties of the parabolic boundary layer. Estimates for the remainder in the expansion are established that are of the order of magnitude of powers of ε.

並列關鍵字

singular perturbation ； boundary layer