In this article, we construct the approximate solutions to the Euler-Poisson system in an annular domain, that arises in the study of dynamics of plasmas. Due to a small parameter (proportional to the square of the Debye length) multiplied to the Laplacian operator, together with unmatched boundary conditions, we find that the solutions exhibit sharp transition layers near the boundaries, which makes the associated limit problem singular. To investigate this singular behavior, we explicitly construct the approximate solutions composed of the outer and inner solutions by the method of asymptotic expansions in appropriate order of the small parameter, turned out to be the Debye length. The equations to single out the boundary layers are determined by the inner expansions, for which we effectively treat nonlinear terms using the Taylor polynomial expansions with multinomials. We can obtain estimates showing that the approximate solutions are close enough to the original ones. We also provide numerical evidences demonstrating that the approximate solutions converge to those of the Euler-Poisson system as the parameter goes to zero.

在本文中，我们构造了环形动力学区域中欧拉-泊松系统的近似解，这是在等离子体动力学研究中出现的。由于将一个小的参数（与Debye长度的平方成正比）乘以Laplacian算子，加上不匹配的边界条件，我们发现解决方案在边界附近表现出尖锐的过渡层，这使相关的极限问题变得奇异。为了研究这种奇异行为，我们通过渐近展开的方法，以小参数的适当顺序显式构造了由外部和内部解组成的近似解，结果证明是Debye长度。选出边界层的方程式由内部展开确定，为此，我们使用带多项式的泰勒多项式展开来有效地处理非线性项。我们可以获得估计值，表明近似解与原始解足够接近。我们还提供了数值证据，证明当参数变为零时，近似解收敛于Euler-Poisson系统的解。