We present a numerical scheme for the solution of two-parameter singularly perturbed problems whose solution has multi-scale behaviour in the sense that there are small regions where the solution changes very rapidly (known as layer regions) otherwise the solution is smooth (known as a regular region) throughout the domain of consideration. In particular, to solve the problems whose solution exhibits twin boundary layers at both endpoints of the domain of consideration, we propose a collocation method based on the quintic $\mathcal{B}$-spline basis functions. A piecewise-uniform mesh that increases the density within the layer region compared to the outer region is used. An $(N+1)\times (N+1)$ penta-diagonal system of algebraic equations is obtained after the discretization. A well-known fast penta-diagonal system solver algorithm is used to solve the system. We have shown that the method is almost fourth-order parameters uniformly convergent. The theoretical estimates are verified through numerical simulations for two test problems.

我们提出了一种求解双参数奇异摄动问题的数值方案，其解具有多尺度行为，即存在解变化非常迅速的小区域（称为层区域），否则解是平滑的（称为一个常规区域）在整个考虑范围内。特别是，为了解决其解决方案在考虑域的两个端点都表现出双边界层的问题，我们提出了一种基于五次的搭配方法$\mathcal{乙}$-样条基函数。使用与外部区域相比增加层区域内的密度的分段均匀网格。一个$(ñ+1)\times (ñ+1)$离散化后得到五对角代数方程组。众所周知的快速五对角系统求解器算法用于求解系统。我们已经证明该方法几乎是四阶参数一致收敛的。通过对两个测试问题的数值模拟验证了理论估计。