Abstract In this paper, a non-uniform mesh optimal B-spline collocation method is presented for the numerical solution of a singular two-point boundary value problem describing electrohydrodynamic flow (EF) of a fluid in a circular cylindrical conduit. The EF problem is highly nonlinear and has a singularity at the point r = 0 . Further, this problem has a boundary layer near right end of the solution domain. We consider a grading function to construct a non-uniform mesh over the problem domain. The non-uniform mesh is constructed in such a way that the mesh is finer near the right end boundary. Convergence of the proposed method is analyzed. The EF problem is solved for small and large values of the two relevant parameters: (i) strength of non-linearity β, and (ii) Hartmann electric number H. The effects of β and H on the velocity field are investigated. The computed results have been compared with those obtained by two other B-spline collocation methods over uniform mesh to show the advantage of the present method. It is shown that the present method provides O ( h 4 ) convergent approximation.

中文翻译：

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求解描述流体电流体流动的强非线性奇异边值问题的四阶非均匀网格优化B样条配置方法

摘要 在本文中，提出了一种非均匀网格优化B样条配置方法，用于描述圆柱管道中流体的电流体动力学流动（EF）的奇异两点边值问题的数值解。EF 问题是高度非线性的，并且在 r = 0 点具有奇点。此外，这个问题在解域的右端附近有一个边界层。我们考虑一个分级函数来在问题域上构建一个非均匀网格。非均匀网格的构造方式使得网格在右端边界附近更精细。分析了所提出方法的收敛性。EF 问题针对两个相关参数的小值和大值得到解决：(i) 非线性强度 β，和 (ii) 哈特曼电数 H。研究了 β 和 H 对速度场的影响。将计算结果与其他两种 B 样条搭配方法在均匀网格上获得的结果进行了比较，以显示本方法的优势。结果表明，本方法提供了 O ( h 4 ) 收敛近似。