Prandtl型积分微分方程的滤波插值法
Filtered interpolation for solving Prandtl's integro-differential equations
摘要：
为了求解Prandtl型方程，我们提出了一种基于Chebyshev节点VP滤波插值的配点求积方法。在局部连续函数的Holder - Zygmund空间中证明了一致收敛性和稳定性.相对于经典的基于Lagrange插值的方法在相同的配置节点，我们成功地再现了最佳的收敛速度的L2情况下，通过切断典型的对数因子，似乎不可避免的处理一致的范数。这样的改进不需要更大的计算工作量。特别是，我们提出了一个快速算法的基础上解决一个简单的2-带宽线性系，并证明，作为其维数趋于无穷大，序列的条件数（在任何自然矩阵范数）趋于有限的限制。
英文摘要：
In order to solve Prandtl-type equations we propose a collocation-quadrature method based on VP filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Holder - Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation at the same collocation nodes, we succeed in reproducing the optimal convergence rates of the L2 case by cutting off the typical log factor which seemed inevitable dealing with uniform norms. Such an improvement does not require a greater computational effort. In particular we propose a fast algorithm based on the solution of a simple 2-bandwidth linear system and prove that, as its dimension tends to infinity, the sequence of the condition numbers (in any natural matrix norm) tends to a finite limit.
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