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A multiple-scale Pascal polynomial triangle solving elliptic equations and inverse Cauchy problems

机译:求解椭圆方程和柯西逆问题的多尺度Pascal多项式三角形

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摘要

The polynomial expansion method is a useful tool to solve partial differential equations (PDEs). However, the researchers seldom use it as a major medium to solve PDEs due to its highly ill-conditioned behavior. We propose a single-scale and a multiple-scale Pascal triangle formulations to solve the linear elliptic PDEs in a simply connected domain equipped with complex boundary shape. For the former method a constant parameter R_0 is required, while in the latter one all introduced scales are automatically determined by the collocation points. Then we use the multiple-scale method to solve the inverse Cauchy problems, which is very accurate and very stable against large noise to 20%. Numerical results confirm the validity of the present multiple-scale Pascal polynomial expansion method.
机译:多项式展开法是解决偏微分方程(PDE)的有用工具。但是,由于其病态严重,研究人员很少将其用作解决PDE的主要媒介。我们提出了单尺度和多尺度的Pascal三角形公式,以解决带有复杂边界形状的简单连接域中的线性椭圆PDE。对于前一种方法,需要一个常数参数R_0,而在后一种方法中,所有引入的比例都是由搭配点自动确定的。然后,我们使用多尺度方法来解决柯西反问题,该方法非常精确且非常稳定,可抵御20%的大噪声。数值结果证实了当前多尺度帕斯卡多项式展开方法的有效性。

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