论文标题
Zernike多项式在求解某些第一和二阶偏微分方程中的应用
Application of Zernike polynomials in solving certain first and second order partial differential equations
论文作者
论文摘要
基于Zernike多项式的集成操作矩阵方法用于确定第一和第二阶的一类非均匀偏微分方程（PDE）的近似解。由于单位磁盘中描述了Zernike多项式的性质，因此该方法在求解圆形区域的PDE方面特别有效。此外，所提出的方法可以用不连续的Dirichlet和Neumann边界条件求解PDE，并且由于这些不连续的函数无法在某些Chebyshev或Gauss-Lobatto点上定义，因此广受好评的伪信号方法并不直接适用于此类问题。解决此类PDE也是Zernike多项式的新应用，因为到目前为止，这些多项式的主要应用似乎是在研究圆形对称光学系统的光学畸变中。在目前的方法中，给定的PDE被转换为形式AX = B的线性方程系统，该系统可以通过L1和L2最小化方法求解，其中L1方法被发现更准确。最后，在Zernike多项式方面的函数扩展时，对于某些类别的功能，给出了系数的衰减速率。
Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike polynomials being described in the unit disk, this method is particularly effective in solving PDEs over a circular region. Further, the proposed method can solve PDEs with discontinuous Dirichlet and Neumann boundary conditions, and as these discontinuous functions cannot be defined at some of the Chebyshev or Gauss-Lobatto points, the much acclaimed pseudo-spectral methods are not directly applicable to such problems. Solving such PDEs is also a new application of Zernike polynomials as so far the main application of these polynomials seem to have been in the study of optical aberrations of circularly symmetric optical systems. In the present method, the given PDE is converted to a system of linear equations of the form Ax = b which may be solved by both l1 and l2 minimization methods among which the l1 method is found to be more accurate. Finally, in the expansion of a function in terms of Zernike polynomials, the rate of decay of the coefficients is given for certain classes of functions.
