二阶导数算子的Jacobi tau近似的谱
Spectrum of the Jacobi tau approximation for the second derivative operator
AI总结：
本文证明在Dirichlet边界条件下，对于一定范围的Jacobi参数，二阶导数算子的Jacobi Tau方法的特征值是实数、负且互异的，并特别分析了Gegenbauer Tau方法（包括Chebyshev和Legendre）的对称情况，给出了特征多项式的递推关系及矩阵结构。
摘要：
证明了对于Dirichlet边界条件下的二阶导数算子，Jacobi Tau方法的特征值在Jacobi参数的一定范围内是实数、负且互异的。特别强调了Gegenbauer Tau方法的对称情况，其中定理中包含的参数范围可以扩展，并且由逐次逼近给出的特征多项式相互交错。这包括常见的Chebyshev和Legendre、Tau和Galerkin方法。证明了Gegenbauer Tau方法的特征多项式满足三项递推关系加上一个常数项，该常数项在Legendre Tau和Galerkin情况下为零。这些递推关系等价于一个三对角加一行矩阵结构。Gegenbauer Tau方法的谱积分公式直接导致了那个基本的且良态的三对角加一行矩阵结构。提供了一个Matlab代码。
英文摘要：
It is proved that the eigenvalues of the Jacobi Tau method for the second derivative operator with Dirichlet boundary conditions are real, negative and distinct for a range of the Jacobi parameters. Special emphasis is placed on the symmetric case of the Gegenbauer Tau method where the range of parameters included in the theorems can be extended and characteristic polynomials given by successive order approximations interlace. This includes the common Chebyshev and Legendre, Tau and Galerkin methods. The characteristic polynomials for the Gegenbauer Tau method are shown to obey three term recurrences plus a constant term which vanishes for the Legendre Tau and Galerkin cases. These recurrences are equivalent to a tridiagonal plus one row matrix structure. The spectral integration formulation of the Gegenbauer Tau method is shown to lead directly to that fundamental and well-conditioned tridiagonal plus one row matrix structure. A Matlab code is provided.
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