Automatic spectral collocation for integral, integro-differential, and integrally reformulated differential equations
Automatic spectral collocation for integral, integro-differential, and integrally reformulated differential equations
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DOI：
10.1016/j.jcp.2010.04.029
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发表时间：
2010-08
期刊：
影响因子：
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通讯作者：
T. Driscoll
中科院分区：
文献类型：
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作者：
T. Driscoll
Fredholm 和 Volterra 积分和积分微分方程的自动切比雪夫谱配置方法已作为 chebfun 软件系统的一部分实现。该系统使符号语法能够应用于数值对象，以便在没有明确引用离散化的情况下提出和解决问题。相同的对象可以用于线性代数中的无矩阵迭代方法，以避免非常大的密集矩阵或允许应用于非光滑系数的问题。 As a further application of the ability to implement operator equations, a method of Greengard [1] for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems.在积分形式中，避免了与高阶问题中的微分矩阵相关的大条件数。实现重铸过程的能力通常来自于 chebfun 中运算符表达式的实现。积分方法也可以扩展到一阶系统，尽管 chebfun 语法目前不允许在这种情况下轻松实现。
Automatic Chebyshev spectral collocation methods for Fredholm and Volterra integral and integro-differential equations have been implemented as part of the chebfun software system. This system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization. The same objects can be used in matrix-free iterative methods in linear algebra, in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients. As a further application of the ability to implement operator equations, a method of Greengard [1] for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems. In the integral form, large condition numbers associated with differentiation matrices in high-order problems are avoided. The ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun. The integral method also can be extended to first-order systems, although chebfun syntax does not currently allow easy implementation in this case.