Analytic and numerical aspects of the observation of the heat equation
Analytic and numerical aspects of the observation of the heat equation
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DOI：
10.1109/cdc.1987.272541
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发表时间：
1987-12
期刊：
影响因子：
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通讯作者：
D. Gilliam;C. Martin;J. Lund
中科院分区：
文献类型：
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作者：
D. Gilliam;C. Martin;J. Lund
我们提出了一个简单的和非常准确的程序近似的初始温度的热方程的线上使用离散的时间和空间采样。该过程是基于“sinc扩展”，在一个特定的类中的功能产生一个统一的指数误差界与指数取决于所选择的空间采样位置的数量。此外，对于每个空间采样点，温度仅需要在一个相同的时间值处采样。对于N个空间采样点，该方法简化为求解一个系数矩阵为（2N + 1）×（2N + 1）的线性方程组。该矩阵是对称的Toeplitz矩阵，因此通过使用求积计算仅2N + 1个值来确定。
We present a simple and extremely accurate procedure for approximating initial temperature for the heat equation on the line using a discrete time and spatial sampling. The procedure is based on the "sinc expansion" which for functions in a particular class yields a uniform exponential error bound with exponent depending on the number of spatial sample locations chosen. Further the temperature need only be sampled at one and the same temporal value for each of the spatial sampling points. For N spatial sample points, the approximation is reduced to solving a linear system with a (2N + 1) × (2N + 1) coefficient matrix. This matrix is a symmetric toeplitz matrix and hence is determined by computing only 2N + 1 values using quadrature.