A Sinc method for an eigenvalue problem of a differential operator with periodic coefficients and its comparison with Hill's method
A Sinc method for an eigenvalue problem of a differential operator with periodic coefficients and its comparison with Hill's method
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具有周期系数的微分算子特征值问题的Sinc方法及其与Hill方法的比较
DOI：
10.1109/itng.2013.31
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Ken'ichiro Tanaka:
中科院分区：
文献类型：
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作者：
T. Arima;S. Taniguchi;T. Ruggeri and M. Sugiyama;Ken'ichiro Tanaka:
考虑了一类具有周期系数的常微分算子的谱计算问题。由于Floquet的理论，这样的问题是减少到一组特征值问题的修改后的运营商与周期性的边界条件。我们对待这样的问题的两个数值方法。首先是希尔的方法，它减少了每个问题的矩阵特征值问题与有限傅立叶级数近似的特征函数的每个运营商。该方法相对于矩阵的大小达到指数收敛速度。然而，随着系数的周期变长，速率变差，这在一些数值实验中观察到。然后，为了在长周期的情况下实现精确计算，我们提出了与Sinc近似相关的第二种方法。基本上，Sinc近似采用由R上的sinc函数sinc（x）= sin（pi x）/（pi x）生成的Sinc基。在这项工作中，采用sinc函数的某种变体来近似周期函数。我们的方法在长周期的情况下保持了很好的精度，这可以在一些数值实验中得到证实。
We consider a problem of computing spectrum of an ordinary differential operator with periodic coefficients. Due to Floquet's theory, such a problem is reduced to a set of eigenvalue problems for modified operators with a periodic boundary condition. We treat two numerical methods for such problems. A first is Hill's method, which reduces each problem to a matrix eigenvalue problem with the finite Fourier series approximation of eigenfunctions of each operator. This method achieves exponential convergence rate with respect to the size of the matrix. The rate, however, gets worse as the period of the coefficients becomes longer, which is observed in some numerical experiments. Then, in order to realize accurate computation in the cases of the long periods, we propose a second method related to Sinc approximation. Basically, Sinc approximation employs Sinc bases generated by the sinc function sinc(x) = sin(pi x)/(pi x) on R. In this work, a certain variant of the sinc function is adopted to approximate periodic functions. Our method keeps good accuracy in the cases of the long periods, which can be confirmed in some numerical experiments.