Solution of Eigenvalue Problems for Spheroidal Wave Equation, Lame Equation, and Ellipsoidal Wave Equation

球波方程、Lame方程和椭球波方程特征值问题的解

• 批准号: 13640128
• 负责人: IKEBE Yasuhiko
• 金额: $ 2.24万
• 依托单位: Meisei University (2002)The University of Aizu (2001)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640128/
• 关键词: Infinite Complex Band Matrix; Eigenvalue; Spheroidal Wave Equation; Lame Equation; Ellipsoidal Wave Equation; スフェロイド波動方程式; 固有値問題; 楕円体方程式

The authors have conducted the research in detail on the eigenvalue problems of infinite complex band matrices, gaining the proof of the computability of approximated eigenvalues as well as its accurate error estimate formula, by an extremely natural truncation method. For more details, refer to1. Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada, The Eigenvalue Problem for Infinite Compact Complex Symmetric Matrices with Application to the Numerical Computation of Complex Zeros of J_0(z) -iJ_1(z) and of Bessel Functions J_m(z) of Any Real Order m, Linear Algebra and Its Applications, Vol. 194(1993), pp. 35-702. Y. Ikebe, N. Asai, Y. Miyazaki, and D. Cai, The Eigenvalue Problem for Infinite Complex Symmetric Tridiagonal Matrices with Application, Linear Algebra and Its Applications, Vol. 241-243(combined volume) (1996), pp.599-618On the other hand, transforming the three-dimensional wave equation Δw + p^2w = 0 by the spheroidal coordinates (there are two kinds, o … More r prolate type and oblate type), and executing the method of separation of variables give w"-tanθ・w' + (λ-μ^2 sec^2θ+γcos^2θ)w = 0, which is widely called spheroidal wave equations. The formulation, error estimate, as well as their visualization were attempted, by matrix method, as an application of the numerical commutation mentioned above.Alternatively, in analogy with the solution of Mathieu differential equation, which produced considerably significant results concerning the computation of eigenvalues up to the present, the authors handled the solution of Lame equation w"(z) ― (a + bk^2 sn^2 z)W(Z) = 0, also from the standpoint of the application stated earlier for its eigenvalues such that periodic Lame function has the period of 2π or 4π. Again, formulation, error estimate along with their visualization were obtained. Lastly, let us note that an ongoing project for solving the eigenvalues of ellipsoidal wave equation is also under investigation, whose related literature is seldom found. Less

对无穷复带矩阵的特征值问题进行了详细的研究，用极自然的截断方法证明了逼近特征值的可计算性，并给出了精确的误差估计公式。有关详细信息，请参阅1。Y.Ikebe，Y.Kikuchi，I.Fujishiro，N.Asai，K.Takanashi，M.Harada，无穷紧复对称矩阵的特征值问题及其在J_0(Z)-IJ_1(Z)和任意实数阶m的贝塞尔函数J_m(Z)的复零点的数值计算中的应用，线性代数，第194卷(1993)，第35-702页.Y.Ikebe，N.Asai，Y.Miyazaki，D.Cai，无穷复对称三对角矩阵的特征值问题及其应用，线性代数及其应用，第241-243卷(合卷)(1996年)，第599-618页。另一方面，用球面坐标变换三维波动方程Δw+p^2w=0(有两种，o…用分离变量法得到w“-tanθ·w‘+(λ-μ^2 s^2θ+γcos^2θ)w=0，即广为人知的球面波动方程。作为上述数值交换的一种应用，尝试用矩阵方法来表示、误差估计和它们的可视化。或者，类似于马修微分方程解在特征值的计算方面产生了相当重要的结果，作者处理了Lame方程w“(Z)-(a+bk^2sn^2z)W(Z)=0的解，也从前面所述的关于其特征值的应用的角度出发，使得周期Lame函数的周期为2π或4π。再次，给出了公式、误差估计及其可视化。最后，让我们注意到，一个正在进行的求解椭球波动方程特征值的项目也在调查中，其相关文献很少找到。较少

项目成果

宮崎佳典, 浅井信吉, 蔡東生, 池辺八洲彦: "Mathieu微分方程式の逆固有値問題"日本応用数理学会論文誌. 8・2. 199-222 (1998)

Yoshinori Miyazaki、Shinkichi Asai、Tongsei Cai、Yasuhiko Ikebe：“Mathieu 微分方程的反特征值问题”日本应用数学学会会刊 8・2（1998 年）。

浅井信吉, 宮崎佳典, 蔡東生, 平澤一紘, 池辺八洲彦: "行列算法によるzJ'_v(z)+HJ_v(z)=0の数値解法"電子情報通信学会論文誌. J79A・7. 1256-1265 (1996)

Shinkichi Asai、Yoshinori Miyazaki、Dongsheng Cai、Kazuhiro Hirasawa、Yasuhiko Ikebe：“使用矩阵运算的 zJ_v(z)+HJ_v(z)=0 的数值解”电子信息通信工程师学会会刊 J79A・。 7. 1256 -1265 (1996)

Y.Ikebe, Y.Kikuchi, I.Fujishiro, N.Asai, K.Takanashi, M.Harada: "The Eigenvalue Problem for Infinite Compact Complex Symmetric Matrices with Application to the Numerical Computation of Complex Zeros of J_0(z)-iJ_1(z) and of Bessel Functions J_m(z) of Any

Y.Ikebe、Y.Kikuchi、I.Fujishiro、N.Asai、K.Takanashi、M.Harada：“无限紧复对称矩阵的特征值问题及其在 J_0(z)-iJ_1 复零点数值计算中的应用

菊池靖, 浅井信吉, 宮崎佳典, 蔡東生, 藤代一成, 池辺八洲彦: "非負実数階第1種ベッセル関数の零点計算アルゴリズム"日本応用数理学会論文誌. 3・4. 425-437 (1993)

Yasushi Kikuchi、Shinkichi Asai、Yoshinori Miyazaki、Tosei Cai、Kazunari Fujishiro、Yasuhiko Ikebe：“第一类非负实数阶贝塞尔函数的零点计算算法”日本应用数学学会杂志3・4。 425-437（1993）

Y. Miyazaki, N. Asai, Y. Kikuchi, D.S. Cai, Y. Ikebe: "Computation of Multiple Eigenvalues of Infinite Tri-Diagonal Matrices"Mathematics of Computation. accepted.

Y. Miyazaki、N. Asai、Y. Kikuchi、D.S. Cai、Y. Ikebe：“无限三对角矩阵的多个特征值的计算”计算数学。