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Pure Mathematics 2024
数值方法求解微分方程的研究——基于切比雪夫多项式的谱方法
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Abstract:
谱方法是处理微分方程的常用方法，本文以理论完善的谱方法为基础，详细介绍了切比雪夫多项式通过S-L问题的由来与切比雪夫多项式的部分性质，并利用这些性质将这些正交多项式作为基对函数进行展开，从而数值求解偏微分方程，我们利用案例来展现其具体的运算过程并验证其方法的有效性。
Based on Spectral Method, we introduce the origin of Chebyshev polynomials via S-L problems and some properties of Chebyshev polynomials. With these properties, we use these orthogonal poly-nomials as basic functions to solve partial differential equations. Also, we use examples to show the detailed operations and verify its effectiveness.
| [1] | 向新民. 谱方法的数值分析[M]. 北京: 科学出版社, 2000. |
| [2] | Gottlieb, D. and Orszag, S.A. (1977) Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611970425 |
| [3] | Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988) Spectral Method in Fluid Dynamics. Springer Verlag, Berlin. https://doi.org/10.1007/978-3-642-84108-8 |
| [4] | Boyd, J.P. (2013) Chebyshev and Fourier Spectral Methods. Courier Corporation. |
| [5] | Butcher, J.C. (1987) The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods. John Wiley & Sons, Chichester. |
| [6] | Hairer, E., Norsett, S.P. and Wanner, G. (1987) Solving Ordinary Differential Equation I: Nonstiff Problems. Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-662-12607-3 |
| [7] | Orszag, S.A. (1980) Spectral Method for Problems in Complex Gemetries. Journal of Computational Physics, 37, 70-92. https://doi.org/10.1016/0021-9991(80)90005-4 |
| [8] | Canuto, C. and Quarteroni, A. (1982) Approximation Results for Orthogonal Polynomials in Sobolev Spaces. Mathematics of Computation, 38, 67-86. https://doi.org/10.1090/S0025-5718-1982-0637287-3 |
| [9] | Canuto, C. and Quarteroni, A. (1982) Error Estimates for Spectral and Pseudo-Spectral Approximations of Hyperbolic Equations. SIAM Journal on Numerical Analysis, 19, No. 3. https://doi.org/10.1137/0719044 |
| [10] | Canuto, C. and Quarteroni, A. (1981) Spectral and Pseudo Spectral Methods for Parabolic Problems with Nonperiodic Boundary Conditions. Calcolo, 18, 197-217. https://doi.org/10.1007/BF02576357 |
| [11] | 郭本瑜. Navier-stokes方程的谱解法[J]. 中国科学(A辑数学物理学天文学技术科学), 1985(8): 715. |
| [12] | Guo, B. (1985) Spectral Method for Solving Navier-Stokes Equation. Science in China, Series A, 28, 1139-1153. |
| [13] | Bar-Yoseph, P.Z., Fisher, D. and Gottlieb, O. (1996) Spectral Element Methods for Nonlinear Spatiotemporal Dynamics of an Euler-Bernoulli Beam. Computational Mechanics, 19, 136-151. https://doi.org/10.1007/BF02824851 |
| [14] | Gear, C.W. and Petzold, L.R. (1984) ODE Methods for the Solution of Differential/Algebraic Systems. SIAM Journal on Numerical analysis, 21, 716-728. |
| [15] | Butcher, J.C. (1964) Implicit Runge-Kutta Processes. Mathematics of Computation, 18, 50-64.
https://doi.org/10.1090/S0025-5718-1964-0159424-9 |
