Kortewego-de Vries方程的Hermite函数谱配置方法
Generalized Hermite Function Spectral-Collocation Method for KdV Equations
摘要: 以Hermite-Gauss节点为配置点，用带松弛因子的Hermite函数谱配置方法求数值解，逼近无界区域上的Kortewego-de Vries方程Cauchy问题的理论解，给出算法格式和相应的数值结果，表明所提算法格式的有效性和高精度。适当地选择松弛因子，使得数值解更好地匹配理论解的渐进行为，所给算法尤其适合于非线性问题。
Abstract: The paper deals with the numerical solutions of initial value problem of KdV equation on un-bounded interval that approximates the solution of the Kortewego-de Vries equation, in which Hermite-Gauss nodes are used to collocate nodes of spectral collocation method. Selecting appro-priately the relaxation factor   involved in the generalized Hermite functions approximation enables us to fit the asymptotic behaviors of exact solutions at infinity closely. Numerical results demonstrate its efficiency and high accuracy of this approach. Especially, it is much easier to deal with nonlinear equation.
文章引用：李冰冰, 王天军. Kortewego-de Vries方程的Hermite函数谱配置方法[J]. 应用数学进展, 2019, 8(4): 631-637. https://doi.org/10.12677/AAM.2019.84070

参考文献

[1] Korteweg, D. and de Vries, G. (1895) On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39, 422-443. [Google Scholar] [CrossRef] 
[2] Kawahara, R. (1972) Oscillatory Solitary Waves in Dispersive Media. Journal of the Physical Society of Japan, 33, 260-264. [Google Scholar] [CrossRef] 
[3] Kichenassamy, S. and Olver, P. (1992) Existence and Nonexistence of Solitary Wave Solutions to Higher-Order Model Evolution Equations. SIAM Journal on Mathematical Analysis, 23, 1141-1166. [Google Scholar] [CrossRef] 
[4] Wang, Z.Q. and Guo, B.Y. (2012) Legendre-Gauaa-Radau Collocation Method for Solving Initial Value Problems of First Order Ordinary Differential Equations. Journal of Sci-entific Computing, 52, 226-255. [Google Scholar] [CrossRef] 
[5] Wang, T.J. 一类线性奇异边值问题的谱配置方法[J]. 河南科技大学学报(自然科学版), 2013, 34(6): 75-78.
[6] Shen, J., Tang, T. and Wang, L.L. (2011) Spectral Mathod: Algo-rithms, Analysis and Applications. Springer Verlag, Berlin. [Google Scholar] [CrossRef] 
[7] 贾红丽, 王中庆. KdV方程的Chebyshev-Hermite谱配置法[J]. 应用数学与计算数学学报, 2013, 27(1): 1-8.
[8] 王天军. 半无界非线性热传导方程的Laguerre拟谱方法[J]. 应用数学与计算数学学报, 2013, 27(1): 9-15.
[9] 王天军, 杨森. 非线性Fokker-Planck方程的Hermite谱配置方法[J]. 安徽工业大学学报, 2012, 29(4): 381-384.
[10] Jie, S. and Tao, T. (2006) Spectral and High-Order Methods with Applications. Science Press, Beijing.
[11] Parkes, E., Zhu, Z., Duffy, B. and Huang, H. (1998) Sech-Polynomial Travelling Solitary-Wave Solutions of Odd-Order Generalized KdV Equations. Physics Letters A, 248, 219-224. [Google Scholar] [CrossRef] 

为你推荐