中国邮电高校学报(英文) ›› 2018, Vol. 25 ›› Issue (4): 94-102.doi: 10.19682/j.cnki.1005-8885.2018.1021

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High precision approximate analytical solutions to ODE using LS-SVM

Zhou Shuisheng, Wang Baojun, Chen Li   

  1. School of Mathematics and Statistics, Xidian University, Xi’an 710071, China
  • 收稿日期:2017-12-07 修回日期:2018-10-03 出版日期:2018-08-30 发布日期:2018-11-02
  • 通讯作者: Wang Baojun,E-mail:wangbaojun0916@126.com E-mail:wangbaojun0916@126.com
  • 作者简介:Wang Baojun,E-mail:wangbaojun0916@126.com

High precision approximate analytical solutions to ODE using LS-SVM

Zhou Shuisheng, Wang Baojun, Chen Li   

  1. School of Mathematics and Statistics, Xidian University, Xi’an 710071, China
  • Received:2017-12-07 Revised:2018-10-03 Online:2018-08-30 Published:2018-11-02
  • Contact: Wang Baojun,E-mail:wangbaojun0916@126.com E-mail:wangbaojun0916@126.com
  • About author:Wang Baojun,E-mail:wangbaojun0916@126.com

摘要: The problem of solving differential equations and the properties of solutions have always been an important content of differential equation the study. In practical application and scientific research, it is difficult to obtain analytical solutions for most differential equations. In recent years, with the development of computer technology, some new intelligent algorithms have been used to solve differential equations. They overcomes the drawback of traditional methods and provide the approximate solution in closed form (i.e., continuous and differentiable). The least squares support vector machine (LS-SVM) has nice properties in solving differential equations. In order to further improve the accuracy of approximate analytical solutions and facilitative calculation, a novel method based on numerical methods and LS-SVM methods is presented to solve linear ordinary differential equations (ODEs). In our approach, a high precise of the numerical solution is added as a constraint to the nonlinear LS-SVM regression model, and the optimal parameters of the model are adjusted to minimize an appropriate error function. Finally, the approximate solution in closed form is obtained by solving a system of linear equations. The numerical experiments demonstrate that our proposed method can improve the accuracy of approximate solutions.

关键词: the kernel function, LS-SVM, ODE, numerical solution, approximate analytical solution

Abstract: The problem of solving differential equations and the properties of solutions have always been an important content of differential equation the study. In practical application and scientific research, it is difficult to obtain analytical solutions for most differential equations. In recent years, with the development of computer technology, some new intelligent algorithms have been used to solve differential equations. They overcomes the drawback of traditional methods and provide the approximate solution in closed form (i.e., continuous and differentiable). The least squares support vector machine (LS-SVM) has nice properties in solving differential equations. In order to further improve the accuracy of approximate analytical solutions and facilitative calculation, a novel method based on numerical methods and LS-SVM methods is presented to solve linear ordinary differential equations (ODEs). In our approach, a high precise of the numerical solution is added as a constraint to the nonlinear LS-SVM regression model, and the optimal parameters of the model are adjusted to minimize an appropriate error function. Finally, the approximate solution in closed form is obtained by solving a system of linear equations. The numerical experiments demonstrate that our proposed method can improve the accuracy of approximate solutions.

Key words: the kernel function, LS-SVM, ODE, numerical solution, approximate analytical solution