The problem of solving differential equations and the properties of solutions have always been an important content of differential equation study. In practical application and scientific research,it is difficult to o...The problem of solving differential equations and the properties of solutions have always been an important content of differential equation 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 overcome the drawbacks 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 precision 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.展开更多
文摘The problem of solving differential equations and the properties of solutions have always been an important content of differential equation 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 overcome the drawbacks 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 precision 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.
