关键词: 近似解, 核函数, 最小二乘支持向量机, 常微分方程, 常微分方程组

Abstract: For linear ordinary differential equations, the analytical solution is convenient for qualitative analysis and practical application. However, most of the differential equations are not available in many cases. The regression method is applied to obtain approximate analytical solution, and the Least Square Support Vector Machine（LS-SVM） method is demonstrated the efficiency of the proposed method over existing methods. But this method requires not only to take the higher derivative of the kernel but also to solve a large linear equation system. Concerning higher order Ordinary Differential Equations（ODE）, the available approach is the reduction of the problem to a system of first-order differential equations and then the system is solved by LS-SVM regression method. These model parameters are adjusted to minimize an error function, finally, the high precision approximate solution（continuous and differentiable） is obtained by solving three system of linear equations. Experimental results verify the effectiveness of the proposed method.

Key words: approximate analytical solution, kernel function, least square support vector machine, ordinary differential equations, system of ordinary differential equations