两点边值问题二尺度小波核LS-SVM解法

Two-scale Wavelet Kernel LS-SVM Method for theTwo-point Boundary Value Problem

针对两点边值问题难以得到解析解,提出了利用二尺度小波核最小二乘支持向量机方法求两点边值问题的近似解;首先将两点边值问题转换为带有两个约束条件的目标优化问题,再利用二尺度小波核函数的组合构造满足边界条件的近似解;其中第一个约束条件用第一尺度小波核函数逼近,第二个约束条件是对第一次逼近的误差函数用第二尺度小波核函数再次逼近,可提高近似解逼近精度;最后将目标优化问题转化为回归问题,进而利用最小二乘支持向量机方法求解回归系数,系数求解过程中核心是将参数回归问题转化为二次规划问题,可避免复杂的微分运算;数值实验表明:方法求解两点边值问题有较高的精度,计算量小,并且具有较好的稳定性,因此二尺度小波核最小二乘支持向量机方法求解两点边值问题的近似解是有效的,并且具有精度高、可微、表达式简单且形式固定等特点。

In practice, it is difficult to obtain analytical solution for two-point boundary value problem, the two-scale wavelet kernel LS-SVM method is proposed to solve the approximate solution of two-point boundary value problem. Firstly, the two-point boundary value problem is transformed into an objective optimization problem with two constraint conditions, and then an approximate solution satisfying the boundary conditions is constructed by using the combination of two-scale wavelet kernel functions. The first constrained condition is approximated by the first scale wavelet kernel function, and the second constraint condition is that the error function of the first approximation is approximated by the second scale wavelet kernel function, which can improve the approximation accuracy of the approximate solution. Finally, the objective optimization problem is transformed into the regression problem, and then the LS-SVM method is used to solve the regression coefficients. In the process of coefficients solving, the key is to transform the parametric regression problem into quadratic programming problem, which can avoid complex differential operation. Numerical experiments show that the proposed method has high accuracy, less calculation and good stability. Therefore, the two-scale wavelet kernel LS-SVM method is effective for solving the two-point boundary value problem, and has the characteristics of high precision, differentiability, simple expression and fixed form.