Reproducing kernel theories have attracted much attention for solving various problems. However, discussions of stability and convergence order are very difficult in the traditional reproducing kernel method by orthogonal expansion because of the randomness of Schmidt's orthogonalization coefficients. Later, the convergence order of the reproducing kernel method is estimated by polynomial interpolation of residuals. But the convergence order is not high. In this paper, taking Duffing equation as an example, noting that the reproducing kernel function is a spline function, a new scheme with much higher convergence order is proposed by constructing spline bases function in the reproducing kernel space and combining polynomial interpolation of residuals. Numerical examples verify the convergence order theories proposed in this paper. It is worth to say that our main results could be applied to construct approximate solutions of various equations.

再生核理论在解决各种问题方面备受关注。然而，由于施密特正交化系数的随机性，在传统的正交展开再现核方法中，稳定性和收敛阶次的讨论非常困难。之后，通过残差的多项式插值来估计再现核方法的收敛阶数。但收敛阶次不高。本文以Duffing方程为例，注意到再生核函数是一个样条函数，通过在再生核空间构造样条基函数并结合残差多项式插值，提出了一种具有更高收敛阶数的新方案。数值算例验证了本文提出的收敛阶理论。