A new reproducing kernel method for Duffing equations. Issue 11 (2nd November 2021) - Chen, Zhong; Jiang, Wei; Du, Hong
Abstract : 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.