A new reproducing kernel method for Duffing equations

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.

Keywords:

• Duffing equation
• reproducing kernel
• spline function
• convergence order
• Newton iterations

2010 Mathematics Subject Classifications:

• 34B05
• 65L10

Acknowledgments

This work has received various grants from National Natural Science Foundation of China under Grant (No.11701124), Project of Enhancing School with Innovation (No.Q18306) and Scientific Research Start-up Funds (No.R20050) of GuangDong Ocean University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work has received various grants from National Natural Science Foundation of China [grant number 11701124], Natural Science Foundation of Heilongjiang Province [grant number E2017059], Project of Enhancing School with Innovation [grant number Q18306] and Scientific Research Start-up Funds (No.R20050) of GuangDong Ocean University.