The renormalization method based on the Taylor expansion and applications for asymptotic analysis

TL;DR

This paper introduces a Taylor expansion-based renormalization method for asymptotic analysis that automatically eliminates secular terms, simplifies mathematical foundation, and improves solutions for complex differential equations.

Contribution

A new renormalization method based on Taylor expansion is proposed, recovering the RG method and enhancing solution accuracy for nonlinear problems.

Findings

Automatically eliminates secular terms in perturbation series

Derives uniform asymptotic solutions for boundary problems

Applies to equations without small parameters, like Duffing and Blasius

Abstract

Based on the Taylor expansion, we propose a renormalization method for asymptotic analysis. The standard renormalization group (RG) method for asymptotic analysis can be derived out from this new method, and hence the mathematical essence of the RG method is also recovered. The biggest advantage of the proposed method is that the secular terms in perturbation series are automatically eliminated, but in usual perturbation theory, we need more efforts and tricks to eliminate these terms. At the same time, the mathematical foundation of the method is simple and the logic of the method is very clear, therefore, it is very easy in practice. As application, we obtain the uniform valid asymptotic solutions to some problems including vector field, boundary layer and boundary value problems of nonlinear wave equations. Moreover, we discuss the normal form theory and reduction equations of dynamical systems. Furthermore, by combining the topological deformation and the RG method, a modified method namely the homotopy renormalization method (for simplicity, HTR) wasproposed to overcome the weaknesses of the standard RG method. In this HTR method, since there is a freedom to choose the first order approximate solution in perturbation expansion, we can improve the global solution. In particular, for those equations including no a small parameter, the HTR method can also be applied. Some concrete applications including multi-solutions problems, the forced Duffing equation and the Blasius equation are given.