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

Based on the Taylor expansion, we propose a renormalization method (TR method, for simplicity) 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 to use in practice. As application, we obtain the uniform valid asymptotic solutions to some typical problems including vector field, boundary layer, boundary value problems of nonlinear wave equations, the normal form theory and reduction equations of dynamical systems. Further, by combining the topological deformation with the TR method, a modified method, namely the homotopy renormalization method (for simplicity, HTR), is proposed to overcome some weaknesses of the standard RG method and TR 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 not including a small parameter, the HTR method can also be applied. Some concrete applications including multi-solution problems, the forced Duffing equation and the Blasius equation are given.