An effective numerical solution of renormalization group equations;

The description of the critical theory,the definition of the renormalization group of the critical theory,the derivation and the sense of the renormalization group equation as well as the functional equation of the group are discussed.

The renormalization group equations for a period-tripling bifurcation in trimodal maps are discussed,and the numerical solutions of their universal functions and scaling factors are obtained.

The renormalization group equations for a periodtripling bifurcation in bimodal maps are discussed, and universal functions and scaling factors are solved.

A study is made on the property of a class of simple and accurate solutions to Feigenbaum functional equation, which is piecewisely and fractionally linear function.

By using the so called renormalization group method,we give a uniformly valid asymptotic expansions of the boundary value problems under consideration.

The present paper presents an uniformly valid asymptotic expansion for a class of singular perturbation boundary value problems via the renormalization group method.

The study using the renormalization group method shows that there is a self-organized criticality characterized by the critical cracking probability of the micro-cells in the process of the progressive cracking of the inner micro-cells of rock.