Renormalization Group Approach To Boundary Layer Problems

The boundary layer problems is originally found in the peculiar phenomenon of high-speed flow through solid interface in hydrodynamics,and have important applica-tions in many fields such as hydrodynamics,aerodynamics and atmospheric oceanog-raphy.Since Prandtl established the boundary layer theory in 1905,many effective methods and techniques have been developed,such as the matched asymptotic expan-sion method,the WKB method and the multi-scales method.In recent years,the renormalization group method developed by Chen et al has been successfully applied to many singular perturbation problems.Now it has become an important tool for obtaining uniformly effective asymptotic solutions of singular perturbation problems.In this paper,several singularly perturbed boundary value problems are studied by renormalization group method.Firstly,we propose a modified renormalization technique based on the group properties of flows of autonomous differential equations,then we apply it to study several kinds of boundary value problems.The main results are as follows.1.The boundary layer problem of second order semilinear differential equation is studied.By using the modified renormalization method,the first order asymptotic solution of the following boundary value problem of second order semilinear differential equation is obtained(?)where ?>0 is a small parameter,?,??are constants,a(x)?Cr[0,1],b(x,y)?Cr 0,1(r? 2),for x ?[0,1],a(x)>0.Furthermore,the asymptotic solutions for three special cases are given and compared with the asymptotic solutions obtained by the matched asymptotic expansion method in terms of rigorous analysis and numerical value.The results show that the asymptotic solutions obtained by the singular renor-malization group method are consistent with those obtained by the matched asymptotic expansion method.2.We studied the following singularly perturbed initial-boundary value problems for quasilinear delay differential equations(?)and singular perturbation initial-boundary value problems for convection-diffusion e-quations(?)In order to solve the problem,the delay differential equation is decomposed into left and right boundary value problems without delay.Then,the asymptotic solutions of left and right problems are constructed by renormalization group method.At last,the left and right solutions are connected by smooth joint condition,and the asymptotic solutions of the original problem are obtained.