Asymptotic Solutions To Nonlinear Differential Equations Based On Renormalization Method

In 1990's,by using renormalization group(RG)method,Goldenfeld et al obtained the global approximate solutions to many important nonlinear differential equations,such as Mathieu equation,Barenblatt's equation,Modified Porous Medium equation and tur-bulent energy balance equation and so on.The results showed that the RG method was more efficient and perhaps more exact than the usual perturbation methods for asymp-totic analysis because it needn't perform asymptotic matching.The RG method has unified some of the classic methods in singular perturbation theory,such as Scaling func-tion Method,Matching method,Multi-scale expansion method,WKB method and so on.However,RG method is formal only which has no a solid mathematical basis.Kunihiro has given a geometrical interpretation to RG method based on the classical theory of en-velop in differential geometry,in which some unnatural assumptions were included.Until recently,Liu proposed the Taylor Renormalization method(TR,for simplicity)which can derived the standard RG method and the Kunihiro's geometrical interpretation.Then,the RG method was formulated a rigorous mathematical foundation by the traditional Taylor series.The biggest advantage of TR method is that the secular terms can be automatically eliminated instead of making a lot of effort to deal with them as in usual perturbation theory.But for some equations,RG method or TR method can't work.To overcome the shortcomings of the RG method,Liu proposed a more powerful method named homotopy renormalization method(HTR)further by combining the homotopy deformation and the RG method.TR method and HTR method are not only having a strictly mathematical foundation,they are also so easy to understand and so convenient to operat that we can obtain the global asymptotic solutions to nonlinear differential equations in mathematics,physics and dynamics by using them.This paper studies several significant nonlinear problems arising in biology,engi-neering mechanics and fluid dynamics.We will obtain the global asymptotic analytical solutions to these problems by using TR method and HTR method.Different means are adopt to deal with these questions so that the solutions we have obtained are as accurate as possible.Some solutions given in the references are contained in our research results,so our results are more general.Especially,in dealing with the nonlinear boundary layer problem,we choose initial homotopy equation by making the best of boundary condi-tions to avoid the defect of the divergency.With the simulation results,we find that our asymptotic solution agrees very well with numerical ones,that shows the superiori-ty of the method and the effectiveness of the results.The chapters are arranged as follows:Chapter 1 is the introduction to the background and the current situation of the study and the main content of this thesis.In Chapter 2 and Chapter 3,we study the Damped Fisher problem arising in biology and the bar vibration problem with perturbation term arising in engineering mechanics separately.The global asymptotic solutions have been obtained by using TR method.One can find the form of the image changes with different parameters.Compared with other perturbation methods,our method is more concise and effective.In Chapter 4,we structure a suitable homotopy equation to the Boussinesq equation firstly,then we get the global asymptotic solution to the Eqs.(??)by making use of HTR method.In Chapter 5,we deal with the nonlinear Schrodinger equation with Cubic-Quintic nonlinearities(?)where,? is a function to be determined with respect to x and t,v(x,t),g3(x,t)and g5(x,t)are functions with both space and time dependence,v(x,t)is the trapping poten-tial,the latter two control the cubic and quintic interactions respectively.In Chapter 5,We combine TR method and the complete discrimination system for polynomial method to get the classification to the asymptotic solutions.Firstly,we get the global asymptotic solutions to Ermakov-Pinney equation.Secondly,we construct the classification to the asymptotic solutions to equation(2).The solutions given by Avelar et al in the references are contained in our research results,so our results are more general.In Chaper 6,we study three boundary layer problems of infinite rotating disk.Thecharacteristics of these problems are nonlinear,high dimensional and with mixed bound-ary conditions.We construct the global asymptotic solutions to these complex problems.In part 1,We study the Schlichting boundary layer problem.Scholars have been keen on the study on obtaining the physical solution to this problem.In this part,by using HTR method,we construct proper homotopy equation to get the physical solution to Schlichting boundary layer problem.In part 2 of this Chapter,we obtain the global ap-proximate solutions to Rasmussen's problem of flow between two infinite rotating disks.Rasmussen gave the analytic approximations solutions to equation(??)in the condition of high Reynolds number.But the solution is too complex and inaccurate.In this part,by using HTR method,we obtain the more accurate global approximate solution.At the end of this part,we make the physical explanations and the graphical representations of the solution with various parameters.According to the figures,we can analysis the periodicity and asymptotic property of the solution.In part 3 of the Chapter 6,we con-sider the reduced Von Karman problem.By using HTR method,we obtain the global approximate solution to the reduced Von Karman problem.With numerical analysis,we find our asymptotic solution has a high precision and the absolute errors are less than 0.03,that is our asymptotic solution agrees very well with numerical result,it indicats our method is more superior and our result is more practical.