Study On The Two Kinds Of Singular Perturbation Problems With Derivatives Of Fractional Order

Fractional derivative is the concept of extension of the integral order.And it has a wide range of real meaning,it can describe many phenomenon of the natural world.Singular perturbation theory and method for solving nonlinear problems is an important tool on the origin of celestial mechanics and nonlinear vibration problems,and its advantage is enough to give the right solution for structural analysis.So singular perturbation theory is receiving increasing international attention to the academic session.And it has become an increasingly widespread application,so as to form a more complete theory gradually.In this paper,We use the singular perturbation theory and method to study the two types of singular perturbation problems with fractional derivative.One is the asymptotic nature with the solution of small fractional damping oscillator.We have discussed the case of linear and nonlinear Duffing oscillator.First,we obtain the asymptotic solution for a small fractional linear damping oscillator by the multi-scale expansion method,and give a remainder estimated,the result is easy to be compared with the corresponding integral situation.Second we study the asymptotic nature of a nonlinear Duffing oscillator's solution by the method of averaging and multi-scale expansion method.The other is the initial value problem about the general,the non-autonomous singular perturbation of fractional differential equations.In the very weak conditions,we use boundary layer function of the original structure of the asymptotic solution of the problem and take advantage of the recent development of the theory of fractional differential inequality whose asymptotic behavior is estimated that an effective unanimously made.Finally,as an example,we make a study of fractional Logistic equation of the asymptotic solution. In conclusion,some of the results in this paper improve and generalize the well-known results,others are new.