An Asymptotic Expansion Method For Solving A Class Of Differential Equations With Local Periodic Structure

Due to the multi-scale characteristics of most composite materials,multi-scale methods are widely used in the field of composite materials.At present,multi-scale method has not only become a very important applied mathematical method in the field of differential equation,but also has a large number of researchers engaged in the application of multi-scale method in various fields such as engineering,mechanics and chemistry.The main contents of this paper are as follows:firstly,the two-point boundary value problem with local periodic structure is solved by using the multi-scale asymptotic expansion method,and the theoretical analysis of this method is given in detail.On the basis of this,a numerical simulation method based on the finite difference method is proposed,and the results of the proposed method are tested with an example,which shows that the method is effective and has some advanced.Secondly,in this paper,the two-point boundary value problem with three scales is studied deeply,and the corresponding multi-scaleasymptotic expansion method is proposed,and the theoretical analysis of this method is given in detail.On the basis of this,a numerical simulation method based on the finite difference method is proposed,and the results of the proposed method are tested with an example,which shows that the method is effective and has some advanced.In this paper,a kind of heat conduction problem with small periodic structure with one-dimensional space direction is studied in depth,and a corresponding multi-scale asymptotic expansion method is given for this problem,and the theory of this method is analyzed in depth.On this basis,the numerical simulation algorithm of the problem is given by combining the finite difference method.The effectiveness of the proposed method is verified by an example,The validity of this method is illustrated and it is advanced.The above work in this paper has certain significance for the numerical simulation of composite materials.In the end,this paper looks forward to the future work.