Asymptotic Solutions Of Several Differential Equations With Alternating Stability

The asymptotic solutions of several differential equations with alternating stabil-ity is studied in the paper.The alternating stability problem refers to the problem of the solution of degenerate equations intersecting,and its degenerate manifold is non-hyperbolic at the intersection point.The problem differs from the general one and it's more complicated.So,stability alternation problems have recently become a research hotspot.The first chapter mainly reviews the process and status of the singular perturbation theory,summarizes the main research of the article,introduces the innovations,and gives the directions for subsequent research.The second chapter of this paper mainly studies the stability of a class of second-order singularly perturbed differential equations with Dirichlet boundary value condi-tions.First and foremost,we divide the problem into two sub-problems,and use the boundary layer function method to construct the asymptotic solutions of the sub-problems respectively.For the left problem,as the solutions of the degenerate equations inter-sect,there will be a situation of alternating stability.We use the regularization method to deal with it,and then use the differential inequality method to prove the existence of the solution.The right problem is handled in the same way.Finally,through numerical simulation,the asymptotic solution image corresponding to different small parameters in the specific problem is obtained.The third chapter of this paper is an extended research based on the second chap-ter,studying the stability of a class of second-order singularly perturbed differential equations with internal layers.Firstly,we used the boundary layer function method to construct the formal asymptotic solution of the sub-problems.For the stability al-ternation situation that appears in the left problem,we use an improved regularization method to deal with it to improve the accuracy,and then use the differential inequality method to prove the existence of the solution to the sub-problems.Afterwards,the seam method is used to smoothly seam the discontinuous points to prove the existence of the solution on the entire interval and give the residual estimate.Finally,the entire problem is perfected by the numerical simulation of the specific problem.