Some Research On Singularly Perturbed Problems Not Meeting The Tikhonov Conditions

Singularly perturbed problems arise in a wide range of mechanics, physics, chemical kinetics, engineering technology and many other problems. Tikhonov theorem is a ground-breaking work in singular perturbation theory and provides a theoretical basis for a large class of singularly perturbed prob-lems in computer programming and simulation. However, with the deepening of research, there are a growing number of singularly perturbed problems that can not meet the conditions of Tikhonov theorem.This paper consists of two parts. The first part considers the Dirich-let boundary value problem that does not meet the differential condition of the Tikhonov Theorem. The second part considers the singularly perturbed problem that does not meet the stability condition of the Tikhonov Theo-rem and discusses the Dirichlet boundary value problem in which eigenvalue identically equal to zero caused by two different equations.In chapter 1, we review the development of singular perturbation and introduce some basic concepts and theorems relevant to our studyIn chapter 2,3,4, we focus on the second order semi-linear system Dirich-let problem, quasi-linear system Dirichlet problem and weakly nonlinear sys-tem Dirichlet problem with discontinuous terms. The asymptotic solutions of these problems are constructed. By sewing orbit smooth, the asymptotic solutions are proved to be uniformly effective in the whole interval and the so-lutions of these problems are proved to be existent and unique. The existence of inner layers is also shown. Numerical results are presented, as illustrations the theoretical results.In chapter 5, the Dirichlet boundary value problem in Tikhonov system with discontinuous terms and both fast variables and slow variables is con-sidered. The appropriate definite condition is given according to the stability condition and the asymptotic solution of the problem is constructed in two intervals. By sewing orbit smooth, the asymptotic solution of this problem is shown and proved to be uniformly effective in the whole interval. The estimation of the remainder is derived.In chapter 6, we consider the Dirichlet boundary value problem in which the eigenvalues in Tikhonov system are identically equal to zero caused by the first-derivative equals to zero. The formal asymptotic solution of the problem is constructed with boundary layer function method. By comparison theo-rem, algebraic decay of the boundary function is concluded. By the theory of differential inequality, we prove that the solution of the problem exists and estimate the remainder.In Chapter 7, we study the boundary problem of semi-linear function e2y"= (y-φ(t))m(y-φ{t))n.In four different case of the parity of m, n, we discuss the phase plane analysis, algebra decay rate of the zero-order bound-ary function, the construction of the asymptotic solution, the existence of the solution and the estimation of the remainder.