Singular Perturbation Problems For A Class Of Three Order Semilinear Equations

As is well known, by some basic laws, many problems arising from physics and engineers can be described by differential equations. But the most of differential equations we encountered can not get its exact solution, so people has to resort the numerical methods, asymptotic analysis and the combination of numerical methods and asymptotic analysis.A variety of physical phenomena can be described by differential equations together with boundary and/or initial conditions. When such problems are nondimensionalized, we arrive at problems containing one or more parameters. It is often the case that such parameters are small. For instance, let us consider a boundary value problem depending on a small parameterε. If the problem has the property that the equations which result from settingεto zero(called the reduced problem) are easily solved, then a solution, say f ( x ,ε), of such a problem can be constructed by means of a power series inε, i.e., where the first term f 0(t ) is the solution of the reduced problem withε= 0. When the result of this perturbation analysis approximates the solution of the original equations and, furthermore, it is uniformly valid in the range of the independent variable under consideration, such an approximation is said to be asymptotic.When the expansion (1) converges uniformly in t asε→0, it is called a regular perturbation problem. On the other hand, when f (t ,ε) does not have uniform limit in some regions o the independent variable t asε→0, the regular perturbation breaks down and f (t ,ε) is said to involve a singular perturbation. The regions in which regular perturbations break down are called regions of nonuniformity.It should be pointed out that expansions of the regular perturbations or straightforward expansion often break down and become nonuniformly valid. It is true to say that this is the rule rather than the exception.Typical examples of singular perturbation problems are classified as follows:1. Sources of nonuniformities appear in relation to an infinite domain, for example, the appearance of secular terms in nonlinear oscillations. In this case, the nonuniformity manifests itself as so-called secular terms, like t kcost and t ksint , which make the expansions unbounded as t tends to infinity. The name secular derives from the early days of celestial mechanics in which this problem first appeared. This type of singular perturbation problem can be classified as being"secular-type"problem.2. A small parameter multiplies the highest-order derivative term in a differential equations. In this case, the perturbation expansion cannot satisfy all the boundary and/or initial conditions, because the perturbed differential equations reduce their order in the perturbation expansions. Thus the straightforward expansion ceases to be valid in some boundary and/or initial layers. This type of problem may be classified as being"layer-type"problem.3. There is a change of type of a partial differential equations. In this case, the classification type of the perturbed equations changes from that of the original equations, nonuniformities might arise.4. The presence (or occurrence) of singularities.In this case, singularities that are not involved in the exact solution appear at a certain stage of the perturbation expansion. In general, such singularities become more intensified as we proceed to successively higher-order approximations.In order to eliminate such nonuniformities, or singularities, and to lead to uniformly valid approximate expressions, the straightforward perturbation procedure should be replaced by other perturbation techniques which are called singular perturbation theory. In these years, many classic methods have been developed, such as asymptotic matching method, multiple scales method, stretched coordinate, averaging, WKB methods and so on.In 1994, Goldenfeld, Oono and their collaborators first showed that the renormalization group equation can used for purely mathematical problems as to improving the global nature of the solutions of differential equations obtained in the perturbation theory. Since then, many people try to using the ideas of renormalization group method to study the singular perturbation problems.In this paper, we consider the following singular perturbation problems of three order semilinear equations Whereε> 0 is a smll parameter,α,β,γare some given constants. By using the renormalization group method, we construct an uniformly valid asymptotic expansion for (3)-(4).Theorem 1. Assume that f ( t ) is twice continuously differentiable and f ( 0 )< 0, g ( t ,y )is continuously about t and infinitely differentiable about , and let y (t ) be solution of the initial value problem (1.3)-(1.4), then A( 0 ) ,B(0 ) ,C(0 ) can be determined by (4),...