Singular perturbation of initial value problem for a nonlinear second order systems

Abstract

In this paper, the singular perturbation of initial value problem for nonlinear second order vector differential equations

$$\begin{gathered} \varepsilon ^\tau x'' = f(t,x,x',\varepsilon ) \hfill \\ x(0,\varepsilon ) = \alpha , x'(0,\varepsilon ) = \beta \hfill \\ \end{gathered}$$

is discussed, where r>0 is an arbitrary constant, ε>0 is a small parameter, x, f, a and β∈Rⁿ. Under suitable assumptions, by using the method of many-parameter expansion and the technique of diagonalization, the existence of the solution of perturbation problem is proved and its uniformly valid asymptotic expansion of higher order is derived.