Asymptotic expansion of the solution of the initial value problem for a singularly perturbed ordinary differential equation |
| |
Authors: | O Yu Khachay |
| |
Institution: | 1.Ural State University,Yekaterinburg,Russia |
| |
Abstract: | We consider the Cauchy problem for the nonlinear differential equation $$\varepsilon \frac{{du}}{{dx}} = f(x,u),u(0,\varepsilon ) = R_0 ,$$ where ? > 0 is a small parameter, f( x, u) ∈ C ∞ (0, d] × ?), R 0 > 0, and the following conditions are satisfied: f( x, u) = x ? u p + O( x 2 + | xu| + | u| p+1) as x, u → 0, where p ∈ ? \ {1} f( x, 0) > 0 for x > 0; f u 2( x, u) < 0 for ( x, u) ∈ 0, d] × (0, + ∞); Σ 0 +∞ f u 2( x, u) du = ?∞. We construct three asymptotic expansions (external, internal, and intermediate) and prove that the matched asymptotic expansion approximates the solution uniformly on the entire interval 0, d]. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|