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Renormalization group second-order approximation for singularly perturbed nonlinear ordinary differential equations. (English) Zbl 1404.34071
In this paper, using the renormalization group method, the authors derived an \(O(\varepsilon^2)\) approximation for the singularly perturbed Cauchy problem with an \(m-\)dimensional slow-variable \(\mathbf{u}_{\varepsilon}\) and a scalar fast-variable \(v_{\varepsilon}:\) \[ \frac{d\mathbf{u}_{\varepsilon}}{dt}=\mathbf{f}(\mathbf{u}_{\varepsilon},v_{\varepsilon}),\;t>0,\;\mathbf{u}_{\varepsilon}(0)=\mathbf{u}^0, \]
\[ \varepsilon\frac{dv_{\varepsilon}}{dt}=-av_{\varepsilon}+\Phi(\mathbf{u}_{\varepsilon},v_{\varepsilon}),\;t>0,\;v_{\varepsilon}(0)=v^0, \] where \(\mathbf{f}\) and \(\Phi\) are defined on \(\mathbb{R}^{m+1},\) \(m\geq 1\), take values, respectively, in \(\mathbb{R}^m\) and \(\mathbb{R}^1,\) and are \(C^2\) functions with bounded derivatives.
As a result, a Tikhonov-type quasi-stationary approximation of order \(\varepsilon^2\) on infinite time interval involving the variables of slow equation, the steady state behavior of the fast equation, and an initial layer term is obtained.

34E05 Asymptotic expansions of solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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