On the application of the method of matching asymptotic expansions to a singular system of ordinary differential equations with a small parameter (Q404689)

The author studies the system of singularly perturbed ordinary differential equations \[ \epsilon^{v_i}\frac{d}{dt}U_i(t,\epsilon)-f_i\left(t,U_1(t,\epsilon),\dots, U_h(t,\epsilon)\right),\;i=1,\dots,h, \] subject to the initial conditions \[ U_i(0,\epsilon)=U_{i,0}, \] where \(f_i\) are smooth functions, \(\epsilon\) is a small positive parameter and \(v_i\) are the non-negative rational numbers satisfying \(\max \{v_i,\;i=1,\dots,h\}=1.\) Using the standard asymptotic expansion techniques, the authors construct an asymptotic expansion of the solution for the problem under consideration which provides its uniform approximation for \(\epsilon\rightarrow 0^+\) in some interval \([0, t_0].\)