Asymptotic analysis of some classes of ordinary differential equations with large high-frequency terms (Q844561)

In this paper, three main results are obtained. The first result consists in the construction and the justification of a complete asymptotic expansion of the solution of the Cauchy problem for a linear differential equation of arbitrary order \(n\) with high-frequency terms proportional to the powers \(\omega^{\frac{k}{2}}\), \(k\leq n\). More precisely, the problem under consideration is: \[ x^{(n)}+\sum_{k=1}^{[\frac{n}{2}]}A_{k0}(t)x^{(n-k)}+\sum_{k=[\frac{n}{2}]+1}^n \sum_{j=0}^{2k-n}\omega^{\frac{j}{2}}A_{kj}(t,\omega t)x^{(n-k)}=\sum_{k=0}^m \omega^{\frac{k}{2}}b_k(t,\omega t),\quad t\in[0,T],\tag{1} \] \[ x(0)=a_0,{\dot{x}}(0)=a_1,\dots,x^{(n-1)}(0)=a_{n-1}. \tag{2} \] The second result pertains to a class of systems of \(n\) linear first-order differential equations with \(n\) unknown quantities. The unknowns \(x_1,x_2,\dots,x_n\) themselves are vector-valued functions. The justification of the result for the system of \(n\) first-order equations is based on the reduction to the Cauchy problem of the type (1), (2) and the application of Theorem 1. Finally, attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers \(\omega^d\). For such equations with \(d\in(\frac{1}{2},1]\), conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.