Vol 240, No 3 (2019)

We consider a boundary-value problem for a system of two ordinary differential equations of the second order with different powers of the small parameter as coefficients of the second derivatives in both equations. One equation of the degenerate system possesses a double root. This leads to qualitative differences between the asymptotics of the boundary-layer solution of the analyzed system and the known asymptotics in the case of simple roots of the equations of degenerate system, namely, the structure of the boundary-layer series changes, the boundary layers are multizone, and the standard algorithm used for the construction boundary-layer functions becomes inapplicable and requires significant modifications.

We consider a double singular perturbation of a boundary-value problem for a nonlinear system of ordinary differential equations. For a formal asymptotic solution constructed by the method of boundary functions and generalized inverse matrices and projectors, we prove an asymptotic property of the formal series.

We consider spectral problems for the Schrödinger operator with polynomial potentials in ℝ^K, K ≥ 2. By using a functional-discrete (FD-)method and the Maple computer algebra system, we determine a series of exact least eigenvalues for the potentials of special form. In the case where the traditional FD-method is divergent (the degree of the polynomial potential exceeds 2 at least in one variable), we propose a modification of the method, which proves to be quite efficient for the class of problems under consideration. The obtained theoretical results are illustrated by numerical examples.

By using the \( {Q}_5^{\ast } \)-representation of numbers

\( \left[0,1\right]\ni x={\beta}_{\alpha_1(x)1}+\sum \limits_{k=2}^{\infty}\left({\beta}_{\alpha_k(x)k}\prod \limits_{j=1}^{k-1}{q}_{\alpha_j(x)j}\right)={\Delta}_{\alpha_1(x){\alpha}_2(x)\dots {\alpha}_k(x)\dots}^{Q_5^{\ast }} \)

determined by the quinary alphabet A₅ ≡ {0, 1, 2, 3, 4} and an infinite stochastic matrix ‖q_ik‖, i ∈ A₅, k ∈ N, with positive elements (q_0k + q_1k + q_2k + q_3k + q_4k = 1) such that \( {\prod}_{k=1}^{\infty}\underset{i}{\max}\left\{{q}_{ik}\right\}=0 \) and β_0k = 0, β_{i + 1, k} = β_ik + q_ik, \( i=\overline{0,4} \), we define a continuous Cantor-type function by the equality

\( f\left({\Delta}_{\alpha_1\dots {\alpha}_k\dots}^{Q_5^{\ast }}\right)={\delta}_{\alpha_1(x)1}+\sum \limits_{k=2}^{\infty}\left({\delta}_{\alpha_k(x)k}\sum \limits_{j=1}^{k-1}{g}_{\alpha_j(x)j}\right)\equiv {\Delta}_{\alpha_1(x)\dots {\alpha}_k(x)\dots}^G, \)

where δ_0n = 0, \( {\delta}_{1n}=\frac{2+{\varepsilon}_n}{4} \), \( {\delta}_{2n}=\frac{2}{4}={\delta}_{3n} \), and \( {\delta}_{4n}=\frac{2-{\varepsilon}_n}{4} \), i.e., δ_{i + 1, n} = δ_in + g_in, n ∈ N, and (ε_n) is a given sequence of real numbers such that 0 ≤ ε_n ≤ 1. We prove that this function is well defined and continuous. Moreover, it does not have intervals of monotonicity, except the intervals where it is constant. A criterion of bounded variation of the function is also established. We are especially interested in the problem of level sets of the function and in the topological and metric properties of the images of Cantor-type sets.