English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

A spectral problem is considered in a thin $3D$ graph-like junction thatconsists of three thin curvilinear cylinders that are joined through a domain(node) of the diameter $\mathcal{O}(\varepsilon),$ where $\varepsilon$ is asmall parameter. A concentrated mass with the density $\varepsilon^{-\alpha}$$(\alpha \ge 0)$ is located in the node. The asymptotic behaviour of theeigenvalues and eigenfunctions is studied as $\varepsilon \to 0,$ i.e. when thethin junction is shrunk into a graph. There are five qualitatively differentcases in the asymptotic behaviour $(\varepsilon \to 0)$ of the eigenelementsdepending on the value of the parameter $\alpha.$ In this paper three cases areconsidered, namely, $\alpha =0,$ \ $\alpha \in (0, 1),$ and $\alpha =1.$ Usingmultiscale analysis, asymptotic approximations for eigenvalues andeigenfunctions are constructed and justified with a predetermined accuracy withrespect to the degree of $\varepsilon.$ For irrational $\alpha \in (0, 1),$ anew kind of asymptotic expansions is introduced. These approximations show howto account the influence of local geometric inhomogeneity of the node and theconcentrated mass in the corresponding limit spectral problems on the graph fordifferent values of the parameter $\alpha.$

analysis and applications

Asymptotic approximations for eigenvalues and eigenfunctions of a  spectral problem in a thin graph-like junction with a concentrated mass in  the node