Asymptotic Expansion of the Solution of a Linear Parabolic Boundary-Value Problem in a Thin Starlike Joint

Abstract

We consider a linear parabolic boundary-value problem in a thin 3D starlike joint that consists of a finite number of thin curvilinear cylinders connected through a domain (node) with diameter O(ε). We develop a procedure for the construction of the complete asymptotic expansion of the solution as ε → 0, i.e., in the case where the starlike joint is transformed into a graph. By using the method of matching of the asymptotic series, we deduce the limit problem (ε = 0) on a graph with the corresponding Kirchhoff-type conjugation conditions at the vertex. We also prove asymptotic estimates, which enable us to trace the influence of the geometric shape of the node and the physical processes running in the node on the global asymptotic behavior of the solution.