Green's function of differential operators on a star-shaped graph with common boundary conditions

Abstract

Differential equations on graphs are one of the new sections of the theory of differential equations
and their fundamental concepts when analyzing models of a wide variety of problems in natural
science. It also arises when analyzing processes in complex systems, allowing as a set of onedimensional
continuum that interact only through the ends. The differential operator on graphs
is currently actively studying by mathematics and is found in many different applications, for
example, chemical kinetics, chemical technology, quantum mechanics, nanotechnology, biology,
organic chemistry, Markov processes, etc. In this paper, we construct the Green function of a
differential operator on a star shaped graph with common boundary conditions. In this paper, a
star shaped graph is a tree with one internal vertex and m leaves. Standard Kirchhoff conditions
are used at the interior vertices and mixed conditions at the boundary vertices. The edges of the
graph is a one-dimensional smooth regular manifold (curve). The top of the graph is a point.
The applicability of the results of this study is high both in theoretical terms - the development of
research in the theory of differential equations with memory on graphs, and in terms of applications
to biological processes, in particular neurobiology, nanotechnology, in the chemical and petroleum
industries.