(untitled) (Journal of Mathematics Mechanics and Computer Science, 2019)
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.