Geodesic
Expansion in eigenfunctions of the Sturm--Liouville problem on a graph bundle
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2008), pp. 50-62

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In this paper we consider the applicability of the Fourier method for partial differential equations on spatial grids (we choose a bundle graph as a model). This leads to an important problem, namely, to the expansion of a given function in eigenfunctions of the corresponding Sturm–Liouville problem on a grid. We study a model problem which describes a symmetric case, when one considers physically identical one-dimensional continuums on the bundle graph. Such problems arise, for example, in the modeling of oscillating processes of an elastic mast with supporting elastic ties.
Keywords: boundary-value problem on a graph, eigenfunctions, the Green function, completeness of system of eigenfunctions, expansion in eigenfunctions. 
@article{IVM_2008_3_a4,
     author = {V. V. Provotorov},
     title = {Expansion in eigenfunctions of the {Sturm--Liouville} problem on a graph bundle},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {50--62},
     publisher = {mathdoc},
     number = {3},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2008_3_a4/}
}
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V. V. Provotorov. Expansion in eigenfunctions of the Sturm--Liouville problem on a graph bundle. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2008), pp. 50-62. http://geodesic.mathdoc.fr/item/IVM_2008_3_a4/