Relationship between the lower frequency spectrum of plates and networks of beams

Solution to the Dirichlet Problem of the Wave Equation on a Star Graph

“…Since then, there have been many papers studying the properties of differential equations on geometric graphs. By applying the method of the separation of variables to partial differential equations on a graph, we obtain a spectral problem of the Sturm-Liouville type on a graph, which has been studied by many authors (we refer the reader to the works of G. Lumer [3], J. von Below [4,5], S. Nicaise [6,7], and other authors [8][9][10][11]. In particular, we note the work of V.V.…”

Solution to the Dirichlet Problem of the Wave Equation on a Star Graph

“…Since then, there have been many papers studying the properties of differential equations on geometric graphs. By applying the method of the separation of variables to partial differential equations on a graph, we obtain a spectral problem of the Sturm-Liouville type on a graph, which has been studied by many authors (we refer the reader to the works of G. Lumer [3], J. von Below [4,5], S. Nicaise [6,7], and other authors [8][9][10][11]. In particular, we note the work of V.V.…”

Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks

“…The problems studied in this paper have some common properties with the above approaches since we will approach a two-dimensional problem by a family of continuous 1-dimensional problems but as each continuous 1-dimensional problem can be approximated by a discrete one, we also examine the limit of these discrete problems. The approximation of the low frequency spectrum of such problems was performed in [13,12] (see also [19] for the plate problem), but to our best knowledge the approximation of the boundary value problem itself was not yet performed. Hence our goal is to fill this gap and to show that indeed the solutions of the continuous and discrete one-dimensional problems converge to the solution of the two-dimensional problem.…”

Solution to the Dirichlet Problem of the Wave Equation on a Star Graph

“…Since then, there have been many papers studying the properties of differential equations on geometric graphs. By applying the method of the separation of variables to partial differential equations on a graph, we obtain a spectral problem of the Sturm-Liouville type on a graph, which has been studied by many authors (we refer the reader to the works of G. Lumer [3], J. von Below [4,5], S. Nicaise [6,7], and other authors [8][9][10][11]. In particular, we note the work of V.V.…”

Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks

“…The problems studied in this paper have some common properties with the above approaches since we will approach a two-dimensional problem by a family of continuous 1-dimensional problems but as each continuous 1-dimensional problem can be approximated by a discrete one, we also examine the limit of these discrete problems. The approximation of the low frequency spectrum of such problems was performed in [13,12] (see also [19] for the plate problem), but to our best knowledge the approximation of the boundary value problem itself was not yet performed. Hence our goal is to fill this gap and to show that indeed the solutions of the continuous and discrete one-dimensional problems converge to the solution of the two-dimensional problem.…”

Solution to the Dirichlet Problem of the Wave Equation on a Star Graph

“…Since then, there have been many papers studying the properties of differential equations on geometric graphs. By applying the method of the separation of variables to partial differential equations on a graph, we obtain a spectral problem of the Sturm-Liouville type on a graph, which has been studied by many authors (we refer the reader to the works of G. Lumer [3], J. von Below [4,5], S. Nicaise [6,7], and other authors [8][9][10][11]. In particular, we note the work of V.V.…”

Approximation of the two-dimensional Dirichlet problem by continuous and discrete problems on one-dimensional networks

“…The problems studied in this paper have some common properties with the above approaches since we will approach a two-dimensional problem by a family of continuous 1-dimensional problems but as each continuous 1-dimensional problem can be approximated by a discrete one, we also examine the limit of these discrete problems. The approximation of the low frequency spectrum of such problems was performed in [13,12] (see also [19] for the plate problem), but to our best knowledge the approximation of the boundary value problem itself was not yet performed. Hence our goal is to fill this gap and to show that indeed the solutions of the continuous and discrete one-dimensional problems converge to the solution of the two-dimensional problem.…”