The network wave equation

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While his paper focuses on the heat equation on networks, given Neumann boundary conditions instead of Dirichlet, the eigenvalues of the heat equation are given to be constant $\mu$ such that

$\frac{{\partial }^{2}u}{\partial {x}^{2}}=\mu u$

Since $\mu$ has to be less than or equal to zero in order for a solution to the heat equation, the eigenvalues of the heat equation are the squares of the eigenvalues of the wave equation

$\frac{{\partial }^{2}u}{\partial {x}^{2}}={\lambda }^{2}u$

Therefore, taking the square roots of the eigenvalues $\mu$ , we can recover the eigenvalues of the wave equation on our network. For the rest of our stated examples, the eigenvalues mentioned are assumed to be eigenvalues of the wave equation.

Von Below studies the network heat equation with Neumann conditions at the endpoints

$\frac{\partial {u}_{j}}{\partial t}={a}_{j}\frac{{\partial }^{2}{u}_{j}}{\partial {x}_{j}^{2}}$

where ${u}_{j}$ is the temperature of the $j$ th edge, ${a}_{j}$ is the positive diffusion coefficient of the $j$ th edge, and ${x}_{j}\in \left[0,{\ell }_{j}\right]$ is position on the $j$ th edge, which has length ${\ell }_{j}$ . The notation is the same as the notation with which the network wave equation was described.

By defining simply a set of nodes and connections between them, Von Below calculates the eigenvalues and eigenmodes of a network with the help of spectral graph theory (a study of the connections between a graph and the eigenvalues of its adjacency matrix). The eigenvalues of the heat equation on networks of strings progress according to several different patterns, some of which are common to every network, some of which are network-specific and dependent on the eigenvalues of the network graph's row-normalized adjacency matrix.

For nonspecific parameter values, the best we can hope to do is to solve a transcendental equation for our eigenvalues. For specific parameter values, such as when ${a}_{j}={\ell }_{j}^{2}$ , we can solve analytically for both the eigenfunctions and eigenvalues of the heat equation, breaking both down into cases. There are, in general, 5 total classes into which eigenpairs of the heat equation on a network can be organized, with each network having at most 4 classes applicable to it.

For each one of these classes mentioned, there is an infinite number of eigenvalues associated with it. Organizing the progression by $k$ instead of by magnitude gives more insight into the nonlinear progression of our network eigenvalues - there is a clear bifurcating/splitting patten of the eigenvalues as $k$ increases. As the progression of eigenvalues splits into two separate patterns, each pattern is linear with different rates.

The fundamental difference here is that the eigenvalues for the one dimensional transverse network wave equation progress with two different rates. While there are two distinct sets of eigenvalues for our three dimensional string, to have eigenvalues progressing at different rates for one dimensional transverse string motion is atypical. This isn't a complete explanation for the irregular progression of the eigenvalues for the three dimensional wave equation; however, it does offer some clues as to the nonlinear ordering of those eigenvalues.

Acknowledgments

Thanks to Jeremy Morell, whose work I built upon, as well as Robert Likamwa, who helped nail down a lot of details concerning visualization and graphics. Thanks to Dr. Cox and Dr. Embree for their guidance their trust that I could actually do something with the project they gave me.

This work was partially supported by NSF DMS Grant 0240058

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