# The network wave equation

 Page 11 / 11

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

#### Questions & Answers

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Got questions? Join the online conversation and get instant answers!