# 10.5 There and back again - an exploration of the liouville  (Page 2/10)

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The remainder of this document is divided as follows. In "Problem Setup" , we provide a short introduction to the forward and inverse problems under consideration. In "The Transformations" , we analyze indetail the transformation used to turn the wave equation into the Sturm-Liouville equation and describe a method for inverting it. In "Numerical Results" , we present the results of the main collection of numerical experiments we conducted, which use a particular Sturm-Liouville potentialfunction from [link] . A discussion of the errors in these results is presented in "Error Analysis: Recovering Eigenvalues" . "Corresponding q(t) and ρ(x) Functions in Closed Form" contains results for other functions in closed form we studied. Finally, we provide a brief description of the numerical methodswe used in "Numerical Methods" .

## The forward problem

We begin by modeling the behavior of a string of length $L$ with spatially-varying mass density $\rho \left(x\right)>0$ for $x\in \left[0,L\right]$ . For a long string experiencing small vibration, let $u\left(x,t\right)$ be the vertical displacement of the string at time $t\ge 0$ . This $u\left(x,t\right)$ satisfies the 1-D wave equation

$\rho \left(x\right){u}_{tt}\left(x,t\right)={u}_{xx}\left(x,t\right),\phantom{\rule{2.em}{0ex}}0\le x\le L,$

subject to the Dirichlet boundary conditions

$u\left(0,t\right)=u\left(L,t\right)=0,\phantom{\rule{2.em}{0ex}}t\ge 0$

where ${u}_{tt}\left(x,t\right)=\frac{{\partial }^{2}u}{\partial {t}^{2}}$ . Assuming the variables can be separated, we write $u\left(x,t\right)=\psi \left(t\right)y\left(x\right)$ and after making the appropriate substitutions, obtain

$\rho \left(x\right){\psi }^{\text{'}\text{'}}\left(t\right)y\left(x\right)=\psi \left(t\right){y}^{\text{'}\text{'}}\left(x\right),$

hence

$\frac{1}{\rho \left(x\right)}\frac{{y}^{\text{'}\text{'}}\left(x\right)}{y\left(x\right)}=\frac{{\psi }^{\text{'}\text{'}}\left(t\right)}{\psi \left(t\right)}.$

As the left side depends only on $x$ and the right side depends only on $t$ , we conclude that both sides must be equal to the same constant, which we denote by $-\lambda$ . Equating the left side with $-\lambda$ and rearranging yields

$-{y}^{\text{'}\text{'}}\left(x\right)=\lambda \rho \left(x\right)y\left(x\right)$

with boundary conditions $y\left(0\right)=y\left(L\right)=0$ . The constant $\lambda$ is said to be an eigenvalue of the boundary-value problem [link] - [link] , and $y\left(x\right)$ is its corresponding eigenfunction . Finding all $\lambda$ such that [link] has a solution satisfying the boundary conditions is called the forward eigenvalue problem , and the collection of all such $\lambda$ is called the problem's spectrum . Physically, the forward problem asks one to determine a string's natural frequencies from a description of its physicalcomposition.

The inverse problem will do the opposite, asking for information about the string from its eigenvalues.

## The inverse problem

In contrast to the forward problem, the inverse eigenvalue problem asks if one can recover information about a string's physical composition fromknowledge of its frequencies. Mathematically, this amounts to asking if we can determine the mass density function $\rho \left(x\right)$ given the spectrum of the problem [link] - [link] .

In determining physical properties of objects, the inverse eigenvalue problem is involved in answering various questions. For a vibrating string one question is, "What is the isospectral set , the mass density of all strings of a given length with a given set of frequencies?" The inverse eigenvalue problem could be used in designing a musical instrument that will produce a desired sound. Also, when a particular object is being designed and it is known that the object will be experiencing vibrations of certain frequencies, the inverse problem can be used to ensure the object will not resonate at these frequencies. Additionally, the inverse problem is involved in structural composition analysis. By collecting acoustic data for a structure, a bridge or building, for example, it can be determined if there are "cracks" or "faults" in the object's components. This allows for analysis of a structure without dismantling the structure.

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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