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

 Page 2 / 10

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.

#### Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
Idrissa Reply
hello
Sherica
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Sherica
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Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
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rolling four fair dice and getting an even number an all four dice
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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China
Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Cesar
I'm interested in nanotube
Uday
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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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
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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
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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