# 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.

what is the condition of a consumer behaviour in the equilibrium under the theory of consumer behaviour
what is equilibrium
Sahr
A point where quantity demand & supply meets called equilibrium
Hasham
a state is said to be equilibrium when there is no tendency of movement.
Nibedita
Pls @Nibedita am confused
Prince
The state of balance achieved by an end user of products that refers to the amount of goods and services they can purchase given their present level of income and the current level of prices. Consumer equilibrium allows a consumer to obtain the most satisfaction possible from their income.
Okwori
where is the calculations?
what are the two conditions for aconsumer to be in the equilibrium under the theory of consumer behaviour in
Sahr
Economic equilibrium is a condition or state in which economic forces are balanced. In effect, economic variables remain unchanged from their equilibrium values in the absence of external influences. Economic equilibrium may also be defined as the point at which supply equals demand for a product,
vinay
Hello there, let's make a time to chat about econimics and its issues.
it's true
hie Sir /Madam l need help when it comes to Economics lm doing it for the first time
Thembelani
So, share your problems that you have in terms of economis and we will discuss on it.
DA
Basic Economic problems
Thembelani
what is the Basic Economic problem
Thembelani
what is the Basic Economic problem
scarcity
Rhaiymornd
a bit of explanation please its my first year doing Economics
Thembelani
rare, limited. economic agents eg You dube, the govt & the business entities wants to maximise their utility/satisfaction but because limited resource or scarcity of such resources they are unable to satisfy their needs.
ian
thank u Sir , l understand what you are saying now
Thembelani
limited resources; you wanna take the most benefits from the minimum resource.
DA
if u ar a fresher, eco has to 2 fundamental parts "micro & macro". micro(small) this is were the economc agents ar discussd, economc systms, dmand & supply, typs of market systms etc and the macro (big) part the elucidates the functns of central bank, typs of employmnt, functns of money & int trade.
ian
there is an old adage that says "a picture is worth a thousand words" economics is full of graphing so it requires on the side of the student to master the art of keeping information in form graphs.
ian
oky Sir
Thembelani
scarcity becomes the fundamental problem of economics because of limited resources, when we take an individual, he or she has many wants, thus unlimited wants but can never satisfy all but only few.
Rhaiymornd
now when we take a firm, a firm maybe willing to produce two or more product into the market but due to limited resources they only produce one. the same way if we take the government, he or she maybe willing to bring development either through infrastructures,
Rhaiymornd
that is when consumer decision making rule comes in
Olusegun
choice arises as a result of scarcity of resources
Olusegun
so if we look through, the individual, firm and government, their wants are unlimited but due limited resources, all of their wants cannot be satisfy. therefore scarcity can be term as limited in supply of resources. scarcity is not lack of resources but insufficient resources
Rhaiymornd
there is a marriage with the following; scarcity, factors of production, opportunity cost curve (occ) or (ppc, ppf, tc) production possibility curve productn possibility frontier transformation curve. The OCC, PPC, PPF & TC explains the decisions made by householders, firms & the govt.
ian
opportunity cost also arises as a result of firm willing to produce a particular commodity but resources use in satisfying or producing such output is limited
Olusegun
wat ar those decisions? the most important is WHY nations economise tht is if they hav abundancy of factors of productn eg land, labour & entreprise? now since all of us have unlimited needs against few resourcs PPC, PPF, TC, OCC walks in to make wise allocatn of resources.
ian
how do those decisions made? eg by economic agents; a. Household (You) - if u have R10 & wish to buy a book & a pen & realise that both commodities seĺl at the same price which of the two (2) can u buy (necesity) and which one can u forgo (not all tht important).
ian
b. firms - they allocat mo resourcs to all thoz commoditz tht they think will yield mo profit. c. Govt - if the govt SA was to come in yo area which 1 would u think they can consider first tht can benefit the majority & the minority. So instead of building football stadium they construct a hospital.
ian
if the SA govt had enough resources they would have built both the stadium and the hospital but because of scarce in terms of resources they had to forgo the construction the stadium to build a hospital which is necessary for the majority to benefit.
ian
Opportunity cost well broken down..
Andres
opportunity cost means the lose of other alternatives when the alternative is chosen
is the benefits that you loose by not selecting a certain alternative.
EDWINY
individual wants maybe unlimited, but means to satisfy them are limited there one has to forgo some alternative in order to acquire other alternative and it must according priority, that is when scale of preference set in for individuals to make choice
Rhaiymornd
hello everyone
Aliyu
Next best alternative forgiven
Shoaib
demand is the amount of goods and services that consumer is willing and able to purchase at a particular prices over given period of time
yep
Abraham
what's demand?
What customers want the most...
Abraham
not only what customers wants, want is just mere desire but demand is backed by purchasing power, ability and willingness
Rhaiymornd
thanks
Abraham
What's opportunity cost?
Abraham
what are the differences between demand and supply
who is called lender of the last resort
Hi
Linda
hlw
Karishma
Central bank
Majeed
hy
Karishma
Hello
Majeed
hy
Karishma
How are you
Majeed
Am gud
Linda
fine
Karishma
Am gud
Linda
hello
Chandra
Well! what's going on
Majeed
r u study in economics
Karishma
anybody there?
Chandra
r u study in economics
Karishma
the central bank
Sessay
Majeed
hey
neha
yes
Abigail
Yesss
Majeed
ok
Karishma
hey
Doctor
yh
Abigail
more questions
Sessay
how ar you
Doctor
split the price effect into income effect and substitution effect
Karishma
fine and u
Abigail
Hi
Godwin
hi
Hey, I am new here. Hope, discussion on Economics will clear our concepts more.
yasir
yes
Abigail
do u speak hindi or english
Karishma
how to consumer equlibrium through ic
Karishma
consumer equilibrium demand equals supply
Kenneth
the consumer is in equilibrium when the indifference curve is tangential to the budget line. or when the BL and IC intersect
Sessay
reasons indifference curve slopes downwards?
Kenneth
fine Abby any good,
Doctor
ur lost
Doctor
hey. im new year. economics teacher how we can discuss some thing interesting.
EDWINY
which one
Doctor
what do u understand the concept of poverty cycle.
EDWINY
hey
Ebong
I'm New here
Ebong
hi
ian
just new here guy's and also an Economics fresher of Kogi State University Anyigba
nelson
wxup
Ayegba
who can tell the laboratory of economic?
Amara
, Dennis Weissman Associates, LLC Laboratory Economics is the monthly business newsletter that gets behind the headlines and press releases.
Ayegba
sooo teah me what an LLC
Emmanuel
what's the topic
economic systems
gracious
hello
Antonio
market
aba
hello where can I find the diagrams
Manu
Hello I am totally out ,I am not understanding why we are here. can someone help me out?
Amara
why Economic is not a pure science can someone help me out
Mohamed
because economics like science put forth a some hypotheses and then do experiments to prove them
Anwesh
but these experiments are not completely controlled
Anwesh
Hello
Comfort
hey
suraj
hi people can you help me out on "demand and supply"
Milton
Am not understanding can someone enlighten me pls
Bertilla
hi people can you help me out on "demand and supply"
Sessay
hello. if Mr.Patrick's income is #900.00 while that of Mr.Shodawe is #1300.00 if Mr.Patrick and Shodowe pay #90.00 and #130.00 as taxes,the tax system is?
Benjamin
Benjamin
regressive tax system
shaikh
OK thanks
Benjamin
Isn't this called proportional tax rate because the rate stays the same - 10%? Tell me if I'm wrong
Ioan
Supply is perfectly elastic and demand increases.
whose there
Waseem
show the demand curve
it slopes downward from left to right
Ama
how resources are allocated in a free economy
explain how discriminating Monopoly increase profits
Charlotte
factors responsible for the emergence of monopoly situation
total output produced by a country over a given period of time .... can someone give me the term plz
GDP
Anjorin
thaks man
TMM
Woman. 👍👍
Anjorin
gross domestic products
janet
GDP
Bertilla
GDP
Prof
GDP
Bertilla
gross domestic product
gross domestic product
GDp
Mohamed
gdp
agboola
what is recession pertaining to GDP
Prince
what is recession pertaining to GDP
Prince
what is recession
Prince
law of demand and supply
Zakariyah
All thing been equal
Temple
no i think recession is pertaining to GNP
owolabi
gross national production
Abraham
what is embago
Peter
all things being equal
Raphael
embargo restriction on trade by government of a country
owolabi
an official ban on trade or other commercial activity with a particular country.
Ayegba
Embargo.....an order by a common carrier or publ regulatory agency prohibiting or restric freight transportation
Ayegba
it's a complete band on important n export
Bertilla
who is Really Good in Mathematics. put me Through. Let Do Group Reading; am Preparing to Write Jamb this Year.
Andrew
Hi. C+I+G+x-m
Shoaib
Gdp aggregate demand are bit same
Shoaib
Dpd = c+i+g+(x-m) is aggregate demand
Shoaib
Gdp=c+i+g+(x-m) is aggregate demand
Shoaib
GDP= c+i+s+g+(x-m)
Saboor
Topic ended waiting for next topic
Shoaib
What is Terms of Trade (TOT)?
At what point a Terms of Trade can be favorable?
Define Expansion path
Girma
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
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
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