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Since it is the dynamic nature of a system that we want to model and understand, the simplest form will be considered. This will involve onestate variable and will give rise to so-called "exponential" growth.

Two examples

First consider a mathematical model of a bank savings account. Assume that there is an initial deposit but after that, no deposits orwithdrawals. The bank has an interest rate r i and service charge rate r s that are used to calculate the interest and service charge once each time period. If the net income (interest less service charge)is re-invested each time, and each time period is denoted by the integer n , the future amount of money could be calculated from

M ( n + 1 ) = M ( n ) + r i M ( n ) - r s M ( n )

A net growth rate r is defined as the difference

r = r i - r s

and this is further combined to define R by

R = ( r + 1 )

The basic model in [link] simplifies to give

M ( n + 1 ) = ( 1 + r i - r s ) M ( n )
= ( 1 + r ) M ( n )
M ( n + 1 ) = R M ( n ) .

This equation is called a first-order difference equation , and the solution M ( n ) is found in a fairly straightforward way. Consider the equation for the first fewvalues of n = 0 , 1 , 2 , . . .

M ( 1 ) = R M ( 0 )
M ( 2 ) = R M ( 1 ) = R 2 M ( 0 )
M ( 3 ) = R M ( 2 ) = R 3 M ( 0 )
· · ·
M ( n ) = M ( 0 ) R n

The solution to [link] is a geometric sequence that has an initial value of M ( 0 ) and increases as a function of n if R is greater than 1 ( r 0 ) , and decreases toward zero as a function of n if R is less than 1 ( r 0 ) . This makes intuitive sense. One's account grows rapidly with a high interest rate and low service charge rate, andwould decrease toward zero if the service charges exceeded the interest.

A second example involves the growth of a population that has no constraints. If we assume that the population is a continuous function of time p ( t ) , and that the birth rate r b and death rate r d are constants ( not functions of the population p ( t ) or time t ), then the rate of increase in population can be written

d p d t = ( r b - r d ) p

There are a number of assumptions behind this simple model, but we delay those considerations until later and examine the nature of the solution ofthis simple model. First, we define a net rate of growth

r = r b - r d

which gives

d p d t = r p

which is a first-order linear differential equation. If the value of the population at time equals zero is p o , then the solution of [link] is given by

p ( t ) = p o e r t p o = p ( 0 )

The population grows exponentially if r is positive (if r b r d ) and decays exponentially if r is negative ( r b r d ). The fact that [link] is a solution of [link] is easily verified by substitution. Note that in order to calculate future values ofpopulation, the result of the past as given by p ( 0 ) must be known. ( p ( t ) is a state variable and only one is necessary.)

Exponential and geometric growth

It is worth spending a bit of time considering the nature of the solution of the difference [link] and the differential [link] . First, note that the solutions of both increase at the same "rate". If we samplethe population function p ( t ) at intervals of T time units, a geometric number sequence results. Let p n be the samples of p ( t ) given by

p n = p ( n T ) n = 0 , 1 , 2 , . . .

This give for [link]

p n = p ( n T ) = p o e r n T = p o ( e r T ) n

which is the same as [link] if

R = e r T

This means that one can calculate samples of the exponential solution of differential equations exactly by solving the difference [link] if R is chosen by [link] . Since difference equations are easily implemented on a digital computer, this is an important result; unfortunately,however, it is exact only if the equations are linear. Note that if the time interval T is small, then the first two terms of the Taylor's series give

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Dynamics of social systems. OpenStax CNX. Aug 07, 2015 Download for free at https://legacy.cnx.org/content/col10587/1.9
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