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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

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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