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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

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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
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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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