Volume||4||Issue||09||September-2016||Pages-5922-5927||ISSN(e):2321-7545
Website: http://ijsae.in
DOI: http://dx.doi.org/10.18535/ijsre/v4i09.15
Growth in Literacy of India by Using Logistic Model and Divided Difference Formula
Authors
Ajay J.Patel , Dr.M.B.Prajapati.2
¹Assistant Professor, Department of Mathematics, Government Science College, Idar, India
²Professor&Head, Department of Mathematics Hemchandracharya North Gujarat University, Patan, India
Email- [email protected], [email protected]
ABSTRACT:
In world Population India demographics occupy second rank among all countries. If we consider literacy
rate of India it stand at 74.04% as per 2011 census, in which Biharstate with very low literacy rate 63.82%
and Kerala state has very good literacy rate 93.91 % among all states of India.Here we are interested to
derive a mathematical model, which give literacy rate of India. In this mathematical model we use Newton’s
divided difference formula andLagrange’s interpolation formula with logistic model approach.
Keywords: Mathematical model, Literacy rate, Newton’s divided difference formula, Logistic model,
Lagrange’s interpolation formula
1
INTRODUCTION
We know that development of the any country is depends on the literacy. Literacy benefits both
communities and individuals. The United Nations Educational Scientific and Cultural Organization
(UNESCO) define literacy as the “Ability to identify, understand, interpret, communicate, compute,create
and use printed and written materials associated with varying contests.
The numbers of illiterates and literates aged seven and above in India asper the provisional population totals
of census 2011 constitute 74.04 percent of the total population aged seven and above as compared to 64.83
percent in 2001
Literacy rate is define as
Crude literacy rate =[ (Number of literate persons)*100] /(Total literate persons).
Effective literacy rate = [(Number of literate persons aged 7 and above)*100] /(Total literate persons aged 7
and above).
We are used effective literacy rate and literacy rate have been used interchangeably.
In 1951 literacy rate was 18.33,in which females literacy rate was 8.86 and males literacy rate was
27.16,means there is a gap in literacy rate in male-female was 18.30. In 1961 gap in literacy rate was
25.05.Similarly, in 1971 gap was 23.98,in 1981 gap was 26.62,in 1991 gap was 24.84,in 2001 gap was
21.59.This shows there is big gap in literacy rate between male and female.
In 1901 crude literacy rate in India was 5.35, now in 2011 crude literacy rate is 64.32.
Table 1
t
k
K
dL/dt
Time in years
Literacy at time t = 0
Literacy rate at time t
Maximum growth rate
Literacy rate carrying capacity
Rate of change of literacy rate with
respect to time t
Ajay J.Patel , Dr.M.B.Prajapati. IJSRE Volume 4 Issue 9 September 2016
Page 5922
II MATERIALS AND METHODS:
Logistic Model
The basic Logistic differential equation is given as
dL/dt = k L(1 – L/K)
(1)
The logistic equation (1) can be solved for population L by taking integration both sides.
dL /L(1 – L/K) =
We get,
–
=∫
∫
dL /L(1 – L/K) = dLK/L(K – L) = dL/L + dL/(K –L)
+∫
–
=∫
∫
| = kt + C
ln| | –ln|
| = - kt -C
ln|
|
|=
K – L/L = A
where A =
or
, a constant
If t =0 and L = Lₒ , then
A = (K - Lₒ)/Lₒ
(2)
L = K /(1 +A
)
(3)
Newton‟s divided difference formula
If (
), ( , ),(
)…………(
) are given with[ , ], [
], ………[
] are unequal
interval then
y(X) = y( ) + y , )*( - ) + y , , )*( - )*( - )
+…………………………………….+y , , ,……, )* ( - )*( - )*( - )*………….*( )
.
Lagrange‟s interpolation formula
If (
), ( , ),(
)…………(
) are given with[ , ], [
], ………[
] are unequal
interval then
Newton‟s Forward difference formula
If (
), ( , ),(
)…………(
interval
) are given with[
,
], [
], ………[
] are equal
then
III MATHEMATICAL MODEL
The actual literacy rate of India from 1951 to 2011 is obtained from state of literacy-education for all in
India [1]
Table 2.
Notation of Year
time t
Actual literacy rate at time t
Notation for actual literacy
rate at time t
t=0
1951
18.33
L₀
t = 10
1961
28.3
L₁₀
t = 20
1971
34.45
L₂₀
Ajay J.Patel , Dr.M.B.Prajapati. IJSRE Volume 4 Issue 9 September 2016
Page 5923
t = 30
1981
43.57
L₃₀
t = 40
1991
52.21
L₄₀
t = 50
2001
64.83
L₅₀
t = 60
2011
74.04
L₆₀
The notation of literacy rate and time for the year 2012 to onwards be continued after the notation for year
2011 in similar manner.
Carrying capacity „K‟ of literacy rate we assume 100.
Value of „A‟ in equation (2) is obtained by carrying capacity and literacy rate of India for the year 1951.The
value of term is 4.45553737.
Now the term „k‟ in logistic equation (3) is constant and it is determined by literacy rate of India for the year
1951 and 1961.Estimated value of k = 0.0564518733.
For intend of literacy rate of India from year 1971 and onwards, the logistic equation (3) can simplified by
putting values of
, K, A.
Hence the logistic equation is
= K/(1 +A
) = 100/(1 + 4.45553737*
).
We have the literacy rate of 1991 to 2011 from [1] .Now literacy rate of 2012 onwards are assumed by
considering the entire factor affected in India,which is depends on literacy.
Table 3.
Time
(Year)
t Year
Actual/assumed literacy rate
Literacy rate by logistic Difference
model
(d)
40
1991
52.21
68.2205776
16.010578
45
1996
59.55835255
74.00419543
14.445843
50
2001
64.83
79.05834672
14.228347
60
2011
74.04
86.90931971
12.86932
65
2016
78.31023684
89.8002458
11.490009
70
2021
82
92.11069821
10.110698
80
2031
88
95.35582728
7.3558273
120
100
80
60
Actual/assumed literacy
rate
40
Literacy rate by logistic
model
20
0
1991 1996 2001 2011 2016 2021 2031
Fig.1 Difference between Logistic Model and Actual Literacy rate
Now we use Newton‟s divided difference formula to remove this difference between logistic model and
actual/assumed literacy rate
Ajay J.Patel , Dr.M.B.Prajapati. IJSRE Volume 4 Issue 9 September 2016
Page 5924
Table 4.
Time
t(Year)
Difference
(d)
First D.D
Second D.D
Third D.D.
Forth D.D.
Fifth D.D.
Sixth D.D.
40
16.0105776
-0.312946944
0.026944771
-0.00165525
5.98692E-05
-1.16224E-06
1.19222E-10
45
14.4458429
-0.04349923
-0.006160231
-0.00015852
2.50021E-05
-1.15747E-06
50
14.2283467
-0.135902702
-0.00933063
0.000466531
-1.55094E-05
60
12.8693197
-0.27586215
0
1.25019E-06
65
11.490009
-0.27586215
2.50038E-05
70
10.1106982
-0.275487093
80
7.35582728
y(t) = y( ) + y , )*( - ) + y , , )*( - )*( - ) +…………………………+y , , ,……, )*
( - )*( - )*( - )*………….*( )
=16.0105776 +(-0.312946944)(
+(0.026944771)*( -40)*( -45)+(-0.00165525))*( -40)*( -45)*
( -50)+(5.98692E-05))*( -40)*( -45)* ( -50)*( -60)+(-1.16224E-06))*( -40)*( -45)* ( -50)*( -60)*( 65)+(1.19222E-10))*( -40)*( -45)* ( -50)*( -60)*( -65)*( -70)
Literacy ratemodel =
- y(t)
=[100/ (1 + 4.45553737*
)] – 16.0105776 +(-0.312946944)(
+(0.026944771)*( -40)*( -45)+(-0.00165525))*( -40)*( -45)* ( -50)+(5.98692E-05))*( -40)*( -45)*
( -50)*( -60)+(-1.16224E-06))*( -40)*( -45)* ( -50)*( -60)*( -65)+(1.19222E-10))*( -40)*( -45)* ( 50)*( -60)*( -65)*( -70)]
= [100/ (1 + 4.45553737*
)] – 16.0105776 +(0.312946944)( +(0.026944771)*( )*( )+(-0.00165525))*( )*( )* ( )+(5.98692E-05))*( )*( )*
( )*( )*+(-1.16224E-06))*( )*( )* ( )*( )*( )+(1.19222E-10))*( )*( )* ( )*( )*( )*( )]
= [100/ (1 + 4.45553737*
)] – 16.0105776 +(-0.312946944)(
)+(0.026944771)*(
+(-0.00165525))*(
+(5.98692E-05))*(
)+(-1.16224E-06))*(
+(1.19222E10))*( ]
Where
,
)*( )* ( )*( )*( )*( )
Where
=
.
Now using Lagrange‟s interpolation formula we can find literacy rate in similar manner .Literacy rate by
using Newton‟s divided difference formula and Lagrange‟s interpolation formula with logistic model
approach is as follows.
Table 5.
Time t
(Year)
Year
literacy rate by Divided difference
Formula = - y(t)
literacy
rate by
interpolation Formula
= - y(t)
40
1991
52.2099976
52.21
41
1992
53.96167793
53.96150534
42
1993
55.54806907
55.54783597
43
1994
56.99479716
56.99458951
44
1995
58.32477982
58.32465507
45
1996
59.55836543
59.55835255
46
1997
60.71347134
60.71357079
Ajay J.Patel , Dr.M.B.Prajapati. IJSRE Volume 4 Issue 9 September 2016
Lagrange‟s
Page 5925
47
1998
61.80572109
61.80590481
48
1999
62.84858059
62.84879204
49
2000
63.85349366
63.8536478
50
2001
64.83001672
64.83
51
2002
65.78595302
65.7856234
52
2003
66.72748625
66.7266732
53
2004
67.65931396
67.65781844
54
2005
68.58478049
68.58237497
55
2006
69.50600986
69.5024383
56
2007
70.42403848
70.41901635
57
2008
71.33894785
71.33216212
58
2009
72.24999729
72.24110643
59
2010
73.15575674
73.14439072
60
2011
74.05423971
74.04
61
2012
74.94303639
74.92549595
62
2013
75.81944699
75.79815026
63
2014
76.68061527
76.6550782
64
2015
77.52366236
77.49337239
65
2016
78.3458208
78.31023684
66
2017
79.14456887
79.10312133
67
2018
79.91776519
79.86985596
68
2019
80.66378354
80.60878598
69
2020
81.38164798
81.31890694
70
2021
82.07116821
82
71
2022
82.7330751
82.65276751
72
2023
83.36915656
83.27896883
73
2024
83.98239348
83.88155632
74
2025
84.57709593
84.46481152
75
2026
85.15903956
85.03448151
76
2027
85.73560204
85.59791543
77
2028
86.31589977
86.16420111
78
2029
86.91092457
86.74430183
79
2030
87.53368061
87.35119318
80
2031
88.19932131
88
100
90
80
70
60
50
40
30
20
10
0
literacy rate by
Divided difference
Formula
2031
2027
2023
2019
2015
2011
2007
2003
1999
1995
1991
literacy rate by
Langrange's
interpolation
Formula
Fig. 2 Year wise literacy rate from 1991 to 2031 by our model
Ajay J.Patel , Dr.M.B.Prajapati. IJSRE Volume 4 Issue 9 September 2016
Page 5926
CONCLUSION:
This paper presents an overview of mathematical model of literacy rate for India. If we use only logistic
model then there is a big difference with actual literacy rate that we can see in Table 2 and Fig. 1.This
mathematical model represented by logistic model and Newton‟s divided difference formula or Lagrange‟s
interpolation formula then difference with actual literacy rate will be negligible. Similarly we can find
literacy rate by Newton forward interpolation formula. A paper represents that if literacy rate going as this
way then in 2021 literacy rate should be 82.0711 and in 2031 literacy rate should be 88.199 by our model. If
government will bring some schemes to improve female literacy or schemes for allpersons then literacy rate
may be increase in future.
REFERENCES:
1. State of literacy-education for all in india(www.educationforallindia.com/chapter6-state-of-literacy2011-census.pdf.
2. Desai, Vaman (2012). Introduction of literacy in India economic growth.
3. Tsoularis, “Analysis of Logistic Growth Models”, Res. Lett. Math. Sci, 2, 23-46, 2001
https://www.math.ust.hk/~machas/numerical-methods.pdf
4. Wali, D. Ntubabare, V. Mboniragira, Mathematical Modeling of Rwanda Population Growth:
Journal of Applied Mathematical Sciences, Vol. 5, no. 53, 2617-2628, 2011.
Ajay J.Patel , Dr.M.B.Prajapati. IJSRE Volume 4 Issue 9 September 2016
Page 5927