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INTERNATIONAL JOURNAL OF CURRENT LIFE SCIENCES
RESEARCH ARTICLE
ISSN: 2249- 1465
International Journal of Current Life Sciences - Vol.5, Issue, 3, pp. 432-435, March, 2015
NUMERICAL SOLUTION OF VOLTERRA'S POPULATION MODEL BY STWS METHOD
Behnam Sepehrian
Department of Mathematics, Faculty of Science, Arak University, Arak , Iran
AR TIC L E
I NF O
Article History:
th
Received 6 , February, 2015
Received in revised form 18th, February, 2015
Accepted 8th, March, 2015
Published online 28th, March, 2015
ABS TR AC T
In this study a numerical technique for solving the Volterra’s population model
by Single-term Walsh series (STWS) method is introduced. First, the problem is
converted to ordinary differential equations. Then Properties of STWS are
utilized to reduce the computation of the Volterra’s population model to some
algebraic equations. The method is computationally attractive, and applications
are demonstrated through an illustrative example.
Key words:
Ordinary differential equations, Numerical
methods, STWS method, Volterra’s population
model.
INTRODUCTION
Walsh
functions (WF)
have
found
wide
applications in signal processing, communication,
and pattern recognition [7]. Rao et al. [6]
presented a method of extending computation
beyond the limit of the initial normal interval in
Walsh series analysis of dynamical systems. In the
last method, various time functions in the system
were first expanded as truncated WF with
unknown coefficients. Then, by using the Kronecker
product [4], the unknown coefficient of the rate
variable was obtained by finding the inverse of a
square matrix. It was shown that this method
involves some numerical troubles if the dimension
of this matrix is large. Hsiao and Chen [3]
introduced the Walsh series operational matrix of
integration to solve linear integral equations. Due
to the nature of the Walsh functions, the solutions
obtained were piecewise constant. Rao et al. [6]
introduced STWS to remove the inconveniences in
WF technique. Furthermore, Balachandran and
Murugesan [1,2] applied STWS technique to the
analysis of linear and nonlinear singular systems.
Sepehrian and Razzaghi [10,11] implemented
STWS method to the bilinear and nonlinear time
varying singular systems. Also, STWS technique
was applied to solve the nonlinear VolterraHammerstein integral equations in [12].
In the present paper we apply the STWS method for
numerical solution of the Volterra model for population
growth of a species within a closed system. This
equation is given in [9,15] as
© Copy Right, IJCLS, 2015, Academic Journals. All rights reserved.
 u ' (t )  u  u
2
u

t
0
u ( x )d x , u  0   u 0 ,
(1)
where u(t) is the scaled population of identical
individuals at time t, and  
c
is a prescribed
ab
nondimensional parameter. Moreover, a > 0, b >
0 and c > 0 are the coefficients of the birth rate,
crowding and toxicity, respectively. The analytical
solution [13]
Shows that
u (t ) >
0 if
u0
> 0 for all t. The
mathematical behavior of the rapid growth, u (t ) ,
was introduced by Scudo [9] and justified by [13]. It
was shown in [9] and [13] that for the case  <<1,
a rapid rise occurs along the logistic curve that will
reach a peak and be followed by a slow
exponential decay where u (t )  0 as t   .
And, for  large, u (t ) has a smaller amplitude
compared to the amplitude of u (t ) for the case 
<<1.
Several approximate and numerical solutions for
Volterra’s population model are known. Wazwaz
[15] implemented the series solution method and
decomposition method independently to Eq. (1)
and to a related ordinary differential equation.
Furthermore, the pade approximations were used
in the analysis to capture the essential behavior
*Corresponding author: Behnam Sepehrian
Department of Mathematics, Faculty of Science, Arak University, Arak, Iran
International Journal of Current Life Sciences - Vol.5, Issue, 3, pp. 432-435, March, 2015
(3) can be approximated by
of the population u (t ) of identical individuals in
t
[15]. In [5], operational matrices of derivative and
(4)
 ( t ' )d t '  E  ( t ) ,
0
product of Rational Chebyshev functions are
introduced and implemented for solving Eq. (1) by
where E m  m is the m  m operational matrix for
using tau method. In [14], three numerical
1 and is given in [3].
algorithms namely the Euler method, the modified
integration with E 11 
Euler method and the forth order Runge-Kutta
2
method are given and pointwise approximations for
Single-Term Walsh Serie.
the solution of Eq. (1) are obtained.
With the STWS approach, in the first interval,
In the present article, we introduce a new
the given function is expanded as single-term Walsh
computational method for solving Eq. (1). The

method presented here is computationally very
attractive. It does not require the kronecker product
of matrices and there is no need for any
operational matrices. One main merit of using this
technique is that it provides both pointwise and
block-Pulse approximations for solution of (1).
The paper is organized as follows. Section 2 is
devoted to the basic formulation of WF and
STWS required for our subsequent development. In
Section 3, the solution of (1) using STWS is
considered. Section 4 is devoted to numerical
examples. In numerical examples we demonstrate
the accuracy of the proposed scheme by comparing
the our numerical findings with the existing
results and the exact solutions.
series in the normalized interval
  0,1 , which
 1
t  0,  by defining   mt , m
 m
being any integer. In STWS, the matrix E in Eq.
1
(4) becomes E  .
2
corresponds to
Let y (t ) expanded by STWS series in the first
interval as
 0 (t ).
y (t )  Y
(1)
(5.a)
Similarly for y ' and y " we get
 0 (t ),
(5.b)
 0 (t ),
(5.c)
y ' (t )  X
(1)
Properties of WF and STWS
y "(t )  Z
(1)
Walsh Function.
Table 1 Computational results of methods
in [15] and [5] together with the exact
values for u m ax .
A function f (t ) , integrable in [0,1) , may be
approximated using WF as

f (t )   f i i (t ),
(2)
i 0
Where i (t ) is the i th WF and f i is the
corresponding coefficient. In practice, only the first
m -term WF are considered, where m is an
integral power of 2. Then from
Eq. (2) we get
m 1
f (t )   f i i (t )  F T  (t ),
i 0
F  ( f 0 , f 1 ,  , f m 1 )T ,
X
 (t )  (0 (t ), 1 (t )1 , , m 1 (t ))T .
(3)
The coefficients f i are chosen to minimize the
mean integral square error
2
1
   (f (t )  F T  (t )) dt ,
0
f i   f (t )i (t )dt .
0
The integration of the vector  (t ) defined in Eq.
0.02
0.04
0.1
0.2
0.5
0.111845
0.210246
0.464476
0.816858
1.626711
t
Method in
[15]
0.903838
0.861240
0.765113
0.657912
0.485282
Method in
[5]
0.923463
0.873708
0.769734
0.659045
0.485188
Exact umax
0.923427
0.873720
0.769741
0.659050
0.485190
(1)
(1)
1
X
2
1
 Z
2

(1)
 y (0),
(6.a)
(1)
 y ' (0).
(6.b)
From Eq. (6.b) we can write
Z
(1)
 2( X
(1)
 y ' (0)).
(7)
Also, according to Sannuti [3] we have
y
(1)
y
'(1)
and are given by
1
Critical
By using Eq. (5.a) and integrating Eqs. (5.b) and
(5.c) with E  1 we get
2
Y
where

1



0
1
0
y ' (t )dt  y (0)  X
(1)
y "(t )dt  y ' (0)  Z
 y (0),
(1)
 y ' (0).
(8.a)
(8.b)
Therefore, from Eqs. (7) and (8.b) we have
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y
'(1)
International Journal of Current Life Sciences - Vol.5, Issue, 3, pp. 432-435, March, 2015
 2X
(1)
 y ' (0).
(9)
y ( )  Y
y ( )  X
(i )
y ( )"  Z
(i )
'
In general for any interval i , i  1, 2,  , Eqs.
(6.a), (7), (8.a) and (9) convert to
Y
(i )
(i )
Z
(i )
y '(i )  2X
(10)
(12)
 y ' (i  1).
In above equations X
(i )
and y ' (i ) give the block-
pulse and the discrete values of y (t ) respectively.
Table 2 Estimated values of
u max
using
STWS with m = 50, 150 and 250 together
with e xac t values.
Critical t
0.1247
0.2235
0.4747
0.8216
1.62671
0.02
0.04
0.1
0.2
0.5
m
 0 ( ),
(21.b)
 0 ( ).
(21.c)
(i )

1
X
m
(i )

1
(X
m
) 
(i ) 2
1
Y
m2
(i )
X
(i )
(22)
.
From Eqs. (10), (11) and (22) the block-pulse value
X ( i ) in the i th interval is given by
(13)
'

Z
(11)
 y (i  1),
(i )
(21.a)
Using Eqs. (20) and (21) we get
1 (i )
X  y (i  1),
2
 2( X ( i )  y ' (i  1)),

y (i )  X
 0 ( ),
(i )
=50
0.924161
0.873674
0.769748
0.658944
0.485173
m
=150
0.923597
0.873766
0.769741
0.659044
0.485187
m
=250
0.923448
0.873721
0.769741
0.659049
0.485190
(
1
1

)(X
m 2m 2
)  (2 
(i ) 2
1 y (i 1)

)X
m
m2
(i )
(23)
By solving Eq. (23) for X ( i ) , i  1, 2,  , and
then by Eq. (10), the block-pulse and by using
Eqs. (12) and (13) discrete approximations of
'
y ( ) and y ( ) can be obtained, respectively.
Solution of Volterra's population model
In this section we solve Volterra’s population model
in Eq. (1) by converting it to an equivalent
nonlinear o r d i nar y d i f f er en ti al equation.
Let
Figure 1 Block-pulse approximation for the cases
=0.02 (Solid), 0.1 (Dashes) and 0.5 (Dots).

(14)
Illustrative Example
y ' (t )  u (t ),
(15)
We applied the methods presented in this paper
and solved Eq. 1 for u (0)  0.1, and  =0.02,
0.04, 0.1, 0.2 and 0.5 with m =50, 150 and 250
y "(t )  u (t ).
(16)
y (t ) 

t
0
u ( x )d x .
This leads to
'
Inserting Eqs. (14)-(16) into Eq. (1) yields the
nonlinear differential equation
 y "(t )  y ' (t )  ( y ' (t )) 2  y (t ) y ' (t ),
(17)
with the initial conditions
y (0)  0,
(18)
y (0)  u 0 ,
(19)
'
and then evaluated
u max and t critical , which also are
evaluated in [5,15]. The resulting values using the
methods in [5] and [15] together with the exact
values obtained by using [14]


1
u m ax  1   ln 
,
1  u0 
obtained by using Eqs. (14) and (15) respectively.
To solve Eq. (17) with initial conditions (18) and
(19), we normalize Eq. (17) in the time scale
  mt as
 y "( ) 
Figure 2 Block-pulse solutions for the cases
(Solid), 0.2 (Dashes).
1 '
1
y ( ) 
( y ' ( )) 2
m
m
1
 2 y ( ) y ' ( ) .
m
Are presented in Table 1.
(20)
y ' ( ) and y "( ) be expressed by
STWS in the i th interval as
Let y ( ) ,

=0.04
In Table 2, our
computational results for approximations of
u max
with m =50, 150, and 250 are shown. So, a
comparison between methods in [5] and [15] and
the present method is made. Figures 1 and 2,
show us the mathematical behavior of the
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International Journal of Current Life Sciences - Vol.5, Issue, 3, pp. 432-435, March, 2015
approximate solutions obtained by STWS method
equations via Walsh functions, Comput. Electr.
with m =100. These graphs agree with the
Engrg. 6 (1979), 279–292.
previous studies in [9] and [13].
6. F. M. Scudo, Volterra and theoritical ecology,
Theoret. Popul. Biol. 2 (1971), 1–23.
CONCLUSION
7. G. P. Rao, K. R. Palanisamy and T. Srinivasan,
Extension of computation beyond the limit of
The properties of STWS are used to solve the
initial normal interval in Walsh series analysis
Volterra’s population model. First, the Volterra’s
of dynamical systems, IEEE Trans. Automat.
population model is converted to a nonlinear
Control 25(2) (1980), 317–319.
ordinary differential equation. Then, the resulted
8.
K. Balachandran and K. Murugesan, Analysis
differential equation is solved by STWS technique.
of nonlinear singular systems via STWS
The method provides both pointwise and blockmethod, Int. J. Comput. Math. 36(1-2) (1990),
pulse approximations for unknown function.
9–12.
Furthermore, unlike the Walsh series approach,
9. K. Balachandran and K. Murugesan,
there is no restriction on m and the methods does
Numerical solution of a singular non-linear
not require the Kronecker product or the inversion of
system from fluid dynamics, Int. J. Comput.
large matrices.
Math. 38(3-4) (1991), 211–218.
References
10.K. G. TeBeest, Numerical and analytical
solutions of Volterra's population model,
1. A. M. Wazwaz, Analytical approximation and
SIAM Rev. 39 (1997), 484–493.
pade
approximation
for
Volterra's
11. K. Parand and M. Razzaghi, Rational
population model, Appl. Math. Comput.
Chebyshev tau method for solving Volterra's
100 (1999), 13–25.
population model, Appl. Math. Comput. 149
2. B. Sepehrian and M. Razzaghi, Solution of
(2004), 893–900.
nonlinear Volterra-Hammerstein integral
12. M. Razzaghi and J. Nazarzadeh, Walsh
equations via single-term Walsh series
functions, Wiley Encyclopedia of Electrical and
method, Math. Prob. Eng. 5 (2005), 547–
Electronics Engineering 23 (1999), 429–440.
554.
13. P. Lancaster, Theory of matrices, Academic
3. B. Sepehrian and M. Razzaghi, Solution of
Press, New York (1969).
time-varying singular nonlinear systems by
14. P. Sannuti, Analysis and synthesis of dynamic
single-term Walsh series, Math. Prob. Eng.
systems via block-pulse functions, Proceedings
3 (2003), 129–136.
IEE 124(6) (1977), 571–596.
4. B. Sepehrian and M. Razzaghi, State
15.R. D. Small, Population growth in a closed
analysis of time-varying singular bilinear
system and Mathematical Modeling, in:
systems by single-term Walsh series, Int. J.
Classroom Notes in Applied Mathematics,
Comput. Math. 80(4) (2003), 413–418.
SIAM, Philadelphia, PA (1989), 317–320.
5. C. H. Hsiao and C. F. Chen, Solving integral
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