A MATHEMATICAL MODELLING AND ANALYTICAL SOLUTIONS OF
NONLINEAR DIFFERENTIAL EQUATIONS MODEL WITH USING
HOMOTOPY PERTURBATION METHOD
S. Mayilvaganan and S. Balamuralitharan
Department of Mathematics, Faculty of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Tamil Nadu, India
E-Mail: [email protected]
ABSTRACT
In this article, we have considered a mathematical modelling and analytical solutions of nonlinear differential equations model with using homotopy perturbation method (HPM). It has been a five compartment model with seven parameters. Homotopy perturbation method (HPM) is executed to give analytical and approximation solutions of non linear ordinary differential equation systems model.
Keywords: HPM, nonlinear differential equations, five compartment model.
1. INTRODUCTION
The general ordinary differential equations model with some effective features is described in [1, 2]. The five compartment model impact has been investigated in [3, 4]. The effect of idea for this nonlinear system has been investigated in [1-14]. The compartment systems modification idea has been discussed in [5]. However the intensity of population’s transmission changes over the evolution of drug resistance is explained in [6]. In [7], the different fluctuation of the density is investigated. In [8], the authors considered the uncovered population and strategies on the transmission of nonlinear systems. A mathematical model for this system and analytical solutions is described in [9]. A model interaction with constant immigration in human population and nonlinear compartment has been depicted in [10]. The dynamics of a model in environment is dealt in [11]. The Modeling of nonlinear transmission by considering the human population is exhibited in [12], where it is assumed that the individuals recovered from system can act as seven parameters. The idea of reservoir class is also incorporated in [13]. This method of control is cheap and is being used in many part of the area. But the level is local. The circle of points to control transmission by decreasing the population is developed in [14]. The optimal control approach is used to minimize the rate of uncovered populations. The paper [5] discussed on the study of optimal control on mathematical modeling program. In this paper, we formulated a non-linear differential equations model by population transmission. We found the HPM and analyzed the approximate solutions [1-6].
In this current work, we discuss the effectiveness of the use of nonlinear in a five compartment model. Modification of the model is done by considering the assumption that humans belong to recovered class have possibility to be accepted, i.e., we consider a nonlinear ordinary differential equations model. Moreover, we also consider the application of parameters and values as introduced in this transmission. Analytical solutions are then performed to reveal the effects of changes on population dynamics. Additionally we also propose the
use of the HPM in providing an approximate solution of the model [5, 6]. The interconnectedness among the general population and places on the planet has made the need to comprehend the present and foresee the future progression of irresistible illnesses. Clearly, the irresistible infection enduring the general population are caused because of pathogenic miniaturized scale life forms, for example, microbes, infections, growths and parasites. The pathogenic miniaturized scale life forms could spread from individual to individual or from creatures and feathered creatures to people in an immediate or aberrant ways and engenders maladies. These irresistible sicknesses have a potential reason prompting passings of individuals worldwide as the restorative headway in not up to the prerequisite level Anderson, R. M. et al (1991) [12]. Likewise, the medicinal manage isn't legitimate and costs included are not reachable to a typical man Martcheva M. et al (2003) and Bernoulli D. et al (2004) [8, 13].
The aim of this paper is to find the approximate analytical solution of the given model utilizing HPM. First time, He [1-4] proposed this technique to discover activities of different sorts of straight and nonlinear system of problems. This technique contains on homotopy and a main idea of topology, and this situation has an opportunity in picking starting approximations and secondary nonlinear operators which frequently help to swap the dishevelled nonlinear easier frame.
2. MATHEMATICAL MODELING
In this examination we have thought about compartmental model for nonlinear differential equations and classified the analytical solutions. The present model has a compartmental structure and planned in light of the suspicions depicted as follows:
model we don't think about transformation of the pathogens. From the uncovered compartment a part of people Rgo into the tainted with side effects compartment
x, while the rest (1R)of the bit of people go into the contaminated without manifestations compartment
y
. A tainted individual of the x compartment will either kick the bucket with the ailment or recoup from the sickness by a few means and go into the expelled compartment. The number of uncovered populations in the xcompartmentwill go into the expelled compartment
z
at a rate of and bite the dust with complaint at a rate of. The number of population in they
compartment recoups from the difficulty and goes into the evacuatedz
compartment at a rate of Arino J., et a (2008) [11]. The compartment structure and stream bearings of population of this model can be delineated as appeared in Figure-1.Table-1. Parameter values for the system of equations (1).
Parameter Description Value Reference
1-2 constant 2.343 [5]
1-2 constant 0.004 [11]
3-5 constant 0.244 [9]
4-5 constant 0.244 [6]
3 constant 0.002 [6]R
2-3 constant 0.056 [3]v
first circle parameter - Assumedw
second circle parameter - Assumedx
third circle parameter - Assumedy
fourth circle parameter - Assumedz
fifth circle parameter - Assumedk
2-4 constant 0.005 [7]Figure-1. Compartmental diagram for the nonlinear differential equation systems.
(
)
dv
v x
y
dt
(
)
dw
v x
y
kw
dt
dx
Rkw
x
x
dt
(1)(1
)
dy
R kw
y
dt
dz
x
y
dt
The notations and physical implications of the parameter utilized as a part of (1) are as following: is the contact rate of people with contamination; kis the inertness time frame in the uncovered class; R is the fraction of uncovered people that go intox compartment;
is the rate of healing from the problems in x
3. HOMOTOPY PERTURBATION METHOD
We demonstrate the basic idea of the homotopy perturbation method (HPM) from He, J.H. (1999, 2000) and also, we considered the following non-linear differential equation:
( )
( ),
A u
f r r
(2)with the following boundary condition:
,
u
0,
B u
r
t
(3)where
A
is a general differential operator,B
is a boundary operator,f(r)
is known analytic function and
is the boundary of the domain
.The operator
A
can be decomposed into a linear part and a nonlinear part, named asL
andN
respectively. Therefore, (2) can be rewritten as the following form:( )
( )
( ),
L u
N u
f r r
(4)By means of homotopy method, He, J.H. (1998, 2000) constructed a Homotopy
v r p
( , ) :
[0,1]
which satisfies0
H( , )=(1- ) [ ( )- ( )]+ [ ( )- (r)]=0
v p
p
L v L u
p A v f
(5)or
0 0
H( , )= ( )- ( ) + ( ) + [ ( )- (r)]=0
v p L v L u
p L v
p N v f
(6)where
p
[0,1]
is an embedding parameter,and
u
0 is an initial approximation of (2) which satisfies the boundary conditions. Obviously, from (5) we get0
H( , 0)= ( )- ( )=0, H( , 1)= ( )- (r)=0.
v
L v L u
v
A v f
(7)By changing the value of
p
from zero to unity,( , )
v r p
changes fromu (r)
0 tou(r)
; in topology, thisis called deformation and
L v L v
( )- ( )=0
0 andA v f
( )- (r)
are called homotopic. According to the homotopy perturbation method, the parameterp
is used as a small parameter, and the solution of (5) can be expressed as a power series inp
and is the form2 3
0 1 2 3
...
v
v
pv
p v
p v
(8)and taking limit
p
1
and we get the finest approximate solutioni.e.,
u
lim
p1v
v
0v
1v
2v
3...
(9)The convergence of the series in Equation (9) is discussed by He, J.H. (1998, 2000).
4. ANALYTICAL SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS
In this section, we apply the Homotopy Perturbation method to non linear ordinary differential equations (1). According to Homotopy Perturbation method He, J.H. (1998, 1999, 2000, 2006), we derive a correct functional as follows:
(
)
0
dv
v x
y
dt
(
)
0
dw
v x
y
kw
dt
0
dx
Rkw
x
x
dt
(10)(1
)
0
dy
R kw
y
dt
0
dz
x
y
dt
We obtain the solution of (10) we first construct a homotopy as follows
(1
p
)
dv
p
dv
vx
vy
0
dt
dt
(1
p
)
dw
kw
p
dw
vx
vy
kw
0
dt
dt
(1 p) dx x x p dx Rkw x x 0
dt
dt
(11)
(1
p
)
dy
y
p
dy
kw
Rkw
y
0
dt
dt
(1
p
)
dz
p
dz
x
y
0
dt
dt
Let
2 0 1 2
...
v
v
pv
p v
2
0 1 2
...
w
w
pw
p w
2 0 1 2
...
x
x
px
p x
(12) 20 1 2
...
y
y
py
p y
2 0 1 2
...
z
z
pz
p z
0
p
:0
0
dv
0
0
0
dw
kw
dt
0
0 0
0
dx
x
x
dt
(13)0
0
0
dy
y
dt
0
0
dz
dt
1p
: 10
dv
dt
1 10
dw
kw
dt
1
1 1 0
0
dx
x
x
Rkw
dt
(14)1
1 0 0
0
dy
y
kw
Rkw
dt
1
0 0
0
dz
x
y
dt
(13) gives
0
1
v
0
0.3
kt
w
e
( ) 00.25
t
x
e
(15)0
0.1
t
y
e
0
1
z
(14) gives 11
v
10.3
ktw
e
( ) 1
0.3
0.3
0.25
Rk
k tRk
ktx
e
e
k
k
(16)1
0.3 (1
)
0.3 (1
)
0.1
k
R
tk
R
kty
e
e
k
k
( ) 10.25
1.1
(1
t) 0.1
tz
e
e
Therefore, the analytical solutions are
1
(
)
0 1 2...
p
lim
v t
v
v
v
1
(
)
0 1 2...
p
lim
w t
w
w
w
1
(
)
0 1 2...
p
lim
x t
x
x
x
(17)
1
(
)
0 1 2...
p
lim
y t
y
y
y
1
(
)
0 1 2...
p
lim
z t
z
z
z
We substitute (17) into (15) and (16). Therefore approximate solutions are
( )
2
ktv t
e
( )
0.6
ktw t
e
( ) 0.3 ( ) 0.3
( ) 0.25 t 0.25 Rk k t Rk kt
x t e e e
k k
0.3 (1 ) 0.3 (1 )
( ) 0.1 t 0.1 k R t k R kt
y t e e e
k k
( )0.25
( )
2.1
(1
t)
0
.1
tz t
e
e
5. CONCLUSIONS
In this study, homotopy perturbation method (HPM) was utilized for finding the solution of nonlinear ordinary differential equation of five compartments model. We expose the precision and proficiency of these strategies by comprehending some nonlinear ordinary differential equation systems. We apply He's homotopy perturbation method to determine certain integrals.
REFERENCES
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[3] He, J.H. 2000. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Non- Linear Mech. 35(1): 37-43.
[4] He JH. 2006. Some Asymptotic Methods for Strongly Non Linear equations. International Journal of Modern Physics B. 20(11): 1141-1199.
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[9] Moghadas S. M. 2006. Gaining insights into human viral diseases through mathematics, European Journal of Epidemiology. 21: 337-342.
[10]Sarah A. and Al-Sheikh. 2012. Modeling and Analysis of an SEIR Epidemic Model with a Limited Resource for Treatment. Global Journal of Science Frontier Research Mathematics and Decision Sciences. 12(14): 11- 19.
[11]Arino J. Brauer F. van den Driessche, P. Watmough J. and Wu J. 2008. A model for influenza with vaccination and antiviral treatment, Journal of Theoretical Biology. 253: 118-130.
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