Analytical Iteration Method Applied to a Class of First Order Nonlinear Evolution Equations in Science

International Journal of Trend
Volume 6 Issue 3, March-April
@ IJTSRD | Unique Paper ID – IJTSRD
Analytical Iteration Method Applied
First Order Nonlinear
Liberty Ebiwareme
1
Department of Mathematics, Rivers State Uni
2
Department of Mathematics, Cross Rivers University of Technology, Calabar, Nigeria
ABSTRACT
In this research paper, we apply the novel Temimi
method to six first order nonlinear partial differential equations
for exact solution namely: Burger’s equation, Fisher’s equation,
Schrodinger equation, wave equation, advection equation
KDV equations respectively. Unlike other semi
iterative methods, this method doesn’t require linearization,
perturbation, discretization, or the calculation of
polynomials for nonlinear terms in the Adomian decomposition
method (ADM). It gives the closed form solution of the problem
if it exists in finite steps of a converging series that’s
computationally convenient, easy to obtained and elegant. It
solves the inherent problem of dealing with the nonlinear term in
a straightforward way without stress. The result obtained
revealed, all the chosen problems give rise to their closed form
solution in simple steps which confirmed the method is
powerful, reliable and has wide applicability to other nonlinear
problems.
KEYWORDS: KDV Equation, Advection equation, Schrödinger
equation, Burger’s equation, Fisher’s equation, Temimi
method
INTRODUCTION
Nonlinear partial differential equations technically
called evolution equations are equations that
constitute the dissipative term and partial derivative
of the dependent variable with respect to one or
more of the independent variables. These equations
feature prominently in most physical phenomena
especially in the physical and applied sciences
(Wazwaz, 2009). They modelled phenomena in the
field of sciences, engineering, Biology,
hydrodynamics, chemistry, physics, optical fibre,
chemical kinetics, plasma physics,
may others, Bekir, Tascan and Unsad (2015),
(Feng, 2002). (Ebiwareme, 2021), Ebiwareme
Ndu (2021), (Liberty, 2021). Academics have
devoted time to extensively studied these equations
for analytical or approximate solution owing to
their importance. All though most of the equations
have closed form solutions, whereas others have
solution which are difficult to obtained however
due to the advent of portable computers, the
solutions can be obtained with less computations.
Trend in Scientific Research and Development
April 2022 Available Online: www.ijtsrd.com e
IJTSRD49491 | Volume – 6 | Issue – 3 | Mar-Apr
Analytical Iteration Method Applied to a Class of
First Order Nonlinear Evolution Equations in Science
Liberty Ebiwareme1
, Edmond Obiem Odok2
Department of Mathematics, Rivers State University, Port Harcourt, Nigeria
In this research paper, we apply the novel Temimi-Ansari
method to six first order nonlinear partial differential equations
Burger’s equation, Fisher’s equation,
Schrodinger equation, wave equation, advection equation and
KDV equations respectively. Unlike other semi-analytical
iterative methods, this method doesn’t require linearization,
perturbation, discretization, or the calculation of an Adomian
polynomials for nonlinear terms in the Adomian decomposition
DM). It gives the closed form solution of the problem
if it exists in finite steps of a converging series that’s
computationally convenient, easy to obtained and elegant. It
solves the inherent problem of dealing with the nonlinear term in
d way without stress. The result obtained
revealed, all the chosen problems give rise to their closed form
solution in simple steps which confirmed the method is
powerful, reliable and has wide applicability to other nonlinear
Equation, Advection equation, Schrödinger
equation, Burger’s equation, Fisher’s equation, Temimi-Ansari
How to cite
Ebiwareme | Edmond Obiem Odok
"Analytical Iteration
a Class of First Order Nonlinear
Evolution Equations
in Science" Published
in International
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and Development
(ijtsrd), ISSN: 2456
6470, Volume
Issue-3, April 2022, pp.132
www.ijtsrd.com/papers/ijtsrd49491.pdf
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Nonlinear partial differential equations technically
called evolution equations are equations that
the dissipative term and partial derivative
of the dependent variable with respect to one or
more of the independent variables. These equations
feature prominently in most physical phenomena
especially in the physical and applied sciences
They modelled phenomena in the
field of sciences, engineering, Biology,
hydrodynamics, chemistry, physics, optical fibre,
medicine and
may others, Bekir, Tascan and Unsad (2015),
(Ebiwareme, 2021), Ebiwareme and
Ndu (2021), (Liberty, 2021). Academics have
devoted time to extensively studied these equations
for analytical or approximate solution owing to
their importance. All though most of the equations
have closed form solutions, whereas others have
which are difficult to obtained however
due to the advent of portable computers, the
solutions can be obtained with less computations.
Some of the methods proposed to study these
equations include: Tanh-
Vinodh (2017), (Wazwaz, 2
Chebyshev collocation method Gupta and Saha
(2015), Vaganan and Asokan (2003), Direct
similarity method, Manafian, Mahrdad and Bekir
(2016), / ′ expansion method
(1998), Homogenous balance method, (Wazwaz,
2007),extended tanh method, Fan and Zhao (2000),
Gomez and Salas (2010), Variational Iteration
method, Nourazr, Mohsen, Nazari
Homotopy perturbation method, Hashim, Norami
and Al-Hadja (2021),Adomian decomposition
method, Necdet and Konural
Kheybari and Khani (2008),spectral collocation
method, Khatter and Temsah (2017), linear
superposition method, Ma and Fan (2011),trial
equation method, Gurefe, Sonmezoglu and Misirli
(2011),Fractional sub-equation method, Zhang and
Zhang (2011),First Integral Method, Zhang, Zhong,
Shashan, Liu, Feng and Gao (2013), (Feng, 2013),
Development (IJTSRD)
e-ISSN: 2456 – 6470
Apr 2022 Page 132
o a Class of
volution Equations in Science
versity, Port Harcourt, Nigeria
cite this paper: Liberty
Ebiwareme | Edmond Obiem Odok
"Analytical Iteration Method Applied to
a Class of First Order Nonlinear
Evolution Equations
in Science" Published
in International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456-
6470, Volume-6 |
3, April 2022, pp.132-139, URL:
jtsrd.com/papers/ijtsrd49491.pdf
© 2022 by author(s) and
International Journal of Trend in
Research and Development
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Access article
under the
the Creative Commons
License (CC BY 4.0)
http://creativecommons.org/licenses/by/4.0)
Some of the methods proposed to study these
Coth method, Asokan and
Vinodh (2017), (Wazwaz, 2008), (Wazwaz, 2009),
Chebyshev collocation method Gupta and Saha
(2015), Vaganan and Asokan (2003), Direct
similarity method, Manafian, Mahrdad and Bekir
expansion method, Fan and Hongqing
Homogenous balance method, (Wazwaz,
2007),extended tanh method, Fan and Zhao (2000),
Gomez and Salas (2010), Variational Iteration
method, Nourazr, Mohsen, Nazari-Golshan (2015),
Homotopy perturbation method, Hashim, Norami
Hadja (2021),Adomian decomposition
method, Necdet and Konuralp (2006), Darvishi,
Kheybari and Khani (2008),spectral collocation
method, Khatter and Temsah (2017), linear
superposition method, Ma and Fan (2011),trial
equation method, Gurefe, Sonmezoglu and Misirli
equation method, Zhang and
g (2011),First Integral Method, Zhang, Zhong,
Shashan, Liu, Feng and Gao (2013), (Feng, 2013),
IJTSRD49491

International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD49491 | Volume – 6 | Issue – 3 | Mar-Apr 2022 Page 133
Ebiwareme and Ndu (2021), Banach contraction
method, (Ebiwareme, 2021).
Most recently, Temimi and Ansari proposed a semi-
analytical iteration method called (TAM) to
successfully solve to linear and nonlinear functional
problems. Unlike the ADM, HAM, VIM and HPM,
this method solved the difficulties that often arise
for nonlinear terms by writing it in the form of
Adomian polynomials (ADM), construction of an
Homotopy (HAM), calculation of the Lagrange
multiplier (VIM) and tedious calculation of an
algebraic calculation from corresponding terms as
in HPM are overcome. TAM has been successfully
applied to solve various problem, such as ODEs
Azeez and Weli (2017), Duffing equation, Al-
Jawary and Al-Razzaq (2016),Chemistry problem,
Al-Jawary and Raham (2016), Nonlinear ODEs,
Temimi and Ansari (2015), Thin film flow
problem, (Al-Jawary, 2017),second order multi-
point boundary value problems, Al-Jawary, Radhi
and Ravnik (2017), Fokker-Planck’s equation,
Temimi and Ansari (2011), Linear and nonlinear
ODEs, Azeez and Weli (2017),Newell-White-head
equation, Latif, Salim, Nasreen, Alifah and
Munirah (2020).
In this present article, our motivation is to apply
this novel semi-analytical iteration method to solve
six differentials nonlinear PDEs. To confirm the
accuracy, reliability, robustness, and efficiency of
this method, we seek to ascertain whether a closed
form or analytical or an approximate solution will
result from this method. The study is organized as
follows. In section, the introduction of the study
detailing PDEs and their applications in physical
phenomena together with methods used hitherto to
seek for their solutions. The fundamentals of the
novel Temimi-Ansari method and the condition for
its convergence is presented in section 2. In section
3, we apply the AIM to six different nonlinear
PDEs and seek closed form solutions and finally the
conclusion of the study is given in section 4.
BASICS OF THE ANALYTICAL ITERATION
METHOD (AIM)
Consider the general functional differential
equation in operator form as follows
+ + = 0, (1)
, = 0, or 0 = and ′
0 = (2)
Where is the independent variable, is an
unknown function, is a given known function,
is a linear operator, is a nonlinear operator and
is a boundary operator.
To implement the TAM method, we first assume
that is an initial guess that satisfy the problem
in Eq. (1) subject to Eq. (2).
+ = 0, , = 0or 0 =
and ′
0 = (3)
The next approximate solution is obtained by
solving the problem
+ + = 0, , =
0, or 0 = and ′
0 = (4)
The next iterate of the problem become
+ + = 0, , =
0, or 0 = and ′
0 = (5)
Continuing the same way to obtain the subsequent
terms, the general equation of the method becomes
+ + =
0, , !
= 0, or 0 = and
′
0 =
Then the solution of the problem in Eq. (11) is
given by
= lim →∞ (6)
From Eq. (6), each & is considered alone as a
solution for Eq. (1). This method easy to
implement, straightforward and direct. The method
gives better approximate solution which converges
to the exact solution with only few members.
NUMERICAL APPLICATIONS
In this section, we apply the AIM to solve six
Partial differential equations that finds usual
applications in Science and Engineering. They
include Fishers’ equation, wave equation, advection
equation, Korteweg-Devries equation (KDV),
Burger’s equation and Schrödinger equation
respectively.
Example 1. Consider the one-dimensional Burger’s
equation of the form
' + − = 0 (7)
Subject to the initial condition
, 0 = 2 (8)
Applying TAM to both sides of the equation, we
get
= ', = − , , * = 0
The initial problem to be solved is of the form
, * + , * = 0, , * = 2 (9)
Integrating both sides of the above equation subject
to the initial condition, we get the initial solution as
, * = 2

The second iterative solution is obtained using the
equation
, * + , * + , * =
0, , * = 2 (10)
Integrating both sides of the above using the initial
condition yield the integral
+ ' , * ,* = + − ,*
'
'
Solving the above integral yield, the second
iterative solution as
, * = 2 − 4 * (11)
The next iterate of the problem is given by
, * + , * + , * = 0, , *
= 2
Taking the inverse operator of both sides of the
above yield
. ' , * ,* = . − ,*
'
'
(12)
Plugging in the derivatives and evaluating, we
obtain the third iterate as
, * = 2 − 4 * + 8* −
0
1
*1
(13)
Similarly, the next iterate is obtained with the
problem
1 , * + , * + , * =
0, 1 , * = 2 (14)
Integrating both sides using the initial conditions,
we get the fourth iterate as follows
. 1' , * ,* = . − ,*
'
'
(15)
Substituting the derivatives, we obtain the solution
as
1 , * = 2 − 4 * + 8 * − 16 *1
+
04
1
*4
−
04
1
*5
+
6
7
*0
− ⋯ (16)
Using the relation, , * = lim ⟶∞ , * , the
closed form solution of the problem is obtained
, * = 2 1 − 2* + 4* − 8*1
+ ⋯
, * =
'
(17)
Example 2. Consider the Fisher’s equation as
follows
' = + 1 − (18)
, 0 = : (19)
To implement TAM, we have the following
= ', = − + 1 − , , *
= 0
The first problem to be solved is given by the
equation
, * + , * = 0, , * = : (20)
Taking the inverse operator of both sides yield the
first iterative solution as
, * = :
relation
, * + , * + , * =
0, , * = : (21)
+ ' , * ,* = + − ,*
'
'
, * = : + : 1 − : * (22)
The third iterate is solved using the relation
, * + , * + , * = 0, , *
= :
Taking the inverse operator of both sides using the
initial condition, we get
. ' , * ,* = . − ,*
'
'
(23)
Solving the above after plugging in the derivatives,
we get the solution as
, * = : + : 1 − : * + : 1 − : : 1 −
2:
';
!
+ =−: 1 − :
'>
1!
? (24)
The fourth iterative solution is obtained using the
problem
1 , * + , * + , * =
0, 1 , * = : (25)
Integrating both sides using the given initial
conditions, we obtained the integral as follows
. 1' , * ,* = . − ,*
'
'
(26)
Solving the integral above yield the iterative
solution below.
1 , * = : + : 1 − : * + : 1 − : : 1 −
2:
';
!
+ : 1 − : 1 − 6: + 6:
'>
1!
+ ⋯ (27)
Continuing in similar fashion the subsequent terms
will be determined, hence the solution of the
problem is obtained using the relation
, * = lim
⟶∞
, *
, * = : + : 1 − : * + : 1 − : : 1 − 2:
*
2!
+ : 1 − : 1 − 6: + 6:
*1
3!
+ ⋯

, * =
ABC
DA ABC (28)
Example 3. Let’s consider the first-order wave
equation in one-dimension as follows
' + E = 0, E > 0 (29)
, 0 = sin
I
J
, 0, * =
J
cos M
−EI*
J
N
0, * = sin
DOP'
Q
, ' , 0 =
DOP
Q
cos
P
Q
(30)
Applying TAM to both sides, we get the following
terms
= ', = E , , 0 = sin
I
J
The first iterate is obtained by solving the problem
of the form
, * + , * = 0, , * = sin
P
Q
(31)
Integrating both sides subject to the initial
condition, we get the initial solution as
, * = sin
I
J
The second iterative solution is obtained by solving
the problem
, * + , * + , * =
0, , * = sin
P
Q
(32)
condition yield the integral below
. ' , * ,* = . − ,*
'
'
(33)
, * = =sin
P
Q
− cos
P
Q
?
OP'
Q
(34)
, * + , * + , * = 0, , *
= sin
I
J
above yield
. ' , * ,* = . − ,*
'
'
(35)
, * =
sin
P
Q
R1 −
!
OP'
Q
S − cos
P
Q
R
OP'
Q
−
1!
OP'
Q
1
S
(36)
Continuing in the same way, the succeeding terms
are obtained, and the closed form solution of the
problem is given by
, * = sin
I
J
cos M
EI*
J
N
− cos
I
J
sin M
EI*
J
N
Using trigonometric identity, the above reduced to
the form
, * = sin =I
DO'
Q
? (37)
Example 4. Let’s consider the homogenous
advection equation of the form
' + = 0 (38)
, 0 = − (39)
Implementing TAM on both sides of the equation,
we have the following terms
= ', = , , * = 0
The first problem to be solved for the first iterate is
given by
, * + , * = 0, , * = − (40)
, * = − (41)
equation
, * + , * + , * =
0, , * = − (42)
condition yield the integral equation
+ ' , * ,* = − + ,*
'
'
, * = − − * (43)
, * + , * + , * = 0, , *
= −
above yield
. ' , * ,* = − . ,*
'
'
(44)
, * = − − * − * − 1
*1
(45)
Similarly, the next iterate is obtained by solving the
problem below
1 , * + , * + , * = 0, 1 , *
= −

. 1' , * ,* = − . ,*
'
'
(46)
as
1 , * == − − * − * −
1
*1
−
1
*4
− ⋯
(47)
, * = − 1 + * + * + *1
+ *4
+ ⋯
, * =
'D
(48)
Example 5. Consider the Korteweg-de Vries
(KDV) equation which takes the form
' − 6 + = 0 (49)
, 0 = −
O;
sec ℎ
O
(50)
Applying TAM to both sides of the equation yield
the expressions
= ', = −6 + , , * = 0
The initial problem to be solved is of the form
, * + , * = 0, , * =
−
O;
sec ℎ
O
(51)
, * = −
O;
sec ℎ
O
(52)
relation
, * + , * + , * =
0, , * = −
O;
sec ℎ
O
(53)
. ' , * ,* = . 6 − ,*
'
'
(54)
, * =
−
O;
sec ℎ
O
−
OV WXY Z; [
;
Tan ℎ
O
(55)
, * + , * + , * =
0, , * = 2 (56)
above yield
. ' , * ,* = . 6 − ,*
'
'
(57)
, * =
−
O;
sec ℎ
O
−
OV WXY Z; [
;
Tan ℎ
O
−
O_
6
`abℎ4
=
O
? 2 − coshdE e * + ⋯ (58)
Continuing in the same, the succeeding term are
obtained in a similar manner
, * = −
O;
`abℎ =
O
− E * ? (59)
Example 6. Consider the Schrödinger equation
in the form
' + f = 0, , 0 = 1 + cos ℎ 2 (60)
Applying TAM to both sides of the equation, we
get the constants
= ', = f , , * = 0
The first problem to be solved is given by
, * + , * = 0, , * = 1 +
cos ℎ 2 (61)
Taking the inverse operator of both sides subject to
the initial condition, we get the initial solution as
, * = 1 + cos ℎ 2 (62)
The next iterative solution is obtained using the
equation
, * + , * + , * = 0, , *
= 1 + cos ℎ 2
. ' , * ,* = . f ,*
'
'
(63)
, * = 1 + cos ℎ 2 + 4f cos ℎ 2 (64)
The third iterative solution of the problem is given
by
, * + , * + , * =
0, , * = 1 + cos ℎ 2 (65)
above yield
+ ' , * ,* = + f ,*
'
'
, * = 1 + cos ℎ 2 + 4f cos ℎ 2 +
4g' ;
!
cos ℎ 2 (66)
Similarly, the next iterate is obtained with the
problem

1 , * + , * + , * =
0, 1 , * = 1 + cos ℎ 2 (67)
. 1' , * ,* = . f ,*
'
'
(68)
as
1 , * = 1 + cos ℎ 2 + 4f cos ℎ 2 +
4g' ;
!
cos ℎ 2 +
4g' >
1!
cos ℎ 2 (69)
closed form solution of the problem is obtainedas
, * = 1 + cos ℎ 2 h1 + 4f* +
4f*
2!
+
4f* 1
3!
+
4f* 4
4!
+ ⋯ N
, * = d1 + cos ℎ 2 ea4g'
(70)
CONCLUDING REMARKS
In this research article, the closed form, or exact
solutions for six different nonlinear partial
differential equation is investigated using the novel
Analytical Iteration Method (AIM). The efficiency
of the method is confirmed by solving the Burgers,
Fisher’s, Advection, Schrödinger, wave and KDV
equations respectively. The method gives an
analytical solution which converges rapidly to the
exact solution with more terms considered. This
solution is easily verifiable in few steps of iteration
subject to the initial condition and is
computationally convenient since it doesn’t require
perturbation, discretization, and linearization. It is
observed the method is reliable, efficient, and
applicable to all class of nonlinear problems in the
fields of Science and Engineering.
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