The aim of this work is to use a semi-analytical method “Reduced Differential Transform Method (RDTM)” for the solution of linear and nonlinear Newell-Whitehead-Segel Equations (NWSE). RDTM does not require linearization, transformation, discretization, perturbation or restrictive assumptions. To determine the performance measure of the RDTM, two illustrative examples were considered. The comparative study of the results obtained via the RDTM was compared with that of the exact solution. Hence, RDTM offers solutions with easily computable components as convergent series and is an alternative approach that overcomes the shortcoming of complex calculations of differential transform method

Fadugba, Sunday Emmanuel

English

PURKH (E-Journals)

MathLAB Journal  Vol 1 No 3  (2018)                                                                                http://purkh.com/index.php/mathlab 
251 
A Semi-Analytical Method for The Solution of Linear And Nonlinear Newell-Whitehead-Segel Equations 
Fadugba Sunday Emmanuel 
Department of Mathematics, Ekiti State University, Ado Ekiti, Nigeria
emmasfad2006@yahoo.com 
Abstract 
This work presents a semi-analytical technique, namely the Reduced Differential Transform Method (RDTM) for 
the solution of linear and nonlinear Newell-Whitehead-Segel Equations (NWSE). The method does not require 
linearization, transformation, discretization, perturbation or restrictive assumptions. To determine the accuracy 
of solutions and the performance measure of RDTM, two illustrative examples were considered. The results 
obtained via RDTM was compared with that of the exact solution. The results show that RDTM is very effective, 
offers solutions with easily computable components as convergent series and quite accurate to systems of linear 
and nonlinear equations. Hence, RDTM is a good alternative approach for the solution of partial differential 
equations that overcomes the shortcoming of complex calculations of differential transform method. 
Keywords: Exact solution, Linear and nonlinear equation, Newell-Whitehead-Segel equation, Reduced 
differential transform, Semi-analytical method   
2010 Mathematics Subject Classification: 34K28, 35A25 
Introduction  
Differential equations play a vital role in various fields such as engineering, mathematics, astronomy, chemistry, 
physics and biology. Partial Differential Equations (PDEs) are mathematical models that are used to describe 
complex phenomena emanating from the world around us.  In the recent years, many researchers  had paid 
attention to the solutions of partial differential equations by means of the various methods,  such as [1], [2], [4], 
[5], [6], [9], [12], [13], just to mention a few. The RDTM was first proposed by [7] and successfully employed to 
solve many types of nonlinear PDEs. Also Keskin and Oturanç [8] used RDTM to obtain the analytical solution of 
linear and nonlinear wave equations. For more details on RDTM see; [3], [10] and [11], just to mention a few.  
In this work, a powerful technique, the “Reduced Differential Transform Method (RDTM)” for the solution of the 
Newell-Whitehead-Segel Equation (NWSE) is presented. The rest of the paper is organized as follows: Section 
Two is the brief review of RDTM. In Section Three, the analysis of the RDTM is presented. Section Four presents 
two illustrative examples and the concluding remarks. 
Brief review of the reduced differential transform 
The reduced differential transform of ( , )u x t at 0t = is defined as  
0
1 ( , )
( )
!
k
k k
t
u x t
U x
k t
=
 
=  
 
                              (1) 
where ( , )u x t  is analytic and continuously differentiable with respect to time t  and space x in the domain of 
interest and ( )kU x is the transformed function. Conversely, the reduced differential inverse transform of ( )kU x  
is defined as 
MathLAB Journal  Vol 1 No 3  (2018)                                                                                http://purkh.com/index.php/mathlab 
252 
0
( , ) ( ) kk
k
u x t U x t

=
=                                           (2) 
Combining (1) and (2) yields 
0 0
1 ( , )
( , )
!
k
k
k
k t
u x t
u x t t
k t

= =
 
=  
 
                                 (3) 
The properties of the reduced differential transform are given below. 
Property 1.  
If ( , ) ( , ) ( , )a x t u x t v x t=  then ( ) ( ) ( )k k kA x U x V x=   
Property 2.  
If ( , ) ( , )a x t u x t= then ( ) ( )k kA x U x=  
Property 3.  
If 
( , )
( , )
u x t
a x t
x

=

then 
( )
( )
k
k
U x
A x
x

=

 
Property 4.  
If 
nmtxtxa =),( then ( ) ( )
m
kA x x k n= − ,  
where the Kronecker delta is given by 
1,        
( )
0,       
k n
k n
k n

=
− = 

 
Property 5.  
If ( , ) ( , )m na x t x t u x t=
 
then ( ) ( )
m
k k nA x x U x−=  
Property 6.  
If 
( , )
( , )
r
r
u x t
a x t
x

=

then 
( )!
( ) ( 1)( 2)...( ) ( ) ( )
!
k k r k r
k r
A x k k k r U x U x
k
+ +
+
= + + + =  
Property 7.  
If ( , ) ( , ) ( , )a x t u x t v x t= then 
0
( ) ( ) ( )
k
k r k r
r
A x U x V x−
=
=  
 
 
 
MathLAB Journal  Vol 1 No 3  (2018)                                                                                http://purkh.com/index.php/mathlab 
253 
Property 8.  
If ( )( , ) ( , )
m
a x t u x t= then 
0
01
( ),                                             0
( )        
( 1)
( ) ( ),            1
( )
k k
n k
n k n
n
U x k
A x
m
U x W x k
kU x
−
−
=


=  +
 


 
Application of the reduced differential transform to the Newell-Whitehead-Segel equation 
Consider the Newell-White-Segel equation of the form 
1 2 3( , ) ( , ) ( , ) ( , )
q
t xxv x t a v x t a v x t a v x t= + −                                (4) 
Subject to the initial condition 
( ,0) ( )v x h x=                               (5) 
where 1 2 3, ,a a a  are real numbers and q is a positive integer. By means of the definition and the fundamental 
properties of RDTM, (4) and (5) yield 
2
1 1 2 32
( )
( 1) ( ) ( ) ( )    
k
k k k
V x
k V x a a V x a H x
x
+

+ = + −

                           (6) 
with 
0
01
( ),                                             0
( )
( 1)
( ) ( ),            1
( )
k k
n k
n k n
n
V x k
H x
q
V x H x k
kV x
−
−
=
=

=  +
 


                        (7) 
and  
0( ) ( ) ( )V x h x k=                            (8) 
respectively. The terms ( )kV x  and ( )kH x  are the transformed values of ( , )v x t and ( , )
qv x t  respectively, 
( ) ( )h x k  is the transformed value of ( )h x . By iterative calculations on (6) and (8), the following values of 
( )kU x are obtained as 
1 1 2 2 3 3( ) ( ), ( ) ( ), ( ) ( ),..., ( ) ( ),...n nV x x V x x V x x V x x   = = = =                          (9) 
Combining (8) and (9) and by means of the reduced  differential inverse transform, the solution of (4) yields 
2 3
1 2 3( , ) ( ) ( ) ( ) ( ) ( ) ... ( ) ...
n
nv x t h x k t x t x t x t x    = + + + + + +                       (10) 
Two Illustrative Examples and Results 
This section consists of two illustrative examples to discuss the performance measure and accuracy of RDTM. 
The results generated are also presented. 
MathLAB Journal  Vol 1 No 3  (2018)                                                                                http://purkh.com/index.php/mathlab 
254 
Example 1: The Linear Newell-Whitehead-Segel equation  
Consider the equation of the form 
( , ) ( , ) 2 ( , ) 0t xxv x t v x t v x t− + =                            (11) 
Subject to the initial condition 
( ,0) xv x e=                              (12) 
whose exact solution is obtained as 
( , ) x tv x t e −=                                         (13) 
By applying the RDTM to (11) and (12) yields 
2
1 2
( )
( 1) ( ) 2 ( )
k
k k
V x
k V x V x
x
+

+ = −

                                        (14) 
and 
0 ( )
xV x e=                                 (15) 
respectively. 
The iterative calculations of (14) and (15) are obtained as follows: 
1 2 3 4 5( ) , ( ) , ( ) , ( ) , ( ) ,..., ( ) ( 1) ,...
2 6 24 120 !
x x x x x
x n
n
e e e e e
V x e V x V x V x V x V x
n
− −
= − = = = = = −                             (16) 
Using (15), (16) and the definition of the reduced differential inverse transform, the analytical approximate 
solution is obtained as 
2 3 4 5
( , ) 1 ... ( 1) ...
2! 3! 4! 5! !
n
x n x tt t t t tu x t e t e
n
− = − + − + − + + − + = 
 
                        (17) 
The results generated by means of the RDTM and exact solution for different values of 0.0(0.1)0.5x = and 
0.0(0.1)0.5t =  are shown in Figures 1 and 2 below respectively. 
MathLAB Journal  Vol 1 No 3  (2018)                                                                                http://purkh.com/index.php/mathlab 
255 
 
Figure 1: The RDTM solution 
 
Figure 2: The Exact solution 
Example 2: The Non-linear Newell-Whitehead-Segel equation 
Consider the equation of the form 
2( , ) ( , ) 2 ( , ) 3 ( , ) 0t xxv x t v x t v x t v x t− − + =                           (18) 
Subject to initial condition 
( ,0)v x =                                            (19) 
MathLAB Journal  Vol 1 No 3  (2018)                                                                                http://purkh.com/index.php/mathlab 
256 
The exact solution of the problem is obtained as 
2
2
2
( , )
2 3 3
t
t
e
v x t
e

 
 
=  
− + 
                               (20) 
Taking the reduced differential transform of (18) and (19), and by means of the fundamental theorems of RDTM, 
one gets 
2
1 2
0
( )
( 1) ( ) 2 ( ) 3 ( ) ( ) 0
k
k
k k i k i
i
V x
k V x V x V x V x
x
+ −
=

+ = + − =

                               (21) 
and 
0( )V x =                              (22) 
Following the same procedures for the solution of the linear case, the iteration calculations on (21) are obtained 
as follows 
2
2
1 2 3
2 (2 3 )(1 3 ) 2 (2 3 )(27 18 2)
( ) 2 3 , ( ) , ( ) ,...
2 6
V x V x V x
      
 
− − − − +
= − = =                   (23) 
From (2), the reduced differential inverse transform is given by 
0
( , ) ( )
          
k
k
k
v x t V x t

=
=                                                                            (24) 
Substituting (22) and (23) into (24) yields 
2
2 2 3
2
2
2 (2 3 )(1 3 ) 2 (2 3 )(27 18 2)
( , ) (2 3 ) ...
2 6
2
          
2 3 3
t
t
v x t t t t
e
e
      
  

 
 − − − − + 
= + − + + +  
   
 
=  
− + 
                  (25) 
Equation (25) is the required approximate solution to (18) subject to (19). 
By varying  , the results generated by means of the RDTM and exact solution for different values of 
0.2(0.1)1.0t =  are shown in Figures 3 and 4 below respectively. 
MathLAB Journal  Vol 1 No 3  (2018)                                                                                http://purkh.com/index.php/mathlab 
257 
 
Figure 3: The RDTM solution 
 
Figure 4: The Exact solution 
Conclusions 
The RDTM is applied to both linear and nonlinear Newell-Whitehead-Segel equations. RDTM does not require 
linearization, perturbation or restrictive assumptions and offers solutions with easily computable components 
as convergent series. The Figures 1, 2, 3 and 4 above are drawn for RDTM as well as the exact solution. It is 
observed from Figures 1 and 2 that the value of the function ( , )v x t increases for different values of 
0.0(0.1)0.5x = and 0.0(0.1)0.5t = . From Figures 3 and 4, it is observed that as  increases, the function 
( , )v x t  increases for different values 0.2(0.1)1.0t = . It also observed from Figures 1, 2, 3 and 4 that the results 
MathLAB Journal  Vol 1 No 3  (2018)                                                                                http://purkh.com/index.php/mathlab 
258 
obtained via RDTM agree with the results of the exact solution. It is concluded that RDTM is efficient and a 
powerful semi-analytical method for solving both linear and nonlinear PDEs emanating from science and 
engineering. 
Conflicts of Interest 
Author declares that there is no conflict of interest 
References 
1. Adomian, G., A new approach to nonlinear partial differential equations, J. Math. Anal. Applic. 102(1984) 
203-434. 
2. Bahşi, K. and Yalçinbaş, S., A new algorithm for the numerical solution of telegraph equations by using 
Fibonacci polynomials. Mathematical and Computational  Application.  21(2016) 1-12. 
3. Fadugba, S.E. and Owoeye, K.O., Reduced Differential Transform for Solving Special Linear and Nonlinear 
Partial Differential Equations, International Journal of Engineering and Future Technology, Vol. 16, No. 3,  
(in Press, 2019), Abu Dhabi, UAE. 
4. He, J.H., A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. 
Simul. 2(1997) 203-205. 
5. He, J.H., Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 
135(2003) 73-79. 
6. He, J.H. and Wu, X.H., Exp-function method for nonlinear wave equations, Chaos Solitons Fract. 30(2006) 
700-708. 
7. Keskin, Y. and Oturanç, G., The reduced differential transform method for partial differential equations, 
Int. J. Nonlin. Sci. Numer. Simul. 10(2009) 741-749. 
8. Keskin, Y. and Oturanç, G., The reduced differential transform method for solving linear and nonlinear 
wave equations, Iran. J. Sci. Technol. 34(2010) 113-122. 
9. Li, Q., Zheng, Z. and Liu, J., Lattice Boltzmann model for nonlinear heat equations, Adv. Neural Netw. 
(2012) 140-148. 
10. Servi, S., Keskin, Y. and Oturanç, G., Reduced differential transform method for improved Boussinesq 
equation, AIP Conference Proceedings 1648, 370012 (2015). 
11. Srivastava, V.K., Mukesh, K.A. and Chaurasia, R.K., Reduced differential transform method to solve two 
and three dimensional second order hyperbolic telegraphic equations, J. King Saud Univ. Eng. Sci. 
17(2014). 
12. Zhang, J.L., Wang, M.L. and Fang, Z.D., The improved F-expansion method and its applications, Phys. Lett. 
A. 350(2006) 103-109. 
13. Zhou, J.K., Differential transformation and its application for electrical circuits, Huazhong University Press, 
Wuhan, China, 1986. 
 
MathLAB Journal  Vol 1 No 3  (2018)                                                                                http://purkh.com/index.php/mathlab 
259 
 


A Semi-Analytical Method for The Solution of Linear And Nonlinear Newell-Whitehead-Segel Equations

https://purkh.com/index.php/mathlab/article/download/140/211

A Semi-Analytical Method for The Solution of Linear And Nonlinear Newell-Whitehead-Segel Equations

Abstract

Similar works

Full text

Available Versions

PURKH (E-Journals)