IHJPAS. 36 (3) 2023 

316 

This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

 
*Corresponding Author: dr.huda_hm2029@yahoo.com 
 

 

 

 

Abstract   

Some nonlinear differential equations with fractional order are evaluated using a novel 

approach Sumudu and Adomian Decomposition technique (STADM). To get the results of the 

given model, the Sumudu transformation and iterative technique are employed. The suggested 

method has an advantage over alternative strategies in that it does not require for additional 

resources or calculations. This approach works well, easy to use, and yield good results. Besides, 

the solution graphs are plotted using MATLAB software. Also, the true solution of the fractional 

Newell-Whitehead equation is shown together with the approximate solutions of STADM. The 

results showed  our approach is  a great way, fantastic, reliable and easy method to deal with 

specific problems in  a variety of applied sciences and engineering fields.  

 

Keywords: Sumudu Transformation, Caputo derivative, Fractional Calculus, Approximate 

solutions, Decomposition  method. 
  

1. Introduction 

Partial differential equations with fractional order (FPDEs) are a modification of integer 

order differential equations. The study of FPDEs has attracted more attention recently. The 

fractional approach has developed into a powerful modeling technique that is frequently used in 

the fields of chaotic dynamics, wave propagation, turbulence, turbulent flow, diffusion processes, 

[1]-[4]. Because some fractional order models cannot be tested analytically and the results for 

FPDEs should be have. Many researchers have focused on developing effective and reliable 

techniques for FPDEs which include Laplace transform [7], Laplace Variational method (LVIM) 

[8], perturbation method [9], and differential transform method [5], Variational iteration method 

[6], and many others. Several analytical and approximation methods using SVIM solving nonlinear 

problems of fractional order [9,10],  and others[11-17] have been proposed. In this Work, we 

applied a new mixture which is a graceful coupling of  two strong approaches STADM for solving 

fractional-order Nonlinear PDES .  

doi.org/10.30526/36.3.3241 

Article history: Received 26 January 2023, Accepted 20 February 2023, Published in  July 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Analytical Solutions to Investigate Fractional Newell-Whitehead Nonlinear 

Equation using Sumudu Transform Decomposition Method  

 

Jasem Hussein  Qays  

 Education of CollegeDepartment of Mathematics, 

University of  ,thamHai-Al Ibn Science Pure for

Baghdad, Baghdad, Iraq. 

 

qaisalsaadm@gmail.com 

 

   Huda Omran Altaie*  

 Education of eCollegDepartment of Mathematics, 

University of  ,thamHai-Al Ibn Science Pure for

Baghdad, Baghdad, Iraq. 

 

dr.huda_hm2029@yahoo.com 

 

 

 

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IHJPAS. 36 (3) 2023 

317 
 

 

2. Idea of Sumudu Transform and Decomposition Approach (STADM):  
The STADM is discussed in this section as it relates to solving FPDEs. The following 

general FPDEs as: 

𝐷𝑡
𝑤y(x, t) + Ly(x, t) + Ny(x, t) = q(x, t) ,  x ≥ 0, t > 0 , n − 1 < 𝑤 ≤ n        ….(1) 

With the initial conditions 

y(x, 0) = g(x)         or    
∂ry(x,0)

∂ tr
= g(r)(x, 0) = gr(x) ,……… (2) 

                                                             r = 0,1, … … . . n − 1 

The Caputo fractional derivative𝐷𝑡
𝑤𝑦(𝑥, 𝑡) of  𝑦(𝑥, 𝑡)  denoted below: 

𝜕𝑤

𝜕𝑡𝑤
  𝑦(𝑥, 𝑡) =

{
 
 

 
 1

Γ(n − w)
  ∫ (𝑡 − 𝑠)

𝑛−𝑤−1   
𝜕𝑛𝑦(𝑥, 𝑠)

𝜕𝑡𝑛
   𝑛 − 1 < 𝑤 < 𝑛

 

𝑡

0

𝜕𝑛𝑦 (𝑥, 𝑡)

𝜕𝑡𝑛
                                                            𝑤 = 𝑛 ∈ 𝑁     

}
 
 

 
 

 

 

Taking Sumudu transform of the Eq. (1), we have:    

𝑆[𝐷𝑡
𝑤𝑦(𝑥, 𝑡)] + 𝑆[𝑅[𝑦(𝑥, 𝑡)]] + 𝑆[𝑁[𝑦(𝑥, 𝑡)]] = 𝑆[𝑞(𝑥, 𝑡)]……….(3) 

The property of Sumudu transform of function derivatives used, then  

𝑆[𝑦(𝑥, 𝑡)]

𝑢𝑤
= ∑

𝑦(𝑥, 0)𝑘

𝑢𝑤−𝑘
+ 𝑆[𝑞(𝑥, 𝑡)] −

𝑛−1

𝑘=0

𝑆[𝐿(𝑦(𝑥, 𝑡)) + 𝑁(𝑦(𝑥, 𝑡))]    

 𝑆[𝑦(𝑥, 𝑡)] = 𝑦(𝑥, 0) + 𝑢𝑤𝑆[𝑞(𝑥, 𝑡)]

− 𝑢𝑤𝑆[𝐿(𝑦(𝑥, 𝑡)) + 𝑁(𝑦(𝑥, 𝑡))]                                (4) 

Application of Sumudu inverse transform on Eq. (4) yielded: 

𝑦(𝑥, 𝑡) = 𝑓(𝑥) + 𝑆−1(𝑢𝑤𝑆 [𝑞(𝑥, 𝑡)])

− 𝑆−1(𝑢𝑤𝑆[𝐿(𝑦(𝑥, 𝑡) + 𝑁(𝑦(𝑥, 𝑡)])                                 (5)        

The  representation of the solution for Eq. (5) as an infinite series is given below: 

                                        𝑦(𝑥, 𝑡) = ∑ 𝑦𝑖(𝑥, 𝑡)

∞

𝑖=0

                                  (6) 

And the nonlinear term is being decomposed as: 

                      𝑁[𝑦(𝑥, 𝑡)] = ∑ 𝐴𝑖(𝑦0, 𝑦1, … . , 𝑦𝑖)

∞

𝑖=0

                           (7) 

Where, 𝐴𝑖 are the Adomian polynomials of functions 𝑦0, 𝑦1, … . , 𝑦𝑖 can be calculated 

by formula given as: 

𝐴𝑖 =
1

𝑖!

𝜕𝑖

𝜕𝜆𝑖
 [𝑁 (∑ 𝜆𝑖𝑦𝑖

∞

𝑖=0

)]

 
 
 
 𝜆=0

 

 

 

Substituting Eqs. (6) and (7) in Eq. (5): 



IHJPAS. 36 (3) 2023 

318 
 

∑ 𝑦𝑖(𝑥, 𝑡)

∞

𝑖=0

= ∑
𝑦(𝑥, 0)(𝑘)

𝑢𝑤−𝑘
 

∞

𝑘=0

+ 𝑆−1(𝑢𝑤𝑆[𝑞(𝑥, 𝑡)])

− 𝑆−1 [𝑢𝑤𝑆 [𝐿 (∑ 𝑦𝑖(𝑥, 𝑡)

∞

𝑖=0

) + ∑ 𝐴𝑖

∞

𝑖=0

]]                           (8)  

Simplification of Eq. (7) as many times as possible resulted into series solution, we 

get: 

𝑦0(𝑥, 𝑡) = ∑
𝑦(𝑥, 0)(𝑘)

𝑢𝑤−𝑘

∞

𝑘=0

+ 𝑆−1(𝑠𝑤𝑆[𝑔(𝑥, 𝑡)]) 

𝑦1(𝑥, 𝑡) = −𝑆
−1[𝑢𝑤𝑆[𝐿(𝑦0(𝑥, 𝑡)) + 𝐴0]] 

𝑦2(𝑥, 𝑡) = −𝑆
−1 [𝑢𝑤𝑆[𝐿(𝑦1(𝑥, 𝑡)) + 𝐴1]] 

⋮ 

𝑦𝑖(𝑥, 𝑡) = −𝑆
−1[𝑢𝑤𝑆[𝐿(𝑦𝑖−1(𝑥, 𝑡)) + 𝐴𝑖]] 

Finally, the iteration 𝑦0, 𝑦1, … . , 𝑦𝑖 were obtained and we approximate the analytical 

solution 𝑦(𝑥, 𝑡) by truncated series 𝑦(𝑥, 𝑡) = ∑ 𝑦𝑖(𝑥, 𝑡)
∞
𝑖=0 . 

 

3. Test Example: 

     The test example shows the reliable and efficient of STADM. All results are calculated using 

the software MATLAB R2021b. 

 

Example 3.1:Consider Newell-Whitehead PDES as follows:  

 

∂w y(x,t)

∂tw
−

∂2 y (x,t)

∂ x2
= y (x, t) − y3 (x, t) ……………..(9) 

With initial condition 

y (x, 0) =
1

2
[1 + tan h (

x

2√2
)]  ……………..(10) 

And the true solution  y (x, t) =
1

2
[1 + tan h (

√2x+3t

4
)] 

 Solution : 

The Adomain polynomials for the nonlinear terms  −y3 Can be computed as follows: 

 
By using Eq.(10), we have 

 

A0 = −y0
3 

A1 = −3 y0
2  y1 

A2 = −3(y0
2 y2 + y0  y1

2) 

A3 = −(3(y0
2 y3 + 6y0 y1  y2 + y1

3 ) 

y0(x, t) =
1

2
 1 + tan h  

x

2√2
   



IHJPAS. 36 (3) 2023 

319 
 

 
We determine the terms below using the same pattern: 

 
 

The  analytical solution is provided by: 

 

y𝑛(x, t) = y0(x, t) + y1(x, t) + y2(x, t) + y3(x, t) 

y𝑛(x, t) =
1

2
 1 + tan h  

x

2√2
  +

3

8
sech2  

x

2 √2
 .  

t𝛼

α
 −

9

4
sin h4  

x

2√2
  csc h3  

x

√2
  .  

tα

α
 
2

+
9

128
.   

tα

α
 
3

.   cosh  
x

√2
 − 2  sech4   

x

2√2
  

 

Where n = 1 , 2 , 3 and 4 for the set of 𝛼 values that applied in this example, which are 0.4, 0.6, 

0.8, and 1. 

4. Results  

 The following  Figures present the absolute error at t=0.002 with various value of x  . We employ 

a few terms to approximate the solution, and the suggested approach, FSTADM, has a high 

convergence order and higher accuracy. Similarly, Figure4.1–Figure4.6 show the 3D exact and 

achieved  results are plotted at 𝛼 =0.4, 0.6, 0.8, and 1. All the  accurate and approximate results 

on the graphs have shown are much closed and indicates the validity of the present technique.  

 

 

 
Figure 4.1: ABS error of the solutions at 𝛼=0.4,0.6,0.8,1. 

y2(x, t) = −
9

4
sin h4  

x

2√2
  csc h3  

x

√2
  .  

tw

w
 

2

 

y3(x, t) =
9

128
.   

tw

w
 

3

.   cosh  
x

√2
 − 2  sech4   

x

2√2
  



IHJPAS. 36 (3) 2023 

320 
 

 
Figure4.2: Approximate solutions ym at 𝛼=0.4,0.6,0.8 and 1 and exact solution . 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.3: The 3D approximate solutions plots at  𝛼 = 0.4 , 0.6 , 0.8 and 1. 

 

 

 

 

 

 

 



IHJPAS. 36 (3) 2023 

321 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
Figure 4.4: The 3D absolute solution plots. 

  

5. Conclusion 

It is difficult to find analytical solutions to some FPDEs with initial and boundary 

conditions, so the solution here rarely exists. Here, the solutions of the time-fractional Newell-

Whitehead equation are made successfully. The results are convergent and much closed to the true 

solutions. The results are shown through 2D and 3D at various fractional-orders. So, the presented 

method has an excellent convergent rate and can be used to solve non-linear applications. 

 

 

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4. H. K. Jassim, ``The approximate solutions of three-dimensional diffusion and wave equations 

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6. F. Evirgen, ``Analyze the optimal solutions of optimization problems by means of fractional 

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