Int. J. Anal. Appl. (2023), 21:33 

 

 

Received: Feb. 26, 2023. 

2020 Mathematics Subject Classification. 80A20. 

Key words and phrases. fractional calculus; reduced differential transform method; diffusion equations. 

 

https://doi.org/10.28924/2291-8639-21-2023-33 © 2023 the author(s) 

ISSN: 2291-8639  

1 

 

Fractional Reduced Differential Transform Method for Solving Mutualism Model with 

Fractional Diffusion 

Mohamed Ahmed Abdallah1,2,*, Khaled Abdalla Ishag3 

1Department of Mathematics, Faculty of Sciences, University of Tabuk, Tabuk, KSA 

2Department of Basic Science, Faculty of Engineering, University of Sinnar, Sinnar, Sudan 

3Department of Basic Science, Faculty of Engineering Science, Omdurman Islamic University, 

Omdurman, Sudan 

*Corresponding author: mohamed.ah.abd@hotmail.com 

ABSTRACT. This study presents the fractional reduced differential transform method for a nonlinear mutualism model with 

fractional diffusion. The fractional derivatives are described by Caputo's fractional operator. In this method, the solution is 

considered as the sum of an infinite series. Which converges rapidly to the exact solution. The method eliminates the need 

to use Adomian's polynomials to calculate the nonlinear terms. To show the efficiency and accuracy of this method, we 

compared the results of the fractional derivatives orders with the ordinary derivative order index α=1 for the nonlinear 

mutualism model with fractional diffusion. Approximate solutions for different values of the fractional derivatives together 

with non-fractional derivatives and absolute errors are represented graphically in two and three dimensions. From all 

numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional 

systems of partial differential equations over existing methods. 

 

1. Introduction 

Recently, it has turned out that many phenomena in engineering and other sciences can be described 

by models using mathematical tools from fractional calculus [1], fractional calculus owes its origin to 

a question of whether the meaning of a derivative to an integer order could be extended to still be 

valid when n is not an integer. Diffusion phenomena is one the most important topic in heat transfer, 

especially in mechanics engineering and biological population. In the earlier literature most of the 

https://doi.org/10.28924/2291-8639-21-2023-33


2 Int. J. Anal. Appl. (2023), 21:33 

 

discussions are devoted to coupled systems of two equations. In the recent years, attention has been 

given to reaction-diffusion systems with three population species, the densities of three populations 

𝑢,𝑣,𝑤 are governed by the following coupled equations: ([2], [3]) 

𝐷𝑡
𝛼𝑢 − d1∇

2u = 𝑢(𝑎1(𝑡,𝑥) − 𝑏1(𝑡,𝑥)𝑢 + 𝑐1(𝑡,𝑥)𝑣) 

𝐷𝑡
𝛼𝑣 − d2∇

2𝑣 = 𝑣(𝑎2(𝑡,𝑥) − 𝑏2(𝑡,𝑥)𝑣 + 𝑐2(𝑡,𝑥)𝑢 + 𝑒(𝑡,𝑥)𝑤)                          (1) 

𝐷𝑡
𝛼𝑤 − d3∇

2𝑤 = 𝑤(𝑎3(𝑡,𝑥) − 𝑏3(𝑡,𝑥)𝑤 + 𝑐3(𝑡,𝑥)𝑣) 

with initial conditions  

𝑢(𝑥,0) = 𝑢0,   𝑣(𝑥,0) = 𝑣0,   𝑤(𝑥,0) = 𝑤0  

where 𝑛 − 1 < 𝛼 ≤ 𝑛, for each 𝑖 = 1,2,3, 𝑑𝑖 is constant and 𝑎𝑖,𝑏𝑖,𝑐𝑖, 𝑒  are smooth functions [2], ∇
2 

denotes Laplacian with respect to the variables 𝑥 = (𝑥1,𝑥2,𝑥3) and 𝑢(𝑥,𝑡),𝑣(𝑥,𝑡),𝑤(𝑥,𝑡)  is solution 

of Eq. (1). If 𝑑𝑖 = 0 for each 𝑖 = 1,2,3 in Eq. (1) we obtain a model of Lotka Volterra for prey-

predator. 

2. Preliminaries and Fractional Calculus 

In this section, gives some important definitions, such as the gamma function and basic definitions 

of the fractional derivatives. 

2.1. Gamma Function  

Gamma function Γ(𝑛) is simply the generalization of factorial to complex and real arguments. The 

gamma function can be defined as ([5], [6])  

Γ(𝑛) = ∫ 𝑡𝑛−1𝑒−𝑡𝑑𝑡 = (𝑛 − 1)!, 𝑛 ∈ 𝐼𝑁
∞

0
                                        (2) 

which is convergent for 𝑛 > 0. A recurrence formula for gamma function is ([5], [6]) 

Γ(𝑛 + 1) = 𝑛Γ(𝑛) 𝑓𝑜𝑟 𝑛 ∈ 𝐼𝑅+                                                  (3) 

Γ(𝑛) =
Γ(𝑛+1)

n
 𝑓𝑜𝑟 𝑛 ∈ 𝐼𝑅−                                                          (4) 

2.2. Fractional Derivatives 

Definition (1): Riemann-Liouville Fractional Integral Operator 

Suppose that 𝛼 > 0, 𝑛 − 1 < 𝛼 ≤ 𝑛, the Riemann-Lioville fractional integral define as [5] 

𝐷𝑡
−𝛼

𝑎
𝑅𝐿 (𝑓(𝑡)) =

1

Γ(𝛼)
∫(𝑡 − 𝑢)𝛼−1𝑓(𝑢)𝑑𝑢

𝑡

𝑎

                                      (5) 

Note: Riemann-Liouville fractional differential operator define as 

𝐷𝛼𝑅𝑙 𝑓(𝑡) = 𝐷𝑛𝐷𝛼−𝑛𝑓(𝑡), 𝛼 < 𝑛                                           (6) 

Definition (2): Caputo Fractional Differential Operator 

Suppose that 𝛼 > 0, 𝑛 − 1 < 𝛼 ≤ 𝑛, the Caputo fractional differential define as [5] 



3 Int. J. Anal. Appl. (2023), 21:33 

 

𝐷𝑡
𝛼

𝑎
𝐶 (𝑓(𝑡)) =

{
 
 

 
 1

Γ(𝑛 − 𝛼)
∫

𝑓𝑛(𝑢)

(𝑡 − 𝑢)𝛼−𝑛+1
𝑑𝑢,    𝑛 − 1 < 𝛼 < 𝑛  

𝑡

𝑎

𝑑𝑛

𝑑𝑡𝑛
𝑓(𝑡)                                                 𝛼 = 𝑛 ∈ 𝑁 

                        (7) 

Riemann-Liouville and Caputo fractional integral operator for polynomial is [5] 

𝐷𝑡
−𝛼

0
𝑅𝐿 (𝑡𝑛) = 𝐷𝑡

−𝛼
0
𝐶 (𝑓(𝑡)) =

Γ(𝑛 + 1)

Γ(𝛼 + 𝑛 + 1)
𝑡𝛼+𝑛                                                   (8) 

Definition (3): The Mittag- Leffler Function 

Suppose 𝛼 > 0,𝛽 > 0, then the Mittag-Leffler function define by [5] 

𝐸𝛼,𝛽(𝑡) = ∑
𝑡𝑘

Γ(𝛼𝑘 + 𝛽)
                                                                               (9)

∞

𝑘=0

 

 

3. Fractional Reduced Differential Transform Method 

Fractional Reduced Differential Transform Method (FRDTM) is iteration method, suppose 

𝑢(𝑡,𝑥1,𝑥2,…,𝑥𝑛) be analytical and continuously differentiable with respect to 𝑛 + 1 variables 

𝑡,𝑥1,𝑥2, . . ,𝑥𝑛 in the domain of interest; then FRDTM in 𝑛 dimensions for the following differential 

equation 

𝐷𝑡
𝛼𝑢 + 𝐿𝑢 + 𝑁(𝑢) = 0                                                                   (10) 

where 𝐷𝑡
𝛼 is differential operator with respect time, 𝐿 differential operator with respect variables 

𝑥1,𝑥2, . . ,𝑥𝑛 and 𝑁(𝑢) is nonlinear term ([7]-[10]). 

𝑢𝑘(𝑥1,𝑥2,…,𝑥𝑛) =
Γ(𝑘𝛼 + 1)

Γ(𝛼(𝑘 + 1) + 1)
[−𝐿(𝑢𝑘)− ∑𝑁(𝑢𝑟)𝑁(𝑢𝑘−𝑟)

𝑘

𝑟=0

]                  (11) 

The approximate solution is given by ([7], [8]). 

𝑢(𝑡,𝑥1,𝑥2,…,𝑥𝑛) = ∑ 𝑢𝑘𝑡
𝛼𝑘 = 𝑢0 + 𝑢1𝑡

𝛼 + 𝑢2𝑡
2𝛼 + ⋯                       (12)

∞

𝑘=0

 

 

4. Numerical Results 

In this section, we assume 𝑑𝑖 = 1,𝑎𝑖 = 1 𝑓𝑜𝑟  𝑖 = 1,2,3,  𝑏1 = 𝑏3 = 1, 𝑏2 = 3, 𝑐1 = 𝑐2 = 𝑐3 = 0.5, 

𝑒 = 0.5 in Eq. (1) 

𝐷𝑡
𝛼𝑢 = uxx + 𝑢 − 𝑢

2 + 0.5𝑢𝑣 

𝐷𝑡
𝛼𝑣 = vxx + 𝑣 − 3𝑣

2 + 0.5𝑢𝑣 + 0.5𝑤𝑣 

𝐷𝑡
𝛼𝑤 = wxx + 𝑤 − 𝑤

2 + 0.5𝑣𝑤 



4 Int. J. Anal. Appl. (2023), 21:33 

 

with initial conditions      

𝑢(𝑥,0) = 𝑒𝑥,   𝑣(𝑥,0) = 𝑥,    𝑤(𝑥,0) = 𝑥 − 𝜋,   0 ≤ 𝑥 ≤ 10 

Applied FRDTM 

𝑢𝑘+1 =
Γ(𝑘𝛼 + 1)

Γ(𝛼(𝑘 + 1) + 1)
((𝑢𝑘)𝑥𝑥 + 𝑢𝑘 − ∑𝑢𝑟𝑢𝑘−𝑟

𝑘

𝑟=0

+
1

2
∑𝑢𝑟𝑣𝑘−𝑟

𝑘

𝑟=0

) 

𝑣𝑘+1 =
Γ(𝑘𝛼 + 1)

Γ(𝛼(𝑘 + 1) + 1)
((𝑣𝑘)𝑥𝑥 + 𝑣𝑘 − 3∑𝑣𝑟𝑣𝑘−𝑟

𝑘

𝑟=0

+
1

2
∑𝑣𝑟𝑢𝑘−𝑟

𝑘

𝑟=0

+
1

2
∑𝑣𝑟𝑤𝑘−𝑟

𝑘

𝑟=0

) 

𝑤𝑘+1 =
Γ(𝑘𝛼 + 1)

Γ(𝛼(𝑘 + 1) + 1)
((𝑤𝑘)𝑥𝑥 + 𝑤𝑘 − ∑𝑤𝑟𝑤𝑘−𝑟

𝑘

𝑟=0

+
1

2
∑𝑤𝑟𝑣𝑘−𝑟

𝑘

𝑟=0

) 

Given                             𝑢0 = 𝑒
𝑥,     𝑣0 = 𝑥,    𝑤0 = 𝑥 − 𝜋 

when 𝑘 = 0 

𝑢1 =
1

Γ(𝛼 + 1)
((𝑢0)𝑥𝑥 + 𝑢0 − 𝑢0

2 +
1

2
𝑢0𝑣0)  

𝑢1 =
(2 − 𝑒𝑥 + 0.5𝑥)𝑒𝑥

Γ(𝛼 + 1)
 

𝑣1 =
1

Γ(𝛼 + 1)
((𝑣0)𝑥𝑥 + 𝑣0 − 3𝑣0

2 +
1

2
𝑣0𝑢0 +

1

2
𝑣0𝑤0) 

𝑣1 =
𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥 + 0.5𝜋𝑥

𝛤(𝛼 + 1)
 

𝑤1 =
1

Γ(𝛼 + 1)
((𝑤0)𝑥𝑥 + 𝑤0 − 𝑤0

2 +
1

2
𝑤0𝑣0) 

𝑤1 =
(x − π)(1 + 𝜋 − 0.5𝑥)

Γ(𝛼 + 1)
 

when 𝑘 = 1 

𝑢2 =
1

Γ(2𝛼 + 1)
((𝑢1)𝑥𝑥 + 𝑢1 − 2𝑢0𝑢1 +

1

2
𝑢0𝑣1 +

1

2
𝑢1𝑣0) 

𝑢2 =
(5 − 9𝑒𝑥 + 2𝑒2𝑥 + 0.25(7 − 𝜋)𝑥 − 0.5𝑥𝑒𝑥 − 𝑥2)𝑒𝑥

Γ(2𝛼 + 1)
 

𝑣2 =
1

Γ(2𝛼 + 1)
((𝑣1)𝑥𝑥 + 𝑣1 − 6𝑣0𝑣1 +

1

2
𝑣0𝑢1 +

1

2
𝑣1𝑢0 +

1

2
𝑣0𝑤1 +

1

2
𝑣1𝑤0) 

𝑣2 =
(1 − 0.5π − 5.5𝑥 + 0.5𝑒𝑥)(𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥 + 0.5𝜋𝑥)− 𝑥3

Γ(2𝛼 + 1)

+
(0.25𝑒𝑥 − 0.5𝜋2 + 0.25𝜋 + 0.5)𝑥2 + (1.5𝑒𝑥 − 0.5𝑒2𝑥)𝑥 + 𝑒𝑥 − 5

Γ(2𝛼 + 1)
 



5 Int. J. Anal. Appl. (2023), 21:33 

 

𝑤2 =
1

Γ(2𝛼 + 1)
((𝑤1)𝑥𝑥 + 𝑤1 − 2𝑤0𝑤1 +

1

2
𝑤0𝑣1 +

1

2
𝑤1𝑣0) 

𝑤2 =
(𝑥 − 𝜋)(π + 1 − 0.5𝑥)(2π + 1 − 1.5𝑥)+ 0.5𝑥(𝑥 − 𝜋)(𝑥 − 0.5𝜋𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥) − 1

Γ(2𝛼 + 1)
 

when 𝑘 = 2 

𝑢3 =
1

Γ(3𝛼 + 1)
((𝑢2)𝑥𝑥 + 𝑢2 − 2𝑢0𝑢2 − 𝑢1

2 +
1

2
𝑢0𝑣2 +

1

2
𝑢1𝑣1 +

1

2
𝑢2𝑣0) 

𝑣3 =
1

Γ(3𝛼 + 1)
((𝑣2)𝑥𝑥 + 𝑣2 − 6𝑣0𝑣2 − 3𝑣1

2 +
1

2
𝑢0𝑣2 +

1

2
𝑢1𝑣1 +

1

2
𝑢2𝑣0 +

1

2
𝑣0𝑤2 +

1

2
𝑣1𝑤1

+
1

2
𝑣2𝑤0) 

𝑤3 =
1

Γ(3𝛼 + 1)
((𝑤2)𝑥𝑥 + 𝑤2 − 2𝑤0𝑤2 − 𝑤1

2 +
1

2
𝑤0𝑣2 +

1

2
𝑤1𝑣1 +

1

2
𝑤2𝑣0) 

. 

. 

. 

𝑢(𝑥,𝑡) = ∑ 𝑢𝑘𝑡
𝛼𝑘 = 𝑢0 + 𝑢1𝑡

𝛼 + 𝑢2𝑡
2𝛼 + ⋯

∞

𝑘=0

 

𝑢(𝑥,𝑡) ≅ 𝑒𝑥 +
(2 − 𝑒𝑥 + 0.5𝑥)𝑒𝑥

Γ(𝛼 + 1)
𝑡𝛼 +

(5 − 9𝑒𝑥 + 2𝑒2𝑥 + 0.25(7 − 𝜋)𝑥 − 0.5𝑥𝑒𝑥 − 𝑥2)𝑒𝑥

Γ(2𝛼 + 1)
𝑡2𝛼 

𝑣(𝑥,𝑡) = ∑ 𝑣𝑘𝑡
𝛼𝑘 = 𝑣0 + 𝑣1𝑡

𝛼 + 𝑣2𝑡
2𝛼 + ⋯

∞

𝑘=0

 

𝑣(𝑥,𝑡) ≅ 𝑥 +
𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥 + 0.5𝜋𝑥

𝛤(𝛼 + 1)
𝑡𝛼

+
(1 − 0.5π − 5.5𝑥 + 0.5𝑒𝑥)(𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥 + 0.5𝜋𝑥) − 𝑥3

Γ(2𝛼 + 1)
𝑡2𝛼

+
(0.25𝑒𝑥 − 0.5𝜋2 + 0.25𝜋 + 0.5)𝑥2 + (1.5𝑒𝑥 − 0.5𝑒2𝑥)𝑥 + 𝑒𝑥 − 5

Γ(2𝛼 + 1)
𝑡2𝛼 

𝑤(𝑥,𝑡) = ∑ 𝑤𝑘𝑡
𝛼𝑘 = 𝑤0 + 𝑤1𝑡

𝛼 + 𝑤2𝑡
2𝛼 + ⋯

∞

𝑘=0

 

𝑤(𝑥,𝑡)

≅ (𝑥 − 𝜋)+
(x − π)(1 + 𝜋 − 0.5𝑥)

Γ(𝛼 + 1)
𝑡𝛼

+
(𝑥 − 𝜋)(π + 1 − 0.5𝑥)(2π+ 1 − 1.5𝑥) + 0.5𝑥(𝑥 − 𝜋)(𝑥 − 0.5𝜋𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥)− 1

Γ(2𝛼 + 1)
𝑡2𝛼 

 



6 Int. J. Anal. Appl. (2023), 21:33 

 

Table 1. Numerical results of variable 𝑢(𝑥,𝑡) 

𝒙                               𝜶 = 𝟏                         𝜶 = 𝟎.𝟖                          𝜶 = 𝟎.𝟓                        

𝜶 = 𝟎.𝟐 

1.0e+014 * 

    0.0000                  -0.0000                        -0.0000                            -0.0000                           

-0.0000 

    0.0000                  -0.0000                        -0.0000                            -0.0000                           

-0.0000 

    0.0000                   0.0000                         0.0000                             0.0000                            

0.0000 

    0.0000                   0.0000                         0.0000                             0.0000                            

0.0000 

    0.0000                   0.0000                         0.0000                             0.0000                            

0.0000 

    0.0000                   0.0000                         0.0000                             0.0000                            

0.0000 

    0.0000                   0.0000                         0.0000                             0.0000                            

0.0000 

    0.0000                   0.0002                         0.0002                             0.0001                            

0.0001 

    0.0000                   0.0042                         0.0034                             0.0021                            

0.0010 

    0.0000                   0.0851                         0.0683                             0.0425                            

0.0209 

    0.0000                   1.7093                         1.3734                             0.8546                            

0.4193 

 



7 Int. J. Anal. Appl. (2023), 21:33 

 

 

Figure 1. Graphical presentation of variable 𝑢(𝑥,𝑡) 

 

Figure 2. Compression derivatives order between non-fractional order with fractional orders of 

variable 𝑢(𝑥,𝑡). 

-10
0

10

-10

0

10
-5

0

5

x 10
8

x axis

alpha=1

t axis

u
 a

x
is

-10
0

10

-10

0

10
-2

0

2

x 10
8

x axis

alpha=0.8

t axis

u
 a

x
is

-10
0

10

-10

0

10
-5

0

5

x 10
7

x axis

alpha=0.5

t axis

u
 a

x
is

-10
0

10

-10

0

10
-1

0

1

x 10
8

x axis

alpha=0.2

t axis

u
 a

x
is

0 1 2 3 4 5 6 7 8 9 10
-2

0

2

4

6

8

10

12

14

16

18
x 10

13

x axis

u
 a

x
is

Comparison between derivative orders

 

 

alpha=1

alpha=0.8

alpha=0.5

alpha=0.2



8 Int. J. Anal. Appl. (2023), 21:33 

 

Table 1 shows the approximate solution of fractional diffusion of variable 𝑢(𝑥,𝑡), it is noted that 

only the Second order of the FRDTM. Figure 1: The surface of diffusion variable 𝑢(𝑥,𝑡) is 

convergence between fractional order and ordinary order, in Figure 2: we get small difference between 

ordinary order with multiple fractional orders. 

Table 2. Numerical results of variable 𝑣(𝑥,𝑡) 

𝒙                               𝜶 = 𝟏                         𝜶 = 𝟎.𝟖                          𝜶 = 𝟎.𝟓                        

𝜶 = 𝟎.𝟐 

1.0e+009 * 

    0.0000                 -0.0000                         -0.0000                           -0.0000                           

-0.0000 

    0.0000                 -0.0000                         -0.0000                           -0.0000                           

-0.0000 

    0.0000                 -0.0000                         -0.0000                           -0.0000                           

-0.0000 

    0.0000                 -0.0000                         -0.0000                           -0.0000                           

-0.0000 

    0.0000                 -0.0000                         -0.0000                           -0.0000                           

-0.0000 

    0.0000                 -0.0003                         -0.0002                           -0.0002                           

-0.0001 

    0.0000                 -0.0023                         -0.0019                           -0.0012                           

-0.0006 

    0.0000                 -0.0182                         -0.0147                           -0.0091                           

-0.0045 

    0.0000                 -0.1473                         -0.1184                           -0.0737                           

-0.0361 

    0.0000                 -1.1998                         -0.9641                           -0.5999                           

-0.2943 

    0.0000                 -9.7643                         -7.8456                           -4.8821                           

-2.3950 

 



9 Int. J. Anal. Appl. (2023), 21:33 

 

 

Figure 3. Graphical presentation of variable 𝑣(𝑥,𝑡) 

 

Figure 4. Compression derivatives order between non-fractional order with fractional orders of 

variable 𝑣(𝑥,𝑡). 

-10
0

10

-10

0

10
-2

0

2

x 10
6

x axis

alpha=1

t axis

v
 a

x
is

-10
0

10

-10

0

10
-1

0

1

x 10
6

x axis

alpha=0.8

t axis

v
 a

x
is

-10
0

10

-10

0

10
-2

0

2

x 10
5

x axis

alpha=0.5

t axis

v
 a

x
is

-10
0

10

-10

0

10
-2

-1

0

x 10
6

x axis

alpha=0.2

t axis

v
 a

x
is

0 1 2 3 4 5 6 7 8 9 10
-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0
x 10

9

x axis

v
 a

x
is

Comparison between derivative orders

 

 

alpha=1

alpha=0.8

alpha=0.5

alpha=0.2



10 Int. J. Anal. Appl. (2023), 21:33 

 

Table 2 shows the approximate solution of fractional diffusion of variable 𝑣(𝑥,𝑡), it is noted that 

only the Second order of the FRDTM. Figure 3: The surface of diffusion of variable 𝑣(𝑥,𝑡) is 

convergence between fractional order and ordinary order, in Figure 4: we get small difference between 

ordinary order with multiple fractional orders. 

Table 3. Numerical results of variable 𝑤(𝑥,𝑡) 

𝒙                               𝜶 = 𝟏                         𝜶 = 𝟎.𝟖                          𝜶 = 𝟎.𝟓                        

𝜶 = 𝟎.𝟐 

1.0e+007 * 

    0.0000                 -0.0001                        -0.0001                            -0.0000                           

-0.0000 

    0.0000                 -0.0000                        -0.0000                            -0.0000                           

-0.0000 

    0.0000                 -0.0000                        -0.0000                            -0.0000                           

-0.0000 

    0.0000                 -0.0000                        -0.0000                            -0.0000                           

-0.0000 

    0.0000                   0.0001                         0.0001                             0.0000                            

0.0000 

    0.0000                   0.0011                         0.0009                             0.0006                            

0.0003 

    0.0000                   0.0077                         0.0062                             0.0038                            

0.0019 

    0.0000                   0.0401                         0.0322                             0.0201                            

0.0098 

    0.0000                   0.1828                         0.1469                             0.0914                            

0.0448 

    0.0000                   0.7647                         0.6144                             0.3823                            

0.1876 

    0.0000                   3.0144                         2.4220                             1.5072                            

0.7394 

 

 



11 Int. J. Anal. Appl. (2023), 21:33 

 

 

Figure 5. Graphical presentation of variable 𝑤(𝑥,𝑡) 

 

Figure 6. Compression derivatives order between non-fractional order with fractional orders of 

variable 𝑤(𝑥,𝑡). 

-10
0

10

-10

0

10
-5

0

5

x 10
5

x axis

alpha=1

t axis

w
 a

x
is

-10
0

10

-10

0

10
-5

0

5

x 10
5

x axis

alpha=0.8

t axis

w
 a

x
is

-10
0

10

-10

0

10
-1

0

1

x 10
5

x axis

alpha=0.5

t axis

w
 a

x
is

-10
0

10

-10

0

10
-5

0

5

x 10
4

x axis

alpha=0.2

t axis

w
 a

x
is

0 1 2 3 4 5 6 7 8 9 10
-0.5

0

0.5

1

1.5

2

2.5

3

3.5
x 10

7

x axis

w
 a

x
is

Comparison between derivative orders

 

 

alpha=1

alpha=0.8

alpha=0.5

alpha=0.2



12 Int. J. Anal. Appl. (2023), 21:33 

 

Table 3 shows the approximate solution of fractional diffusion of variable 𝑤(𝑥,𝑡), it is noted that 

only the Second order of the FRDTM. Figure 5: The surface of diffusion of variable 𝑤(𝑥,𝑡) is 

convergence between fractional order and ordinary order, in Figure 6: we get small difference between 

ordinary order with multiple fractional orders. 

 

5.  Conclusions 

The fractional reduced differential transform method has been successfully applied to obtain an 

analytical approximate solution for the mutualism model with fractional diffusion. It is easy to 

recognize that FRDTM is powerful mathematical tool for solving different kinds of linear and/or 

nonlinear fractional partial differential equations the FRDTM is no need to use Adomian's polynomials 

to calculate the nonlinear terms. We have concluded that the fractional derivative of diffusion 

mutualism model is more accurate than ordinary derivative order. From all numerical results, we can 

conclude the efficiency of the proposed method for solving different types of nonlinear fractional 

partial differential equations so we recommended researchers would use Fractional reduced 

differential transform method when derivation the mathematical models (biological phenomena) for 

fractional derivatives. 

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the 

publication of this paper. 

 

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