A Study of Applied Reduced Differential Transform Method Using

Volterra Integral Equations in Solving Partial Differential Equations

NGUYEN MINH TUAN

Department of Mathematics, Faculty of Applied Science

King Mongkuts University of Technology North Bangkok

1518 Pracharat 1 Road,Wongsawang, Bangsue

Krungthep Mahanakorn 10800, THAILAND.

Abstract: - Nowadays, integration is one of the trending fields applied in calculus, especially in partial

differential equations. Researchers are contributing to support useful utilities to solve partial differential

equations in many kinds of methods. In this paper, we perform an application of Volterra Integral

Equations in a reduced differential transform method (we call VIE-RDTM) to find the approximate

solutions of partial differential equations. The aim is to find the approximate solutions approach to the

exact solutions with more general forms. We also extend some new results for basic functions and

compare the solutions using the reduced differential transform method and VIE-RDTM by depicting

the approximate solutions in some partial differential equations. The results showed that the VIE-RDTM

method gets the state-of-the-art general form of the solutions when the errors approach zero.

Key-Words: - Volterra Integral, Reduced Differential Transform Method, RDTM, VIE-RDTM.

Received: December 17, 2022. Revised: August 13, 2023. Accepted: September 9, 2023. Published: October 3, 2023.

1 Introduction

Recently, linear and nonlinear differential

equations have been applied in many kinds of

life, especially in the real world of computer

science, and economic problems (readers could

see more in [1, 2, 3, 4, 5]). Whilst applying in

economics looked at a newfangled approach to

reveal the nature of the solutions or control the

models in optimal problems, partial equations

are developed with more and more various types

of form, and more complete structures (shown in

[6]). The next generation of partial equations in

fractional partial equations contributed to the

development of technology, and economics.

Fractional calculus appeared as the consequence

of modern technology development and has been

expanded in different aspects of science (Shown

in [7], and [8]). Fractional calculus has rapidly

become a significant tool to control the models

in economics such as controlling price problems,

controlling options, or controlling inflation

problems. Besides that, fractional calculus also

inherited all the powerful properties from partial

differential calculus, and its application in

variation structure resembling majors (See in

[9]). However, partial differential equations have

played an important role and have not completed

the stages in the future. The development from

the partial differential equation will create the

hard roots and basic preliminaries for construing

the high floors of calculus. With the state-of-the-

art contribution of fractional partial differential

calculus, the scenario of various types of

methods have created variants to apply easier to

specific problems. The homotopy perturbation

method (see in [10]) is one of the most popular

methods applied to solve partial differential

equations. This method is useful in finding

approximate solutions to economic problems.

Similar to the homotopy perturbation method,

the Variational Iteration method is also applied

in many kinds of partial differential equations

and has got a good performance in approximate

solution results (see in paper [11]). Besides that,

the Laplace transform method is a classical

method that contributed to the development of

calculus (in paper [12]). Compared to other

methods, the Laplace method is useful in solving

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Nguyen Minh Tuan

E-ISSN: 2732-9976

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Volume 3, 2023

related integral equations, and some equations

about economics, and finance (see more in [6]).

The Adomian Decomposition method (see in

paper [13, 14]) is used in solving Volterra

equations, by using an iterative way, the

Adomian Decomposition method establishes the

sequences based on initial values to find

approximate solutions. In general, most of the

methods have shown good ways to directly or

indirectly illustrate the advantages and powerful

tools to find the solutions. The Reduce

Differential Transform method was introduced

by Y. Kensin et al. (See the series [15, 16, 17,

18]) and has been extended with effective

application immensely in some different

branches of partial differential equations (shown

in [19, 20, 21, 22, 23]). After that, the Reduce

Differential Transform method was expanded

into the fractional Reduce Differential

Transform method to follow the fractional

differential equations (see more in [24, 25, 26,

27, 28]). The Fractional differential derivative is

the hottest trend variation to connect the real

world and is applied in the majority of calculus.

Many books and papers have focused on

stochastic problems wherein integrals are a good

presentation with respect to the components in

stochastic optimization problems. With the hope

to combine integration and derivative, the

purpose is to find exact solutions based on initial

conditions, we applied Volterra Integral

Equations as the facility to reveal the exact

solutions. This paper considers the differential

partial equations written formed (1).

Af(x,t)+Bf(x,t)+ Mf(x,t)+Lf(x,t) = g(x,t) (1)

with the initial condition f(x,0) = h(x), where A

=

m

x



, and B =

n

x



are partial differential

derivative, M, L represent the linear or nonlinear

terms having partial derivative derivatives, and

g(x,t), h(x) are given functions. In some

fractional differential partial equations, the

constant values are hidden and the integration

will reveal the exact solution under terminal

conditions.

2 Methodology

2.1 Some Basic Definitions for Calculus

To see more details about the notations and

definitions, readers could read in [29, 30, 31, 32].

Now we would remind some related basic

reports in partial differential derivative theory.

First, we consider some definitions

Definition 1 (Gamma function) (Shown in [29],

P.8) For z ∈ C, and Re(z) > 0, the integral as

follows is defined

 

1

0

zt

z t e dt







(2)

Proposition 1 Base on the equation 2, we have

some specific properties related to the gamma

function using integration by part and direct

integration Γ(1) = 1:

Γ(z+1) = zΓ(z) (3)

Γ(n+1) = n! (4)

Definition 2 (Convert to Volterra Integral

equations)

     

1

0 0 0

int

1

0

( , )

1,

1!

x x x

n

n fold egration

xn

f w t dw dw

x w f w t dw

n



 





  



To understand more clearly equation (5), we

could start practicably from the equation (6) and

(7):

     

0 0 0

,,

x x x

f w t dwdw x w f w t dw

  

(6)

     

2

0 0 0 0

1

,,

2!

x x x x

f w t dwdwdw x w f w t dw

   

(7)

Now we can embark form the first step of

equation (6): From the right-hand side, set

     

0,

x

h x x w f w t dt



, and we have

differentiation

   

0,

x

h x f w t dt

x





. After

that, integrating by x and get the equation as the

following:

   

00 ,

xx

h x f w t dtdw

(8)

and the equation (8) is equal the left-hand side of

the equation (6). So, the equation (6) has been

proved. Similar to the equation (7), and using the

inductive method to get the transformation

formula (5).

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Nguyen Minh Tuan

E-ISSN: 2732-9976

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2.2 Reduced Differential Transformation

Method

The reduced Differential Transformation

Method is very useful in solving linear and

nonlinear equations. This method has shown a

reduced differential transform to find the

approximate solution in the differential equation

(See more in [31, 32, 34, 35, 36, 37]. Now we

will summarize method as follows. Let’s

examine the two-variable function expressed as

f(x,t), we can write as the following

f(x,t) = w(x)v(t) (9)

Now, we can express equation (9) be formed

       

0 0 0

,,

i j k

kk

i j k

F x t w i x v j t H i j t

  

  









  

(10)

where Hk(i,j) = w(i)v(j) denotes the spectrum of

F(x,t). Then the of F(x,t) is formed by

     

0

1,

1

k

kk

k

tt

H x F x t

kt 







  



(11)

where k is the order of time derivative. Then the

inverse transformation of Hk is defined by

    

0

,k

kk

k

F x t H x t t









(12)

Combine equation (11) and equation (12), we

have

       

0

1

,,

1

kk

k

ktt

F x t F x t t t

kt













  





(13)

If we choose t0 = 0, from equation (13) we have

     

0

1

,,

1

kk

k

ktt

F x t F x t t

kt











  





(14)

By applying inductive method for equation (14),

we have solution:

   

, lim ,

k

f x t F x t





(15)

2.3 Using Volterra Integral Equations -

Reduced Differential Transformation

Method

With the reduced differential transform method,

the approximate solution has been found when

approaching exact solutions. In some cases, the

solution could be expressed more generally

when it could be different by a constant. Now we

perform a new extension of this method, which

we call a Volterra Integral Equations of reduced

differential transform method (VIE-RDTM).

Let’s examine the two-variable function

expressed as f(x,t), we can write as the following

f(x,t) = w(x)v(t) (16)

we can integrate the equation (16) respect to x

from 0 to x, and then integrate the equation (16)

respect to t from 0 to t (or respect to x one more

time):

     



0 0 0 0

, , ,

x t x x

F x t f w t dtdw or f w t dtdw   

(17)

Now, we can express equation (17) by formed

     



000

00

0

,

xij

kij

xx

k

kk

k

F x t u i x v j t dw

H i j t dw or x w H t dw



















(18)

where Hk(i, j) = u(i)v(j) is the spectrum of F(x,t).

Then the fractional reduced differential

transform method of F(x,t) is formed by

     

0

1,

1

k

kk

k

tt

H x F x t

kt 







  



(19)

Then the inverse transformation of Hk is defined

by

    

0

,k

kk

k

F x t H x t t









(20)

Combine equation (19) and equation (20), we

have

       

0

1

,,

1

kk

k

ktt

F x t F x t t t

kt













  





(21)

If we choose t0 = 0, from equation (21) we have

     

0

1

,,

1

kk

k

ktt

F x t F x t t

kt











  





(22)

By applying the inductive method for equation

(22):

   

, lim ,

k

f x t F x t





(23)

Finally, the solution is called from equation (22),

and equation (23) is also the original solution of

the equations.

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Nguyen Minh Tuan

E-ISSN: 2732-9976

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Volume 3, 2023

2.4 Basic Functions Using Transformation

Results

Now we will illustrate the fundamental function

results of the extension of composed of Volterra

Integral equations and reduced differential

transform method (VIE-RDTM) (shown in [38,

39, 40, 41, 42]). Let Tf be the transformation of

this method and g(x,t), h(x,t) are fundamentally

analytic functions. By setting Gk = Tf(g), Hk =

Tf(h) is the results of the transform after

integrating by x, we have some results as shown

in Table 1 (See more in [43],[44],[45]). We will

prove some specific terms as the following

properties:

Proposition 2

   

     

0

00

,,

1,

!

x

kxx

kk

k

f x t g w t dw then

F x g w t dw G w dw

kt













(24)

Proposition 3

   

     

00

1

100

,,

11

,

!

xt

kxx

kk

k

f x t g w t dw then

F x g w t dw G w dw

k t k













(25)

Proposition 4

   

0

1

10

1

0

,,

1,

!

1

x

kx

kk

x

k

f x t g w t dw then

t

F x g w t dw

kt

k G w dw















(26)

Proposition 5

   

     

    

00

0

1

0

,,

1,

!

1

xx

kx

kk

x

k

f x t g w t dw then

t

F x x w g w t dw

kt

x w k G w dw













  





(27)

Proposition 6

   

    

2

0

2

20

2

0

,,

1,

!

12

x

kx

kk

x

k

f x t g w t dw then

t

F x g w t dw

kt

k k G w dw











  



(28)

Proposition 7

   

    

2

00

2

20

2

0

,,

1,

!

12

xx

kx

kk

x

k

f x t g w t dw then

t

F x g w t dw

kt

k k G w dw











  





(29)

Proposition 8

   

     

0

00

,,

xm

k

x

k k r

r

f x t t g w t dw then

F x r m G w dw















(30)

3 Numerical Results

In this section, we illustrate the solutions by

using Volterra Integral Equation-Reduced

Differential Transform Method compared to the

solutions using the Reduced Differential

Transform Method ([46, 47, 48]) through the

examples as the following:

Example 1 We consider the Black-Scholes

equation (See in [6]) formed

   

2

,,

0

f x t f x t

x

tx



  



(31)

satisfy the terminal condition f(x,0) = x2.

Using RDTM method Applying RDTM method

(see more detailed in [49]), from equation (31),

we turn the equation (31) into new form

       

2

12

,

1

11

kk

f x t

k H x x H x

kx









(32)

Using the condition, we have the following

equations (33), (34):

H0(x)=x2; H1(x)=2x2 (33)

H2(x)=2x2; H1(x)=

4

3

x2 (34)

By inductive method, we have the specific

solution as (35).

     

23

222

, 1 2 2! 3!

tt

f x t x t



    





(35)

This result approach to exact solution

u(x,t) = x2 e2t .

Using VIE-RDTM method: From equation (31),

we integrate both sides of the equation using

integration by part respect to x from 0 to x, and

then integration respect to t from 0 to t to turn

(31) into Volterra integration equation, we

rewrite the equation (31) as equation (36):

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DOI: 10.37394/232021.2023.3.11

Nguyen Minh Tuan

E-ISSN: 2732-9976

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Volume 3, 2023

     

 

2

00

, , 2 ,

2,

xx

f w t dw x f x t xf x t d

x

d w dwd





















(36)

Apply transformation in Table 1 by using VIE-

RDTM, from equation (36), we have the new

form of equation (37):

     

 

2

11

0

1

0

12

2

x

k k k

x

k

H w dw x H x xH x

k

H w dw

k















(37)

From (37), we will construe the inductive step by

counting values H1, H2, H3, ...

H0(x)=x2; H1(x)=2x2 (38)

H2(x)=2x2; H1(x)=

4

3

x2 (39)

We got the exact solution using the VIE-RDTM

method using the formula equation:

     

23

222

, 1 2 2! 3!

tt

f x t x t



    





(40)

Finally, the solution in equation (40) lead to

another form of exact solution f(x,t) = x2e2t , and

is illustrated in Figure 1 and Figure 2.

Example 2 In the second example, a partial

differential equation for a European option is

given (See in [6]) by:

     

2

, , , 0

f x t f x t f x t

xx

t x x

  

  

  

(41)

subject to initial condition u(x,0) = x2.

Using RDTM method Applying the

transformation using RDTM, we have equation

(42):

     

2

12

1

k k k

H x x H x x H x

k x x











  



(42)

By using inductive step, we have equation (43),

(44):

H0(x) = x2; H1(x) = 4x2 (43)

H2(x) = 8x2; H3(x) = 32x2/2; H4(x) = 64x2/3;···

(44)

Finally, using RDTM, we have the solution

shown in (45):

     

23

244

, 1 4 2! 3!

tt

f x t x t



    





. (45)

The equation (45) will lead to the exact solution

f(x,t) = x2e4t .

Using VIE-RDTM method From equation (41),

integrating both sides respect to x then applied

the formula we have equation 46:

     

2

0

00

,,

3 , ,

x

xx

x w f x t dw x f x t

t

wf w t dw x w f w t dw







  





(46)

       

   

2

0

3

1

x

kk

x

kx

k

x H x wH w dw

x w H w dw kx w H w dw





















(47)

We will construe the inductive values as

equation (48), (49):

H0 = x3/3; H1 = 4x3/3 (48)

H2 = 8x2; H3 = 32x2/2; H4 = 64x2/3; ··· (49)

Finally, we have the exact solution using VIE-

RDTM as equation (50):

     

23

244

, 1 4 2! 3!

tt

f x t x t



    





. (50)

The equation (50) will lead to the exact solution

f(x,t) = x2e4t and will be graphed in Figure 3 and

Figure 4.

Example 3 In this example, a fractional

diffusion PDE is to demonstrate the relation

between variable stable distribution (See in [6])

given by

     

22

,,

,0

f x t f x t

t x t f x t t

tx



   



(51)

subjected to the term condition u(x,0) = 1, and

 

,0u x x

t





.

Using RDTM method Apply RDTM method,

we have the form:

(k +1)Hk+1 = x

t



Hk+1 −Hk−1 +δ(k −1) (52)

By iterative method, we have the following

terms in equation (53):

H0(x) = 1; H1(x) = x; H2(x) = 0 (53)

H3(x) = x/2; H4(x)= 0 (54)

H5(x) = x/8; H6(x)= 0 (55)

The solution of equation (51) has form:

 

24

, 1 1 28

tt

f x t xt

    





. (56)

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The equation (56) will be convergent to exact

solution

 

2

,1

t

f x t xte

.

Using VIE-RDTM method: From equation

(51), integrating both sides using integration by

part respect to x, we have

     

 

00

22

00

, , ,

,

xx

t f w t dw xf x t f w t d

t

t f w t dw t dw











(57)

Applying transformation using Table 1 and we

have equation (58):

       

   

     

1

0

00

0

11

1

11

x

kk

k

x

kr

r

k

xx

kr

r

k H w dw xH x k

r H w dw

r H w dw k dw















  



   











(58)

Applying inductive for equation (58), we have

equation (59), (60), (61):

H0(x) = 1; H1(x) = x. (59)

H2(x) = 0; H3(x)= x/2; H4(x)= 0. (60)

H5(x) = x/8; H6(x) = 0. (61)

Similarly, using VIE-RDTM we have the exact

solution expressed as equation (62):

 

24

, 1 1 2 8!

tt

f x t xt

    





(62)

The result will lead to exact solution form

 

2

,1

t

f x t xte

and has been depicted in

Figure 5 and Figure 6.

Example 4 In this example, we illustrate a

diffusion process defined by the Burgers

equation as the following (see in [9]):

   

 

,,

1 , 0

f x t f x t

f x t

tx



  



(63)

satisfy the initial condition: u(x,0) = x−1.

Using RDTM method

Similarly, from equation (63), we have

 

1

0

1

k

k k k r r

r

H x x H H H

kx









  









(64)

By taking inductive for the terms, we have

equation (65), (66):

H1(x) = −x; H2(x) = x; (65)

H3(x) = −x; H4 = x; ··· (66)

Finally, we have the exact solution using the

RDTM method shown as equation (67):

 

2 3 4

, 1 1f x t x t t t t       

(67)

The equation (67) will be convergent to the exact

solution

 

,1

1

x

f x t t

   

.

Using VIE-RDTM method From equation (63),

we integrate both sides:

   

0

,,

x

f w t dw f x t

t

f w t f w t dw

x









(68)

Applied transformation by using RDTM, we

have equation (69):

   

0

1

,

k

x

Hx

f w t dw

tk

H w f w dw

x





















(69)

Counting the terms by an inductive method, we

have equation (70), (71):

H0(x) = x−1; H1(x) = −x; H2(x) = x. (70)

H3(x) = −x; H4(x) = x; ··· (71)

Finally, we have the exact solution using the

VIERDTM method shown as equation (72):

 

2 3 4

, 1 1f x t x t t t t       

(72)

The equation (72) will lead to exact solution

 

,1

1

x

f x t t





and delineated in Figure 7 and

Figure 8.

Example 5 In this example, we illustrate a linear

differential equation defined by the following

(see in [50]):

     

2 2 2

22

, , ,

4f x t f x t f x t

t x x t

  



   

(73)

satisfy the initial condition:

u(x,0) = x2; ut(x,0) = ex.

Using VIE-RDTM method From equation (73),

we integrate both sides:

     

2

0 0 0

,,

4,

x x x

f w t f t

dw f x t dw

tt







  

(74)

Applied transformation by using RDTM, we

have equation (69):

      

   

00

0

4 1 2

1

xx

kk

x

k

k k H w dw H t

k H w dw

  







(75)

Counting the terms by an inductive method, we

have equation (70), (71):

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H0(x) = x2; H1(x) = ex; H2(x) = 1/4 − 3ex/8. (76)

H3(x) = 13ex/96; H4(x) = − 17ex /512; ··· (77)

Finally, we have the exact solution using the

RDTM method shown as equation (72):

 

2 3 4

,

41 ...

5 4 32 384 6144

41 ...

5 2 6 24

xx

x

f x t e e t

t t t t

e

t t t

et





     







     





(78)

The equation (78) will lead to exact solution

 

2

24

4

,45

t

xxt

t

f x t x e e





   





. compared to

[50] and performed in Figure 9 and Figure 10.

4 Conclusion

Partial differential calculus is usefully applied in

many fields of applied mathematics and

technology science. It is also applied in a huge

number of branches of finance, physics,

viscoelastic mechanics, or economics [51, 52]).

In this paper, we suggested the combination of

integration and differential method, and this

method showed the most convenient shortcut to

find the exact solution. The method is omitting

the derivative terms and replacing them with

reduced terms and integration. The method has

performed successfully to get the analytic

solution approaching the exact solution.

Acknowledgement:

The author Nguyen Minh Tuan acknowledges

the comments of anonymous referees, which

improved the manuscript qualitatively, and is

indebted to the editor for illuminating advice and

valuable discussions

References:

[1] A. Kilbas, H. M. Srivastava, and J. J. Trujillo,

Theory and applications of fractional differential

equations theory and applications of fractional

differential equations, (Elsevier B.V., 2006) 1st ed.

[2] A. B. Malinowska, T. Odzijewicz, and D. F.

Torres, Advanced methods in the fractional calculus

of variations, SpringerBriefs in Applied Sciences and

Technology, 2015.

[3] S. G. Georgiev, Fractional dynamic calculus and

fractional dynamic equations on time scales, Springer

International Publishing AG part of Springer Nature,

2018.

[4] V. E. Tarasov, Fractional dynamics applications

of fractional calculus to dynamics of particles, fields

and media, (Springer is part of Springer

SClence+Busmess Media, 2010) Chap. 1-2.

[5] S. G. Samko, A. A. Kilbas, and O. I. Marichev,

Fractional integrals and derivatives: Theory and

applications, (Gordon and Breach Science Publishers

S .A., 1993) Chap. 1-4.

[6] H. Pham, Continuous-time stochastic control and

optimization with financial applications, (Springer

Dordrecht Heidelberg London New York, Springer-

Verlag Berlin Heidelberg, 2009) Chap. 1-3. 1st ed.

[7] Y. Zhou, J. Wang, and L. Zhang, Basic theory of

fractional differential equations, (World Scientific

Publishing Co. Pte. Ltd, Printed in Singapore, 2016)

2nd ed.

[8] H. A. Fallahgoul, S. M.Focardi, and F. J.Fabozzi,

Fractional calculus and fractional processes with

applications to financial economics theory and

application, (Elsevier Limited, Academic Press is an

imprint of Elsevier, 2017) Chap. 1-2. 1st ed.

[9] V. Capasso and D. Bakstein, An introduction to

continuous-time stochastic processes, (Springer New

York Heidelberg Dordrecht London, 10.1007/978-1-

4939-2757-9, 2015) 3rd ed.

[10] H. Ji-Huan and Y. O. El-Dib, Homotopy

perturbation method with three expansions, Springer

International Publishing (2021).

[11] J. saberi nadjafi and F. Akhavan, Variational

iteration method for solving nonlinear differential

difference equations (nodes), Australian Journal of

Basic and Applied Sciences (2010).

[12] M. Shaeel, A. Khan, and S. A. Hasnain, Laplace

transformation and inverse Laplace transform

involving generalized incomplete hypergeometric

function, Pakistan Journal of Statistics (2021).

[13] D. A. Maturi, Adomian decomposition method

for solving heat transfer Lighthill singular integral

equation, International Journal of GEOMATE

(2022).

[14] D. J. EVANS and K. R. RASLAN, The

Adomian decomposition method for solving delay

differential equations, International Journal of

Computer Mathematics (2004).

[15] F. Ayaz, On the two-dimensional differential

transform method, Applied Mathematics and

Computation (2003).

[16] Y. Keskin and G. Oturanc, Reduced differential

transform method for solving linear and nonlinear

wave equations, Iranian Journal of Science &

Technology (2010).

[17] Y. Keskin and G. Oturanc, Reduced differential

transform method for partial differential equations,

International Journal of Nonlinear Sciences and

Numerical Simulation (2009).

EQUATIONS

DOI: 10.37394/232021.2023.3.11

Nguyen Minh Tuan

E-ISSN: 2732-9976

99

Volume 3, 2023

[18] Y. Keskin and G. Oturanc, Reduced differential

transform method for generalized kdv equations,

Mathematical and Computational Applications

(2010).

[19] H. Jafari, H. K. Jassim, S. P. Moshokoa, V. M.

Ariyan, and F. Tchier, Reduced differential transform

method for partial differential equations within local

fractional derivative operators, Advances in

Mechanical Engineering (2016).

[20] Y. Keskin and G. Oturanc, Application of

reduced differential transformation method for

solving gas dynamics equation, Int. J. Contemp.

Math. Sciences (2010).

[21] A. Taghavi, A. Babaei, and A. Mohammadpour,

Application of reduced differential transform method

for solving nonlinear reaction diffusion-convection

problems, Applications & Applied Mathematics

(2015).

[22] S. R. M. Noori and N. Taghizadeh, Study of

convergence of reduced differential transform

method for different classes of differential equations,

International Journal of Differential Equations

(2021).

[23] M. Sohail and S. T. Mohyud-Din, Reduced

differential transform method for solving a system of

fractional pdes, International Journal of Modern

Mathematical Sciences (2012).

[24] S. Das, Functional fractional calculus,

(Springer-Verlag Berlin Heidelberg, 2011) Chap. 1-

5.

[25] M. Riahi, E. Edfawy, and K. E. Rashidy, New

method to solve partial fractional differential

equations, Global Journal of Pure and Applied

Mathematics (2017).

[26] C. Milici, G. Draganescu, and J. T. Machado,

Introduction to fractional differential equations,

(Springer Nature Switzerland AG, 2019) Chap. 1-3.

[27] U. N. KATUGAMPOLA, A new fractional

derivative with classical properties, JOURNAL OF

THE AMERICAN MATHEMATICAL SOCIETY

(2010).

[28] C. Goodrich and A. C. Peterson, Discrete

fractional calculus, (Springer International

Publishing Switzerland, 2015) Chap. 1-4.

[29] R. Herrmann, Fractional Calculus An

Introduction For Physicists (World Scientific

Publishing Co. Pte. Ltd, 2014).

[30] I. Podlubny, Fractional Differential Equations

(Academic Press, 1999).

[31] S. Abuasad, A. Yildirim, I. Hashim, S. A. A.

Karim, and J. Gómez-Aguilar, Fractional multistep

differential transformed method for approximating a

fractional stochastic sis epidemic model with

imperfect vaccination, Environmental research and

public health (2019).

[32] M. S. Rawashdeh, A reliable method for the

space-time fractional burgers and time-fractional

cahn-Allen equations via the frdtm, Rawashdeh

Advances in Difference Equations (2017).

[33] S. Mukhtar, S. Abuasad, I. Hashim, and S. A. A.

Karim, Effective method for solving different types

of nonlinear fractional burgers equations, MDPI

(2020).

[34] B. K. Singh and V. K. Srivastava, Approximate

series solution of multi-dimensional, time fractional-

order (heat-like) diffusion equations using frdtm,

Royal Society Open Science (2015).

35] D. Lu, J. Wang, M. Arshad, Abdullah, and A. Ali,

Fractional reduced differential transform method for

space-time fractional order heat-like and wave-like

partial differential equations, Journal of Advanced

Physics (2017).

[36] M. S. Mohamed and K. A. Gepreel, Reduce

differential transform method for a nonlinear integral

member of kadomtsevpetviashvili hierarchy

differential equations, Journal of the Egyptian

Mathematical Society (2016).

[37] V. K. Srivastava, N. Mishra, S. Kumar, B. K.

Singh, and M. K. Awasthi, Reduced differential

transform method for solving (1+n)-dimensional

burgers equation, Egyptian Journal of basic and

applied sciences, 5 (2014).

[38] C. F. Lorenzo and T. Hartley, The fractional

trigonometry with applications to fractional

differential equations and science, (JohnWiley &

Sons, Inc., Hoboken, New Jersey, Printed in the

United States of America, 2017) Chap. 1-3.

[39] M. Z. Mohamed, T. M. Elzaki, M. S. Algolam,

E. M. A. Elmohmoud, and A. E. Hamza, New

modified variational iteration Laplace transform

method compares Laplace Adomian decomposition

method for solution time-partial fractional

differential equations, Hindawi Journal of Applied

Mathematics (2021).

[40] B. Benhammouda, H. Vazquez-Leal, and A.

Sarmiento-Reyes, Modified reduced differential

transform method for partial differential algebraic

equations, Hindawi Publishing Corporation Journal

of Applied Mathematics (2014).

[41] S. Abuasad, I. Hashim, and S. A. A. Karim,

Modified fractional reduced differential transform

method for the solution of multiterm time-fractional

diffusion equations, Hindawi Advances in

Mathematical Physics (2019).

[42] K. A. Gepreel, A. M. S. Mahdy, M. S.

Mohamed, and A. Al-Amiri, Reduced differential

transform method for solving nonlinear

biomathematics models, Computers, Materials &

Continua (2019).

EQUATIONS

DOI: 10.37394/232021.2023.3.11

Nguyen Minh Tuan

E-ISSN: 2732-9976

100

Volume 3, 2023

[43] A. Ziqan, S. Armiti, and I. Suwan, Solving three-

dimensional Volterra integral equation by the

reduced differential transform method, International

Journal of Applied Mathematical Research (2016).

[44] L. Oussama and M. Serhani, Bifurcation

analysis for the prey-predator model with Holling

type iii functional response incorporating prey

refuge, Applications and Applied Mathematics: An

International Journal (2019).

[45] S. R. M. Noori and N. Taghizadeh, Application

of reduced differential transform method for solving

two-dimensional Volterra integral equations of the

second kind, Applications and Applied Mathematics:

An International Journal (2019).

[46] B. K. Singh and P. Kumar, Frdtm for numerical

simulation of the multi-dimensional, timefractional

model of Navier-Stokes equation, ENGINEERING

PHYSICS AND MATHEMATICS (2018).

[47] G. E. F. Liu, M. M. Meerschaert, S. Momani, N.

N. Leonenko, W. Chen, and O. P. Agrawal,

Fractional differential equations, Hindawi Publishing

Corporation International Journal of Differential

Equations (2010).

[48] B. K. Singh, Fractional reduced differential

transform method for numerical computation of a

system of linear and nonlinear fractional partial

differential equations, Int. J. Open Problems Compt.

Maths (2016)

[49] A. Eman, F. Asad, A.-S. Mohammed, K.

Hammad, and K. R. Ali, Approximate series solution

of nonlinear, fractional klein-gordon equations using

fractional reduced differential transform method,

Journal of Mathematics and Statistics (2017).

[50] F. Ayaz, Application of reduced differential

transform method for solving two-dimensional

Volterra integral equations of the second kind,

Applied Mathematics and Computation (2003).

[51] E. C. de Oliveira and J. A. T. Machado, A

Review of Definitions for Fractional Derivatives and

Integral, Hindawi Publishing Corporation

Mathematical Problems in Engineering 2014 (2014).

[52] Y. TIAN and J. LIU, A modified exp-function

method for fractional partial differential equations,

THERMAL SCIENCE (2021).

[53] T. Abdeljawad, A. Atangana, J. Gtomez Aguilar,

and F. Jarad, On a more general fractional integration

by parts formulae and applications, Elsevier B.V. 51

(2019).

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Nguyen Minh Tuan carried out all the content of

the paper consisting of conceptualization, data

curation, investigation, formal analysis,

methodology, software, visualization, writing-

original draft, and writing review and editing.

Please visit Contributor Roles Taxonomy (CRediT)

that contains several roles:

www.wseas.org/multimedia/contributor-role-

instruction.pdf

Alternatively, the following text will be published:

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Figure 1: The first example in 2D

Figure 2: The first example in 3D

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Figure 3: The second example in 2D

Figure 4: The second example in 3D

Figure 5: The third example in 2D

Figure 6: The third example in 3D

Figure 7: The fourth example in 2D

Figure 8: The fourth example in 3D

EQUATIONS

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Figure 9: The fifth example in 2D

Figure 10: The fifth example in 3D

Alternatively, in case of no funding the following

text will be published:

No funding was received for conducting this study.

Alternatively, in case of no conflicts of interest the

following text will be published:

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to declare that

is relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Nguyen Minh Tuan carried out all the content of

the paper consisting of conceptualization, data

curation, investigation, formal analysis,

methodology, software, visualization, writing-

original draft, and writing review and editing.

Please visit Contributor Roles Taxonomy (CRediT)

that contains several roles:

www.wseas.org/multimedia/contributor-role-

instruction.pdf

Alternatively, the following text will be published:

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.