A Fractional Reduced Differential Transform Method for Solving

Multi-Fractional Telegraph Equations

NGUYEN MINH TUAN1ID , PHAYUNG MEESAD2∗ID , PIWAN WONGSASHINCHAI1ID

1Faculty of Information Technology,

Posts and Telecommunications Institute of Technology,

122 Hoang Quoc Viet, Cau Giay district, 11300, Ha Noi,

VIETNAM

1Department of Mathematics and Computer Science,

FernUniversitat in Hagen, UniversitatsstraSSe 1/11, 58097 Hagen,

GERMANY

2Department of Information Technology and Management,

King Mongkut’s University of Technology North Bangkok,

1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800,

THAILAND

3Department of Mathematics, Faculty of Science and Technology,

Rambhai Barni Rajabhat University, 2200, Chanthaburi,

THAILAND

* Corresponding author.

Abstract: - This paper presents a novel modiﬁcation of the Fractional Reduced Differential Transform Method

(FRDTM) to solve space-time multi-fractional telegraph equations. The telegraph equation is crucial in modeling

voltage and current distribution in electrical transmission lines, and its solutions have applications in physics,

economics, and applied mathematics. The proposed method effectively simpliﬁes the fractional differential

equations by omitting one fractional derivative term, allowing for the transformation of the remaining terms using

the FRDTM. The solutions demonstrate the method’s accuracy and efﬁciency in fractional partial differential

equations. This study advances the analytical solutions of fractional telegraph equations by providing a

straightforward yet powerful approach to fractional differential problems.

Key-Words: - Fractional Reduced Differential Transform Method; FRDTM; Fractional telegraph Equation.

Received: April 13, 2024. Revised: August 15, 2024. Accepted: October 11, 2023. Published: November 18, 2024.

1 Introduction

In the new technology phase, the fractional derivative

has been strongly applied in applied science to

solve the fractional controlled price equations,

discretize the time-independent space-fractional

models, nonlinear mechanism by dynamical

complexity, and fractional controlled diffusion

processes. Some controlling problems have been

analyzed and performed in nonlinear differential

problems, especially the Markov process. The

fractional derivative has also been construed as the

state-of-the-art performance for the problems in

physics science, economics, or other aspects, [1], [2],

[3], [4], [5], [6]. Large-scale papers about fractional

derivative branches were published, which played a

signiﬁcant role in research and applied to different

aspects of life. Besides that, the fractional derivative

also showed a broad and efﬁcient use in investigating

the behavior of real groups, [7]. Moreover, fractional

calculus could become an effective tool to express

the real world by representing arguments and rate

diffusion changes. Fractional calculus is also applied

in the utilization of computer science, and probability

distribution, especially in ﬁnancial risk management.

The solutions of fractional partial differential

equations focus on fat-tailed, and stable distribution.

The ﬁnancial business is essentially worldwide for

the government to set up algorithms for adjusting

and controlling inﬂation related to market price,

total income, or debasement. Mathematicians have

supported sustained efforts to produce more and more

mathematical tools for establishing the fundamental

foundations. The purpose is to support managers,

ofﬁcial agents, businessmen, and the government

in solving the ﬁnancial distribution models and

time-continuous systems, [8].

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Various methods are created to ﬁnd the best solutions

for nonlinear problems, especially by combining

other algorithms to seek the solutions. For example,

the homotopy perturbation method combining

the expansion method is applied in solving many

aspects of economic problems, [9], [10]. The

method is based on the inductive algorithm to ﬁnd

the approximate solutions. Another performance

showing a variation iteration method is to ﬁnd the

approximate solutions relying on integrating the

equations approaching exact solutions, [11]. This

method also has the best performance when the errors

approach zero. Moreover, a combination of a Laplace

transform method has depicted effective results in

applied physics and biology science, [12]. Laplace

transforms have been demonstrated successfully

when solving the economic-ﬁnancial equations, and

space-time fractional telegraph equations, [13], [14].

In addition, the Adomian decomposition method

is applied to ﬁnd the zeroes of Volterra equations

and depict the approximate solutions on Mapple.

The results showed the important summary of

solving some complicated Lighthill singular integral

equations, [15].

Based on the effective methods mentioned above,

the reduce transform method was introduced and

performed with the effective application, [16], [17],

[18], [19], [20], [21], [22]. The technique is applied

to solve many kinds of partial equations composed

of heat and wave equations using linear or nonlinear

terms in normal or fractional derivative, [23], [24],

[25], [26], [27], [28], [29]. The technique has also

illustrated the approximate solution and approach to

the exact solution when nth terms come to inﬁnity

where fractional integration problems are considered,

[30]. Many researchers are supporting the facilities

to ﬁnd the best illustration of the solutions in many

kinds of fractional differential equations, [31]. Two

terms of space and time fractional derivative are

considered for applying the integration of fractional

derivative of the fractional term and the left one

keeping on for FRDTM, [32], [33], [34], [35], [36],

[37], [38], [39], [40], [41], [42]. This paper will

integrate FRDTM and propositions of fractional

derivatives to ﬁnd solutions to the one-dimensional

fractional differential equations, [43], [44], [45], [46],

[47], [48], [49], [50], [51], [52]. The new integration

will be expressed using the new modiﬁcation for the

fractional reduced derivative transformation method

and the analytic solutions of the fractional telegraph

equations written (1.1) as the following, [31], [53],

[54], [55], [56], [57], [58], [59], [60], [61], [62]:

xf(x,t) =C

tf(x,t) + N[f(x,t)] + L[f(x,t)],(1.1)

with the initial condition as the following f(0,t) =

g(t),ft(0,t) = h(t), where C

∂α

∂

, and C

∂β

∂

, 0 <

≤1, 1 <

≤2 are Caputo’s derivatives.

L,Ndenote the linear, and nonlinear operator existing

partial derivatives, f(x,t)is a given function, [14],

[30], [63], [64], [65], [66], [67].

2 Fundamental Functions

We will summarize some related propositions in

fractional derivative theory, [27], [28]. First of all,

we consider some deﬁnitions as follows

Deﬁnition 1 (Gamma function).Given values z ∈C,

and Re(z)>0, the integration is deﬁned

Γ(z) = Z∞

tz−1e−tdt.

We have some speciﬁc identities related to the

gamma function

Γ(z+1) = zΓ(z),Γ(n+1) = n!.

Deﬁnition 2. [45] Mittag-Lefﬂer functions are

usefully deployed in the form of solutions given below

(z) =

∞

∑

k=0

Γ(k

>0,

>0.

Based on the deﬁnition, we have some useful

identities as follows

E1,1(z) =

∞

∑

k=0

Γ(k+1)=ez;E1,2(z) =

∞

∑

k=0

Γ(k+2)=ez−1

E1,3(z) =

∞

∑

k=0

Γ(k+3)=ez−1−z

z2;

E2,1(z) =

∞

∑

k=0

Γ(2k+1)=cosh(z);

E2,2(z2) =

∞

∑

k=0

z2k

Γ(2k+2)=sinh(z)

Deﬁnition 3. [28] Considering a continuous function

y=f(x)with an arbitrary constant n −1<

≤n,

the Caputo’s fractional derivative of the order

given by

xf(x) = 1

Γ(n−

)Zx

(x−s)n−

−1f(n)(s)ds.(2.1)

Based on Deﬁnition 3, we have derivative of

function f(x) = C, (C is a constant)

xf(x) = 0.

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The inverse operator of C

x, called J

xis fractional

integral operator of order

is given as the following

f(x) = 1

Γ(

)Zx

(x−s)

−1f(s)ds.(2.2)

We have some of Caputo’s fractional derivative

properties

C

xf(x) = f(x)−

n−1

∑

k=0

f(k)(0)xk

k!,x>0.(2.3)

For particular values, we have some useful

propositions

Deﬁnition 4. For m −1<

≤m, we have the

following propositions, [29], [59]:

Caputo’s fractional derivatives of f(x) = x

xf(x) = Γ(1+

)

Γ(1+

−

>−1,

∈R.

Caputo’s fractional derivatives of f(x) = e

xf(x) =

mxm−

E1,m−

+1(

x).(2.4)

Caputo’s fractional derivatives of f(x) = sin

xf(x) = 1

2i(i

)mxm−

(E1,m−

+1(i

−(−1)mxm−

E1,m−

+1(−i

x)).(2.5)

Caputo’s fractional derivatives of f(x) = cos

xf(x) = 1

2(i

)mxm−

(E1,m−

+1(i

+ (−1)mxm−

E1,m−

+1(−i

x)).(2.6)

Caputo’s fractional derivatives of the function given

by f(x)=(x−a)

−1Ep,q

(

(x−a)

)is

x,a+f(x)=(x−a)

−

−1Ep,q

−

(

(x−a)

),(2.7)

where

Ep,q

(x) =

∞

∑

k=0

pqkxk

Γ(k

)k!,pqk =Γ(p+qk)

Γ(p),

,p,q∈C,Re(

)>0,Re(p)>0,q∈N.

3 Methods

In this section, we will illustrate the results of the

FRDTM, [55]. Given Tfbe the transformation of this

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

functions corresponding to Gk=Tf(g),Hk=Tf(h)

the output of the transform. Using the FRDTM,

some basic functions are created shown in Table 1,

[42]. FRDTM is to ﬁnd the approximate solution

for fractional differential equations, [34]. Regarding

two-variable functions expressed as f(x,t), we set up

the following steps, [32]

Step 1: Integrating the Eq. (1.1), and applying the

Caputo’s fractional derivatives properties Eq. (2.3) to

transform the space fractional differential terms.

Step 2: Express the terms in the form

Fk(x)=(

∞

∑

i=0

u(i)xi)(

∞

∑

j=0

v(j)tj) =

∞

∑

k=0

Hk(i,j)tk,

where Hk(i,j) = u(i)v(j)is the compression of

F(x,t).

The term of fractional reduced differential transform

method of F(x,t)is formed by

Hk(x) = 1

Γ(k

+1)[

∂

Fk(x,t)

∂

]t=t0,(3.1)

where

denotes the order of time fractional

derivative.

Step 3: Then the inverse transformation of Hkis

deﬁned by

Fk(x,t) =

∞

∑

k=0

Hk(t)(t−t0)k

.(3.2)

Combine Eq. (3.1) and Eq. (3.2), we have

Fk(x,t) =

∞

∑

k=0

Γ(k

+1)[

∂

Fk(x,t)

∂

]x=x0(t−t0)k

.(3.3)

We can choose t0=0, from Eq. (3.3), we have

Fk(x,t) =

∞

∑

k=0

Γ(k

+1)[

∂

Fk(x,t)

∂

]tk

.(3.4)

Applying the inductive method for equation (3.4), we

have

f(x,t) = lim

k→∞Fk(x,t).

In a special case, the fractional derivative is the

component of the integer and fractional derivative, we

suppose

f(x,t) =

∞

∑

k=0

Fk(x)(t−t0)k

where

is the order of the fractional derivative and

Fkis the transformation of f(x).

With k=0,1, ...,(

q−1), we apply the

transformation formula

Fk(x,t) = 





0 if k

/∈Z+

)!(dk

dt k

f(x,t))t=t0if k

∈Z+.

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Table 1: Basic results using FRDTM method.

Original Functions FRDTM for some fundamental functions

f(x,t) = ag(x,t)±bh(x,t)Fk(x) = aGk(x)±bHk(x)

f(x,t) = g(x,t)h(x,t)Fk(x) = ∑k

r=0Gr(x)Hk−r(x)

f(x,t)=(x−x0)pFk(x) =

(k−

p) = 1 if k=

0 if k6=

f(x,t) =C

xg(x,t)Fk(x) = Γ(

(k+1)+1)

Γ(1+k

)Gk+1(x)

f(x,t) = sint Hk(x) = sin(k

f(x,t) = axmtnHk(x) = axm

(k−m)

f(x,t) = etHk(x) = 1

4 Applications

In this section, we illustrate the solutions of

one-dimensional fractional differential equations by

using the extension of FRDTM via the examples as

the following, [26], [46], [47], [48], [52]:

Example 1. Consider the equation, [49], in the form

tf(x,t)−C

xf(x,t)−f(x,t) = 0,(4.1)

satisfy the terminal conditions

f(x,0) = 1+sinx,f(0,t) = et,fx(0,t) = 1,

where 0<

≤1,1<

≤2.

Transform the J

xon both sides Eq. (4.1) using

Eq. (2.3), we have

xC

tf(x,t) = f(x,t)−(x+et) + J

xf(x,t).(4.2)

Using FRDTM on both sides for Eq. (4.2), we have

xHk+1(x) = Γ(k

+1)

Γ(

(k+1) + 1)[Hk(x)−(

(k)x+1

k!)

xHk(x)].

(Derivative of sin xand xby Caputo’s fractional

derivatives of the order

)

Transform the initial condition H0(x) = 1+sin x,

using iterative method, Eq.(2), Eq. (2.5), we have

H1(x) = 1

Γ(

+1)[C

x(sinx−x) + 1−sinx];

H2(x) = Γ(1+

)

Γ(2

+1)[C

x(H1−1) + H1];

H3(x) = Γ(2

+1)

Γ(3

+1)[C

x(H2−1

2) + H2];

H4(x) = Γ(3

+1)

Γ(4

+1)[C

x(H3−1

3) + H3];···.

The solution gained is

f(x,t) =

∞

∑

k=0

Hk(t)xk

=sinx+1+H1(x)t

+H2(x)t

+H3(x)t

+···.

This solution will be convergent to the exact solution

when

f(x,t) = sinx+

∞

∑

k=0

Γ(k

+1)=sinx+E1,1(t

When

=1,

=2, the exact solution attained is

f(x,t) = sinx+et. The solutions are performed in

Figure 1, Figure 2, and Table 2 in Appendix. Table

2 (Appendix) shows the solution by comparing the

values

1=0.5,

2=0.7,

=1. error_1 shows

the difference between

, and error_2 shows the

difference between

Example 2. Consider the equation formed, [50],

xf(x,t) =C

0D2

tf(x,t)+

∂

tf(x,t)+ f(x,t),(4.3)

satisfy the terminal condition f (0,t) = e−t,fx(0,t) =

e−t,0<

≤1,1<

≤2.

Transform Eq. (4.3) :

=2(p=3,q=

2),:

Apply Table 1, and using condition fx(0,t) = e−t,

taking FRDTM, we have

Hk+p(t) = Γ(kq +1)

Γ(kq +pq +1)[

∂

t2Hk(t)+

∂

tHk(t)+Hk(t)].

We establish the inductive method to calculate the

terms of the transform as follows

H0(t) = e−t;H1(t) = 0; H2(t) = e−t;

H3(t) = e−t

Γ(5

2);H4(t) = 0;H5(t) = e−t

Γ(7

2);···.

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Table 1. Basic results using FRDTM method

The analytic solution will be gathered as the

following

f(x,t) =

∞

∑

k=0

Hk(t)xkq =e−t+e−tx+e−t

Γ(5

2)x1.5+e−t

Γ(7

2)x2.5+···.

Example 3. Consider the equation formed

xf(x,t) =C

tf(x,t) +

∂

tf(x,t) + f(x,t),(4.4)

satisfy the terminal condition f (0,t) = e−t,0<

≤

1,1<

≤2.

Transform the Eq. (4.4) by taking the J

on both

sides, we have

tC

xf(x,t) = f(x,t)−f(x,0)−t ft(x,0)

t[ft(x,t) + f(x,t)].(4.5)

Apply the J

xtransform formula on Table 1 for the

Eq. (4.5), we have

tHk+1(t) = Γ(k

+1)

Γ(

(k+1) + 1)[Hk(t)−1

k!+t

∂

tHk(t) + Hk(t)]].(4.6)

From the initial conditions, using the inductive

method and Eq. (2), Eq. (2.4), we have

H0(t) = e−t;H1(t) = 1

Γ(

+1)

t[e−t−1+t];

H2(t) = Γ(

+1)

Γ(2

+1)[C

t(H1−1−t) +

∂

tH1(t) + H1(t)];

H3(t) = Γ(2

+1)

Γ(3

+1)[C

t(H1−1

2! −t

2!) +

∂

tH1(t) + H1(t)];

H4(t) = Γ(2

+1)

Γ(3

+1)[C

t(H3−1

3! −t

3!) +

∂

tH3(t) + H3(t)];

H5(t) = Γ(2

+1)

Γ(3

+1)[C

t(H4−1

4! −t

4!) +

∂

tH4(t) + H4(t)];···.

The analytic solution is demonstrated as follows

f(x,t) = e−t+H1(t)x

+H2(t)x2

+H3(t)x3

+H4(t)x4

+H5(t)x5

+···.(4.7)

When

=1,

=2, the exact solution is f(x,t) =

ex−t. The solutions are depicted in Figure 3, Figure 4,

and Table 3 in Appendix. Table 3 (Appendix) shows

the numerical solutions by comparing the values

0.2,

2=0.5,

=1. error_1 shows the difference

between

, and error_2 shows the difference

between

Example 4. We consider the equation formed [51]:

xf(x,t) =C

tf(x,t) +

∂

tf(x,t)

−x2−t+1,(4.8)

satisfy the terminal condition

f(0,t) = t,fx(0,t) = 0,f(x,0) = x2,

ft(x,0) = 1,1<

≤2,1<

≤2.

Taking the J

xtransformation on both sides using

Eq. (2.3) for Eq. (4.8), we have:

tf(x,t) = f(x,t)−f(x,0)−t

∂

tf(x,0)

[

∂

tf(x,t)−x2−t+1].

Apply the properties in Table 1, we have

tHk+1(t) = Γ(k

+1)

Γ(k

+1){Hk(t)−

(k−2) +t

(k)

t[Hk(t)−

(k−2)] + (1−t)

(k)}.

Using the inductive method, we calculate the

following terms

H0(t) = t;H1(t) = 1

Γ(

+1)[C

x(t+1) + 1];

H2(t) = Γ(

+1)

Γ(2

+1)[C

x(H1) + H1];

H3(t) = Γ(2

+1)

Γ(3

+1)[C

x(H2−1) + H2−1];

H4(t) = Γ(3

+1)

Γ(4

+1)[C

x(H3) + H3];

H5(t) = Γ(4

+1)

Γ(5

+1)[C

x(H4) + H4];···.

The analytics solution of the equation is

f(x,t) = t+H1(t)x

+H2(t)x2

+H3(t)x3

+H4(t)x4

+H5(t)x5

+···.

When

=2,

=2, the exact solution becomes

f(x,t) = t+x2.

The graphs of solutions are portrayed in Figure

5, Figure 6, and Table 4 in Appendix. Table 4

(Appendix) shows the solution by comparing the

values

1=1.2,

2=1.5,

=2. error_1 shows

the difference between

, and error_2 shows the

difference between

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Example 5. We consider the equation formed, [50]:

tf(x,t)+x

∂

xf(x,t)+

∂

x2f(x,t) = 2t

+2x2+2,

satisfy the terminal condition f (x,0) = x2,t>0,0<

≤1.

Taking the J

xtransformation on both sides, we

have the expression

f(x,t)−f(x,0)−J

t[x

∂

xf(x,t)+

∂

x2] = J

t[2t

+2x2+2],

Applying the Table 1, we have the following equation

Hk(t)−

(k−2) + J

∑

r=0

(r−1)Hk+1−r(t)

+ (k+1)(k+2)Hk+2(t)]

=2Γ(

+1)

Γ(2

+1)t2

t[2

(k−2) + 2

(k)].

Using the identity method and solving simultaneous

equations, we have the following terms

H0(t) = 2Γ(

+1)

Γ(2

+1)t2

;H1(t) = 0;

H2(t) = 1;H3(t) = 0; H4(t) = 0; H5(t) = 0;···.

The solution is formed to the exact solution

f(x,t) = 2Γ(

+1)

Γ(2

+1)t2

+x2.

The analytic solutions are illustrated in Figure 7,

Figure 8, and Table 5 in Appendix. Table 5

(Appendix) shows the solutions by comparing the

values

1=0.4,

2=0.5,

=1 where error_1 shows

the difference between

, and error_2 shows the

difference between

5 Discussion

This study introduces an enhanced version of the

FRDTM method to solve a class of multi-fractional

space-time telegraph equations. The results show

that by eliminating one of the fractional derivative

terms, the modiﬁed FRDTM can effectively simplify

the problem and lead to accurate analytical solutions.

Compared to existing methods, such as the Adomian

Decomposition Method and the Laplace Transform

Method, the modiﬁed FRDTM offers a more direct

and computationally efﬁcient approach to solving

fractional differential equations . However, the

method also has some limitations. In cases where the

fractional derivatives involve transcendental terms

or anomalous series, the FRDTM’s performance

can be hindered by difﬁculties in approximating

the fractional series. This suggests that while the

FRDTM is effective for many types of fractional

telegraph equations, further reﬁnement may be

needed when applied to more complex or highly

nonlinear equations. Future research could explore

hybrid methods or extensions of the FRDTM to

better address these challenges. The versatility

of the FRDTM in fractional calculus applications,

particularly in engineering and physics, highlights

its potential for broader applications. Speciﬁcally,

the methods ability to handle fractional time-space

equations suggests promising opportunities in ﬁelds

requiring long-term behavior predictions, such as

ﬁnance, electrical transmission systems, and ﬂuid

dynamics.

6 Conclusions

In conclusion, this paper demonstrates the

effectiveness of a modiﬁed Fractional Reduced

Differential Transform Method (FRDTM) for

solving multi-fractional telegraph equations. The

method simpliﬁes complex fractional differential

equations by isolating one variable and transforming

the remaining terms. The resulting analytical

solutions highlight the potential of FRDTM for

solving large-scale fractional partial differential

equations. Although this method provides a more

efﬁcient route to exact solutions, its limitations

in handling anomalous series and transcendental

terms suggest that further work is needed. Future

research should focus on enhancing the FRDTM for

more complex fractional problems and exploring

its applications in diverse ﬁelds such as economics,

physics, and engineering. This study contributes

to the ongoing development of fractional calculus

and provides a strong foundation for future work in

solving space-time fractional equations in various

scientiﬁc and industrial applications.

Acknowledgment:

The authors acknowledge the reviewer’s comments

that contribute to improving the paper’s quality.

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APPENDIX

Table 2: Numerical comparison of the example 1.

1=0.5

2=0.7

0.0 1.4794 1.4794 1.4794

0.1 1.9363 1.7347 1.5846

0.2 2.1846 1.9362 1.7008

0.3 2.4008 2.1387 1.8293

0.4 2.6053 2.3486 1.9713

0.5 2.8106 2.5681 2.1281

0.6 3.0290 2.7985 2.3015

0.7 3.2746 3.0406 2.4932

0.8 3.5643 3.2947 2.7050

0.9 3.9181 3.5611 2.9390

1.0 4.3601 3.8401 3.1977

Table 3: Numerical comparison of the example 3.

=0.2

=0.5

0.0 1.0000 1.0000 2.0138

0.1 1.4078 1.3567 1.8221

0.2 1.6494 1.5651 1.6487

0.3 1.8734 1.7585 1.4918

0.4 2.0884 1.9445 1.3499

0.5 2.2955 2.1241 1.2214

0.6 2.4937 2.2963 1.1052

0.7 2.6815 2.4598 1.0000

0.8 2.8573 2.6130 0.9048

0.9 3.0192 2.7541 0.8187

1.0 3.1655 2.8816 0.7408

Table 4: Numerical comparison of the example 4.

=1.2

=1.5

0.1 0.4260 1.2508 0.3500

0.2 0.4089 0.5918 0.4500

0.3 0.4722 0.5340 0.5500

0.4 0.5535 0.5715 0.6500

0.5 0.6419 0.6387 0.7500

0.6 0.7338 0.7186 0.8500

0.7 0.8277 0.8049 0.9500

0.8 0.9229 0.8949 1.0500

0.9 1.0190 0.9873 1.1500

1.0 1.1157 1.0812 1.2500

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Table 5: Numerical comparison of the example 5.

=0.4

=0.5

0.1 0.5520 0.4272 0.2600

0.2 0.7757 0.6045 0.2900

0.3 0.9772 0.7817 0.3400

0.4 1.1654 0.9590 0.4100

0.5 1.3443 1.1362 0.5000

0.6 1.5161 1.3135 0.6100

0.7 1.6823 1.4907 0.7400

0.8 1.8438 1.6680 0.8900

0.9 2.0012 1.8452 1.0600

1.0 2.1553 2.0225 1.2500

1.1 2.3062 2.1997 1.4600

Figure 1: Solution performance of example 1

for x=0.5.

Figure 2: Exact solution of example 1 using FRDTM,

=1,

=2.

Figure 3: Solution performance of example 3

for x=0.5.

Figure 4: Exact solution of example 3 using FRDTM,

=1,

=2.

Figure 5: Solution performance of example 4

for x=0.5.

Figure 6: Approximate solution of example 4,

=2.

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Figure 7: Solution performance of example 5

for x=0.5,

=1.

Figure 8: Exact solution of example 5 using FRDTM,

=1.

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Contribution of Individual Authors to the

Creation of a Scientiﬁc Article (Ghostwriting

Policy)

Nguyen Minh Tuan: Conceptualization,

data curation, investigation, methodology,

software, visualization, writing-original draft and

writing-review and editing, validation, visualization,

writing-original draft and writing-review and editing.

Phayung Meesad, Piwan Wongsashinchai, Nguyen

Hong Son: methodology, resources, supervision,

validation, visualization, and writing review and

editing.

Sources of Funding for Research Presented in a

Scientiﬁc Article or Scientiﬁc Article Itself

No funding was received for conducting this study.

Conﬂicts of Interest

The authors have no conﬂicts of interest to

declare that are relevant to the content of this

article.

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(Attribution 4.0 International , CC BY 4.0)

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