Modification of Sumudu Decomposition Method for Nonlinear
Fractional Volterra Integro-Differential Equations
NONGLUK VIRIYAPONG
Mathematics and Applied Mathematics Research Unit
Department of Mathematics, Faculty of Science, Mahasarakham University
Maha Sarakham, 44150
THAILAND
Abstract: In this paper, modification of Sumudu decomposition method is successfully applied to find the ap-
proximate solution of nonlinear Volterra integro-differential equation of fractional order. The proposed method is
based on the combining of two powerful techniques, Sumudu decomposition method and Daftardar-Gejji and Ja-
fari (DGJ) method. Several illustrative examples are given to demonstrate the validity, reliability, and efficiency
of the proposed technique.
Key-Words: Volterra integro-differential equation, Caputo fractional derivative, Sumudu transform, Adomian
decomposition method, DGJ method
Received: May 23, 2021. Revised: February 21, 2022. Accepted: March 22, 2022. Published: April 18, 2022.
1 Introduction
The fractional integro-differential equations (FIDEs)
are in general form of integer order integro-
differential equations. In this study concerns with the
approximate analytical solution of the nonlinear Ca-
puto fractional integro-differential equation of the fol-
lowing form:
Dαy(t) = p(t)y(t) + g(t) + t
0
K(t, τ )F(y(τ))dτ,
(1)
with the initial condition
y(i)(0) = βi;i= 0,1,2, . . . , m −1,(2)
where Dαis the Caputo fractional differential opera-
tor of order α,m−1< α ≤m, f (t)∈L2([0,1]),
p(t)∈L2([0,1]) and K(t, τ )∈L2([0,1]2)are
known functions, y(t)is unknown function.
Such kind of equations are the focus of research
due to their pivotal role in the mathematical model-
ing of many physical problems in several fields of
physics, engineering, and economics, such as arising
in heat conduction in materials with memory, signal
processing and fluid mechanics [1, 2, 3]. However,
the fractional integro-differential equations are usu-
ally difficult to solve analytically and may not have
exact or analytical solutions, so approximate and nu-
merical methods for approximate solutions to integro-
differential equation of integer order are extended to
solve fractional integro-differential equations.
In recent years, many methods have been devel-
oped to solve fractional integro-differential equations,
especially nonlinear, which are receiving a lot of at-
tention. For instance, we can mention the follow-
ing works. Momani and Noor [4] applied the Ado-
mian decomposition method (ADM) to solve fourth-
order FIDEs, Mittal and Nigam [5] applied ADM to
find the approximate solutions for the FIDEs, Yang
and Hou [6] developed and applied the Laplace de-
composition method to solve linear and nonlinear
FIDEs, Tate and Dinde [7] presented a new modifi-
cation of Adomian decomposition method for non-
linear Volterra FIDEs, Hamoud and Ghadle [8] ap-
plied ADM and modified Laplace Adomian decom-
position method to find the approximate solution for
Voterra integro-differential equation of fractional or-
der, and Al-Khaled and Yousef [9] applied Sumudu
decomposition method to solve the fractional nonlin-
ear Volterra-Fredholm integro-differential equation.
In addition, the applications of the homotopy analy-
sis method [10] and CAS wavelets method [11] for
solution of fractional differential equations.
This article aims to introduce a new method for
solving fractional nonlinear integro-differential equa-
tions, called the modified Sumudu decomposition
method (MSDM) which is a modification to the
Sumudu decomposition method (SDM). The MSDM
is a combination of the two powerful techniques,
Sumudu transform and the iterative DGJ method
which is presented by Daftardar-Gejji and Jafari [12]
in which the Adomian polynomials in SDM have
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been replaced by Daftardar-Jafari polynomials. The
Sumudu transform was used to avoid calculation of
fractional derivative and integration of some difficult
functions while the nonlinear term can be easily han-
dled by the use of the approach given by Daftardar-
Gejji [12], and there is no need to calculate the Ado-
mian polynomials as compared with SDM.
2 Preliminaries
Some useful definitions and properties of fractional
calculus and Sumudu transform, are presented in this
section.
2.1 Fractional calculus
Definition 1 [13] The Riemann-Liouville fractional
integral of order α > 0, of a real valued function
f(t)is defined as:
Iαf(t) = 1
Γ(α)t
0
(t−τ)α−1f(τ)dτ.
Definition 2 [13] The fractional derivative Dαof
f(t)in the Caputo’s sense is defined as
Dαf(t) = 1
Γ(m−α)t
0
(t−τ)m−α−1f(m)(τ)dτ,
for m−1< α ≤m, m ∈N.
2.2 Sumudu transform
The Sumudu transform is an integral transform, which
was first introduced by Watugala and applied to solve
differential equations and control engineering prob-
lem [14].
Definition 3 [14] The Sumudu transform over the fol-
lowing set of functions
A=f(t) : ∃M, τ1, τ2>0,|f(t)|< Me|t|/τj
if t∈(−1)j×[0,∞)
is defined for u∈(−τ1, τ2)as
G(u) = S[f(t)] = ∞
0
e−tf(ut)dt
=∞
0
1
ue−t/uf(t)dt. (3)
Function f(t)in (3) is called inverse Sumudu trans-
form of F(u)and is denoted by f(t) = S−1[F(u)].
In Belgacem et al. [15], the Sumudu transform
was shown to be the duality of the Laplace transform.
Hence, one should be able to compete to a great extent
in problem-solving. The Sumudu and Laplace trans-
forms exhibit a duality relation expressed as follows:
G(u) = F(1
u)
u, F (s) = G(1
s)
s,
where G(u) = S[f(t)] and F(s) = L[f(t)]. For
further detail and properties about Sumudu transform
can found in [15]. Some basic transform of the func-
tions related to present work are as follow:
1. S[1] = 1
2. S[tn] = n!un;n= 1,2, . . .
3. S[tα] = Γ(α+ 1)uα;α > 0
4. St
0
f(τ)dτ=uS[f(t)]
Definition 4 Let f(t)and g(t)are continuous func-
tions and exponential order, the convolution of f(t)
and g(t)is defined as
(f∗g)(t) = t
0
f(τ)g(t−τ)dτ
Theorem 1 [15] Let f(t)and g(t)are continuous
function and exponential order. If S[f(t)] = F(u)
and S[g(t)] = G(u)then
S[(f∗g)(t)] = St
0
f(τ)g(t−τ)dτ
=uF (u)G(u)
Theorem 2 [15] The Sumudu transform of the Ca-
puto fractional derivative id defined as
S[Dαf(t)] = u−αS[f(t)] −
m−1
k=0
u−α+kf(k)(0),
for m−1< α < m, m ∈N.
3 Analysis of the method
Firstly, we consider the fractional integro-differential
equation of Volterra type. According to modified
Sumudu decomposition method, we apply Sumudu
transform first on both side of (1), we get
S[Dαy(t)] =S[p(t)y(t)] + S[g(t)]
+S[t
0
K(t, τ)F(y(τ))dτ].(4)
Using the property of Sumudu transform and simpli-
fying, we can obtain
S[y(t)] =
m−1
k=0
uky(k)(0) + uαS[g(t)] + uαS[p(t)y(t)]
+uαS[t
0
K(t, τ)F(y(τ))dτ].(5)
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Operating the inverse Sumudu transform on both
sides of (5), we get
y(t) = S−1m−1
k=0
uky(k)(0)+S−1[uαS[g(t)]]
+S−1[uαS[p(t)y(t)]]
+S−1uαSt
0
K(t, τ )F(y(τ))dτ.(6)
Next assume that
f(t) =S−1m−1
k=0
uky(k)(0)+S−1[uαS[g(t)]],
R(y(t)) =S−1[uαS[p(t)y(t)]],
N(y(t)) =S−1uαSt
0
K(t, τ)F(y(τ))dτ.
(7)
Thus, equation (6) can be written in the following
form
y(t) = f(t) + R(y(t)) + N(y(t)),(8)
where fis a known function, Rand Nare given linear
and nonlinear operator of y, respectively.
The second step in modified Sumudu decomposi-
tion method is that we represent solution as in form of
infinite series, given as follow:
y(t) = ∞
n=0
yn,(9)
where the term ynare to be recursively computed.
Then we have
R∞
n=0
yn=∞
n=0
R(yn),(10)
The nonlinear operator Nis decomposed as
N∞
n=0
yn=N(y0)+ ∞
n=1 Nn
i=0
yi−Nn−1
i=0
yi
(11)
Thus, equation (8) is given as
∞
n=0
yn=f+∞
n=0
R(yn) + N(y0)
+∞
n=1 Nn
i=0
yi−Nn−1
i=0
yi,(12)
then a recurrence relation is defined as follow:
y0=f, (13)
yn+1 =R(yn) + N(y0+y1+. . . +yn)
−N(y0+y1+. . . +yn−1),(14)
and we have
y0+y1+. . . +yn+1 =R(y0+y1+. . . +yn)
+N(y0+y1+. . . +yn).
(15)
Thus, we obtain
∞
n=0
yn=f+R∞
n=0
yn+N∞
n=0
yn.(16)
The m-term approximate solution of (8) is given by
y=y0+y1+. . . +ym−1.
However, when αis an integer, the exact solu-
tion may be obtained. The m-term approximation
ya=m−1
n=0 yncan be used to approximate the so-
lution. The zeroth component is very important, the
choice of (13) as the initial solution always leads to
noise oscillation during the iteration procedure [6].
Moreover, the selection of y0to contain a minimal
number of terms is giving more flexibility to solve
complicated nonlinear equations. The modification
of MSDM is based on the assumption that the function
fthat arises from the source term and prescribed ini-
tial conditions can be divided into two parts, namely,
f0and f1. Under this assumption, we set
f=f0+f1,(17)
and on applying a slight variation to the component
y0and y1, the modified recursive relation defined as
follows,
y0=f1,
y1=f2+R(y0) + N(y0),
yn+1 =R(yn) + N(y0+y1+. . . +yn)
−N(y0+y1+. . . +yn−1).
(18)
The slight variation in reducing the number of terms
of y0will help in a reduction of computation and will
accelerate the convergence. This slight variation in
the component y0and y1may provide the exact solu-
tion by using two iterations only. It should be noted
that the success of this method depends on proper
choice of the part f0and f1.
4 Applications
In this section, to demonstrate the applicability and
validity of the proposed method, we have applied it to
nonlinear Volterra integro-differential equations with
fractional order and the result obtained will be com-
pared with the exact solution.
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Example 4.1 First, we consider the following non-
linear fractional Volterra integro-differential equa-
tion :
Dαy(t) = 2t−t6
6+5 t
0
(t−τ)y2(τ)dτ, 0< α ≤1,
(19)
with the following initial condition
y(0) = 0,(20)
which has the exact solution in the case of α= 1 is
y(t) = t2.
To solve this problem by the proposed method, we ap-
ply Sumudu transform on both side of (19) and using
the inverse Sumudu transform, we have
f(t) =S−1uαS2t−t6
6,
N(y(t)) =5S−1uα+1S[t]S[y2].
(21)
By using the modified recursive relation (18) and
(21), we get
y0=S−1uαS[2t],
y1=−1
6S−1uαSt6+ 5S−1uα+2S[y2
0],
yn+1 =5S−1uα+2S[(y0+y1+. . . +yn)2
−(y0+y1+. . . +yn−1)2].
(22)
By using the recursive relation (22), we get
y0=2tα+1
Γ(α+ 2),
y1=20Γ(2α+ 3)
Γ2(α+ 2)Γ(3α+ 5)t3α+4 −5!
Γ(α+ 7)tα+6
y2=400Γ(2α+ 3)Γ(4α+ 6)
Γ3(α+ 2)Γ(3α+ 5)Γ(5α+ 8)t5α+7
−20 ·5!Γ(2α+ 8)
Γ(α+ 2)Γ(α+ 7)Γ(3α+ 10)t3α+9
+2000Γ2(2α+ 3)Γ(6α+ 9)
Γ4(α+ 2)Γ2(3α+ 5)Γ(7α+ 11)t7α+10
−200 ·5!Γ(2α+ 3)Γ(4α+ 11)
Γ2(α+ 2)Γ(3α+ 5)Γ(α+ 7)Γ(5α+ 13)t5α+12
+(·5!)2Γ(2α+ 13)
Γ2(α+ 7)Γ(3α+ 15)t3α+14
.
.
.
Hence, the 3-term approximate solution of problem
(19) and (20) is
ya(t) =y0+y1+y2
=2tα+1
Γ(α+ 2) +20Γ(2α+ 3)t3α+4
Γ2(α+ 2)Γ(3α+ 5)
−5!tα+6
Γ(α+ 7) +400Γ(2α+ 3)Γ(4α+ 6)t5α+7
Γ3(α+ 2)Γ(3α+ 5)Γ(5α+ 8)
−20 ·5!Γ(2α+ 8)t3α+9
Γ(α+ 2)Γ(α+ 7)Γ(3α+ 10)
+2000Γ2(2α+ 3)Γ(6α+ 9)t7α+10
Γ4(α+ 2)Γ2(3α+ 5)Γ(7α+ 11)
−200 ·5!Γ(2α+ 3)Γ(4α+ 11)t5α+12
Γ2(α+ 2)Γ(3α+ 5)Γ(α+ 7)Γ(5α+ 13)
+(·5!)2Γ(2α+ 13)t3α+14
Γ2(α+ 7)Γ(3α+ 15) (23)
In the case of α= 1, we get
y0=t2, y1= 0,
and
yn= 0, n ≥2.
Therefore, ya(t) = t2, which is the same of exact
solution. For α= 1, it is the only case that the ex-
act solution is known, and the obtained solution is in
good agreement with the exact solutions for only two
iterations of MSDM.
Figure 1 shows the behavior of the approximate
solution of problem (19) and (20) using the MSDM
for different values of α.
Figure 1: Approximate solutions for Example 4.1 for different
values of α.
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Example 4.2 Consider the following nonlinear frac-
tional Volterra integro-differential equation:
Dαy(t) = 1 −t2y(t)+3t
0
y2(τ)dτ, 0< α ≤1,
(24)
with the following initial condition
y(0) = 0 (25)
which has the exact solution in the case of α= 1 is
y(t) = t.
To solve this problem by the proposed method, we ap-
ply Sumudu transform on both side of (24) and using
the inverse Sumudu transform, we have
f(t) =S−1uαS[1],
R(y(t)) = −S−1uαS[t2y],
N(y(t)) =3S−1uα+1S[y2].
(26)
By using the recursive relation (14) and (26), we get
y0=tα
Γ(α+ 1),
y1=−S−1uαS[t2y0]+ 3S−1uα+1S[y2
0],
yn+1 =−S−1uαS[t2yn]
+ 3S−1uα+1S[(y0+y1+. . . +yn)2
−(y0+y1+. . . +yn−1)2].
(27)
By using the recursive relation (27), we get
y1=−(α+ 2)t2α+2
2Γ(2α+ 2) +6Γ(2α)t3α+1
αΓ2(α)Γ(3α+ 2)
y2=−S−1uαS[t2y1]+ 3S−1uα+1S[2y0y1+y2
1]
=(α+ 2)Γ(2α+ 5)
2Γ(2α+ 2)Γ(3α+ 5)t3α+4
−6Γ(2α)Γ(3α+ 4)
αΓ2(α)Γ(3α+ 2)Γ(4α+ 4)t4α+3
−3(α+ 2)Γ(3α+ 3)
αΓ(α)Γ(2α+ 2)Γ(4α+ 4)t4α+3
+36Γ(2α)Γ(4α+ 2)
α2Γ3(α)Γ(3α+ 2)Γ(5α+ 3)t5α+2
+36Γ(2α)Γ(4α+ 5)
α2Γ3(α)Γ(3α+ 2)Γ(5α+ 6)t5α+5
−36Γ(2α)Γ(5α+ 4)
α2Γ3(α)Γ(3α+ 2)Γ(6α+ 5)t6α+4
+108Γ2(2α)Γ(6α+ 3)
α2Γ4(α)Γ2(3α+ 2)Γ(7α+ 4)t7α+3
.
.
.
The 3-term approximate solution of (24) is given by
ya(t) = y0+y1+y2.(28)
For α= 1, it is the only case which the exact solu-
tion is known. When α= 1, then ya(t) = twhich is
the same of exact solution. Thus, the approximate so-
lution using only two-steps of MSDM is in excellent
agreement with the exact solutions.
The approximate solution of problem (24) and (25)
for α= 0.2,0.4,0.6,0.8and 1are shown in Figure 2.
Figure 2: Approximate solutions for Example 4.2 for distinct
values of α.
Example 4.3 Consider the following nonlinear frac-
tional Volterra integro-differential equation [6]:
Dαy(t) = 1+ t
0
y(τ)y′(τ)dτ,
0≤t < 1,0< α ≤1,(29)
with the initial condition
y(0) = 0.(30)
Similar to the previous example, to solve this problem
by the proposed method, we apply Sumudu transform
on both side of (29) and using the inverse Sumudu
transform, we have
f(t) =S−1uαS[1],
N(y(t)) =S−1uα+1S[yy′].(31)
By using the recursive relation (14) and (31), we get
y0=tα
Γ(α+ 1),
y1=S−1uα+1S[y0y′
0],
yn+1 =S−1uα+1S[(y0+. . . +yn)(y′
0+. . . +y′
n)
−(y0+. . . +yn−1)(y′
0+. . . +y′
n−1)].
(32)
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By using the recursive relation (32), we get
y1=Γ(2α)
3α2Γ2(α)Γ(3α)t3α,
y2=S−1uα+1S[(y0+y1)(y′
0+y′
1)−y0y′
0]
=S−1uα+1S[y0y′
1+y1y′
0+y1y′
1]
=4Γ(2α)Γ(4α)
15α3Γ3(α)Γ(3α)Γ(5α)t5α
+Γ2(2α)Γ(6α)
21α4Γ4(α)Γ2(3α)Γ(7α)t7α,
.
.
.
Hence, the 3-term approximate solution of problem
(29) and (30) is
ya(t) =y0+y1+y2
=tα
Γ(α+ 1) +Γ(2α)
3α2Γ2(α)Γ(3α)t3α
+4Γ(2α)Γ(4α)
15α3Γ3(α)Γ(3α)Γ(5α)t5α
+Γ2(2α)Γ(6α)
21α4Γ4(α)Γ2(3α)Γ(7α)t7α.(33)
In the integer case, α= 1, the exact solution is
y(t) = √2tan(t
√2)and 3-term approximate solution
by MSDM are plotted in Figure 3. It can be found that
the obtained approximate solution is close to the ex-
act solution and the result show good agreement with
the result of Ref. [6]. The accuracy of the obtained
solution can be improved by taking more terms in the
series approximate solution.
Figure 3: The comparison between the approximate and exact
solution of Example 4.3, in the case α= 1.
Figure 4 shows the behavior of approximate solu-
tion of problem (29) and (30) using the MSDM for
different values of α.
Figure 4: Approximate solutions for Example 4.3 for distinct
values of α.
Example 4.4 Consider the following nonlinear frac-
tional Volterra integro-differential equation [6]:
D6
5y(t) = 5t4
5
2Γ(4
5)−t9
252 +t
0
(t−τ)2y3(τ)dτ, (34)
with the following initial conditions
y(0) = 0, y′(0) = 0.(35)
Applying the proposed algorithm of MSDM, we ob-
tain
f(t) =S−1u6
5S5t4
5
2Γ(4
5)−t9
252,
N(y(t)) =S−1u6
5+1S[t2]S[y3].
(36)
By using the modified recursive relation (18) and
(36), we get
y0=S−1u6
5S5t4
5
2Γ(4
5)=t2
y1=−S−1u6
5St9
252
+S−1u11
5S[t2]S[y3
0],
yn+1 =S−1u11
5S[t2]S[(y0+. . . +yn)3
−(y0+. . . +yn−1)3].
(37)
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By using the recursive relation (37), we get
y1=−9!
252S−1u51
5+S−1u11
5S[t2]S[t6]
=−2·6!S−1u51
5+ 2 ·6!S−1u51
5
= 0,
y2=S−1u11
5S[t2]S[(y0+y1)3−y3
0]
= 0,
yn= 0, n > 2.
Therefore, the obtain solution is
y(t) = ∞
n=0
yn=t2,
which is the exact solution.
Example 4.5 Consider the following initial value
problem of nonlinear fractional Volterra integro-
differential equation [4, 11]:
Dαy(t) = 1 + t
0
e−τy2(τ)dτ, (38)
for 0≤t < 1,3< α ≤4and with the following
initial conditions
y(0) = 0, y′(0), y′′(0), y′′′(0) = 1.(39)
The exact solution of this problem for α= 4 is y(t) =
et. To solve this problem by the proposed method,
we apply Sumudu transform on both side of (38) and
using the inverse Sumudu transform, we have
f(t) =S−13
k=0
uky(k)(0)+S−1uαS[1],
N(y(t)) =S−1uα+1S[e−ty2].
(40)
Following the same procedure as the previous exam-
ple, we take the truncated Taylor expansions for ex-
ponential term in (40), that is e−t≈1−t+t2
2! −t3
3! .
By using the recursive relation (14) and (40), the
recursive MSDM algorithm is
y0=1,
y1=t+t2
2! +t3
3! +tα
Γ(α+ 1)
+S−1uα+1S(1 −t+t2
2! −t3
3!)y2
0,
yn+1 =S−1uα+1S(1 −t+t2
2! −t3
3!)·
(y0+. . . +yn)2−(y0+. . . +yn−1)2
(41)
By using the recursive relation (41), we get
y0= 1,
y1=t+t2
2! +t3
3! +tα
Γ(α+ 1) +tα+1
Γ(α+ 2)
−tα+2
Γ(α+ 3) +tα+3
Γ(α+ 4) −tα+4
Γ(α+ 5)
.
.
.
Hence, the 2-term approximate solution of problem
(38) and (39) is
ya(t) = 1 + t+t2
2! +t3
3! +tα
Γ(α+ 1) +tα+1
Γ(α+ 2)
−tα+2
Γ(α+ 3) +tα+3
Γ(α+ 4) −tα+4
Γ(α+ 5).(42)
Figure 5: The comparison between the approximate and exact
solution of Example 4.5, in the case α= 4.
In the integer case, α= 4, the exact solution is
y(t) = etand 2-term approximate solution by MSDM
are plotted in Figure 5. It can be found that the ob-
tained solution is close to the exact solution. It is re-
markable to note that the accuracy of the obtained so-
lution can be improved by taking more terms in the
series approximate solution.
Figure 6 shows the behavior of the approximate
solution of this problem using the MSDM for dif-
ferent values of α. The numerical results for some
3< α < 4, are shown in Table 1 with a compari-
son to Ref.[4] and [11]. Table 1 presents the MSDM
numerical solutions to be good agreement with the nu-
merical solution of ADM in [4] and CAS method in
[11].
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t
α= 3.25 α= 3.5α= 3.75
ADM CAS MSMD ADM CAS MSMD ADM CAS MSMD
0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.1 1.10655 1.10526 1.10524 1.10675 1.10520 1.10519 1.10615 1.10518 1.10518
0.2 1.22393 1.22189 1.22204 1.22432 1.22159 1.22165 1.22323 1.22145 1.22148
0.3 1.35320 1.35231 1.35207 1.35376 1.35099 1.35085 1.35231 1.35027 1.35020
0.3 1.49560 1.49676 1.49735 1.49627 1.49408 1.49443 1.49464 1.49254 1.49276
0.5 1.65255 1.66349 1.65988 1.65327 1.65647 1.65421 1.65162 1.65218 1.65075
0.6 1.82566 1.84380 1.84185 1.82635 1.83337 1.83211 1.82482 1.82670 1.82589
0.7 2.01669 2.04438 2.04552 2.01930 2.02935 2.03024 2.01602 2.01941 2.02007
0.8 2.22763 2.27759 2.27329 2.22808 2.25366 2.25080 2.22718 2.23720 2.23530
0.9 2.46069 2.52650 2.52763 2.46093 2.49493 2.49614 2.46046 2.47265 2.47375
Table 1: Numerical results for Example 4.5 with comparison to ADM and CAS
Figure 6: Approximate solutions for Example 4.5 for distinct
values of α.
5 Conclusion
In this article, the modification of Sumudu decompo-
sition method based on combination of the Sumudu
transform method and the iterative DGJ method
is introduced to solve nonlinear fractional Volterra
integro-differential equations. Several examples con-
cerning nonlinear Volterra integro-differential equa-
tions of fractional order are tested to confirm the
applicability and the advantages of the proposed
method. The comparison made between exact solu-
tion and obtained solutions by MSDM. The achieved
results are being well in agreement with exact solu-
tions. The results show that the proposed method is
powerful and efficient one for solving nonlinear frac-
tional integro-differential equation in terms of its sim-
plicity, implementation and high accuracy.
We expect that the proposed method can be ap-
plied to solve different type of fractional integro-
differential problems and other fractional problems
arising in science.
Acknowledgment
This research project was financially supported by
Mahasarakham University. The author is very grate-
ful to the reviewers for their valuable comments and
suggestions towards the improvement of this paper.
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Sources of funding for research
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scientific article itself
This research project was financially supported by
Mahasarakham University.
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