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Article
A new modified Adomian decomposition method for
nonlinear partial differential equations
Djelloul Ziane1,2, Rachid Belgacem3,∗and Ahmed Bokhari3
1Laboratory of mathematics and its applications (LAMAP), University of Oran1 Ahmed Ben Bella, Oran, 31000,
Algeria.; djeloulz@yahoo.com
2Department of Physics, University of Hassiba Benbouali, Ouled Fares, Chlef 02180, Algeria.
3Department of Mathematics, University of Hassiba Benbouali, Ouled Fares, Chlef 02180, Algeria.;
belgacemrachid02@yahoo.fr (R.B); bokhari.ahmed@ymail.com (A.B)
*Correspondence: belgacemrachid02@yahoo.fr
Received: 12 September 2019; Accepted: 30 October 2019; Published: 3 November 2019.
Abstract: In literature, there are many methods for solving nonlinear partial differential equations. In this
paper, we develop a new method by combining Adomian decomposition method and Shehu transform
method for solving nonlinear partial differential equations. This method can be named as Shehu transform
decomposition method (STDM). Some examples are solved to show that the STDM is easy to apply.
Keywords: Adomian decomposition method, Shehu transform method, nonlinear partial differential
equation.
MSC: 44A05, 26A33, 44A20, 34K37.
1. Introduction
The use of integral transforms (Laplace, Sumudu, Natural, Elzaki, Aboodh, Shehu and other transforms)
in solving linear differential equations as well as integral equations has developed significantly as a
result of the advantages of these transformations. Through these transforms, many problems of engineering
and sciences have been solved. However, it was found that these transforms remain limited in solving
equations that contain a nonlinear part.
To take advantage of these transformations and to use them to solve nonlinear differential equations,
researchers in the field of mathematics were guided to the idea of their composition with some methods such
as: Adomian decomposition method (ADM) ([1–4]), homotopy analysis method ([5–8]), variational iteration
method (VIM) ([9–12]), homotopy perturbation method (HPM) ([13–16]) and DJ iteration method ([17–20]).
The objective of the present study is to combine two powerful methods, Adomian decomposition method
and Shehu transform method to get a better method to solve nonlinear partial differential equations. The
modified method is called Shehu transform decomposition method (STDM). We apply our modified method
to solve some examples of nonlinear partial differential equations.
2. Basics of Shehu transform
In this section we define Shehu transform and gave its important properties [21].
Definition 1. The Shehu transform of the function v(t)of exponential order is defined over the set of functions:
A=v(t):∃N,k1,k2>0, |v(t)|<Nexp |t|
ki,i f t ∈(−1)j×[0, ∞), (1)
Open J. Math. Anal. 2019,3(2), 81-90; doi:10.30538/psrp-oma2019.0041 https://pisrt.org/psr-press/journals/oma
Open J. Math. Anal. 2019,3(2), 81-90 82
by the following integral
ˆ
S[v(t)]=[V(s,u)]=
∞
Z
0
exp(−st
u)v(t)dt
=lim
α−→∞
α
Z
0
exp(−st
u)v(t)dt,s>0, u>0. (2)
It converges if the limit of the integral exists, and diverges if not.
The inverse Shehu transform is given as:
ˆ
S−1[V(s,u)]=v(t), for t>0. (3)
Equivalently
v(t) = ˆ
S−1[V(s,u)]=1
2πi
α+i∞
Z
α−i∞
1
uexp(st
u)V(s,u)ds, (4)
where sand uare the Shehu transform variables, and αis a real constant and the integral in Equation (4)is
taken along s=αin the complex plane s=x+iy.
Theorem 2. (The sufficient condition for the existence of Shehu transform [21]). If the function v(t)is piecewise
continues in every finite interval 06t6βand of exponential order αfor t >β. Then its Shehu transform V(s;u)
exists.
Theorem 3. (Derivative of Shehu transform [21]). If the function v(n)(t)is the nth derivative of the function v(t)∈A
with respect to 0t0then its Shehu transform is defined as:
ˆ
Shv(n)(t)i=sn
unV(s,u)−
∞
∑
n=0s
un−(k+1)v(k)(0). (5)
Taking n =1, 2 and 3in Equation (5), we obtain the following derivatives with respect to t:
ˆ
Sv0(t)=s
uV(s,u)−v(0). (6)
ˆ
Sv00(t)=s2
u2V(s,u)−s
uv(0)−v0(0). (7)
ˆ
Sv000 (t)=s3
u3V(s,u)−s2
u2v(0)−s
uv0(0)−v00(0). (8)
Now, we summarize some important properties of this transform [21].
1. Linearity: ˆ
S[(αf(t) + βg(t)]=αˆ
S[f(t)]+αˆ
S[g(t)].
2. Change of scale: ˆ
S[f(βt)]=u
βVs
β,u.
Other properties are given in the Table 1.
Table 1. Some important properties of Shehu transform
v(t)ˆ
S[v(t)]v(t)ˆ
S[v(t)]
1u
ssin at au2
s2+a2u2
tu2
s2cos at us
s2+a2u2
tn
n!,n=0, 1, 2, .. u
sn+1sinh at au2
s2−a2u2
tnexp(at)
n!un+1
(s−au)n+1cosh at us
s2−a2u2
Open J. Math. Anal. 2019,3(2), 81-90 83
Proposition 4. If ∂v(x,t)
∂tand ∂2v(x,t)
∂t2exist, then
ˆ
S∂v(x,t)
∂t=s
uV(x,s,u)−v(x, 0). (9)
ˆ
S∂2v(x,t)
∂t2=s2
u2V(x,s,u)−s
uv(x, 0)−∂v(x, 0)
∂t. (10)
Proof. By means of integration by parts, we get
ˆ
S∂v(x,t)
∂t=
∞
Z
0
e−st
u∂v(x,t)
∂tdt =lim
τ−→∞
τ
Z
0
e−st
u∂v(x,t)
∂tdt
=lim
τ−→∞
hv(x,t)e−st
uiτ
0+s
u
τ
Z
0
e−st
uv(x,t)dt
=s
uV(x,s,u)−v(x, 0).
Let ∂v(x,t)
∂t=w(x,t), then, by using Equation (2)and Equation (9), we get:
ˆ
S∂2v(x,t)
∂t2=ˆ
S∂w(x,t)
∂t=s
uˆ
S[w(x,t)]−w(x, 0)
=s
uˆ
S∂v(x,t)
∂t−∂v(x, 0)
∂t
=s2
u2V(x,s,u)−s
uv(x, 0)−∂v(x, 0)
∂t.
Proposition 5. Let V(x,s,u)is the Shehu transform of v(x,t),then
ˆ
S∂nv(x,t)
∂tn=sn
unV(x,s,u)−
n−1
∑
k=0s
un−(k+1)∂kv(x, 0)
∂tk. (11)
Proof. We use use mathematical induction to prove (11). By means of Equation (9), the formula (11) is true
for n=1 and suppose
ˆ
S∂nv(x,t)
∂tn=sn
unV(x,s,u)−
n−1
∑
k=0s
un−(k+1)∂kv(x, 0)
∂tk, (12)
Let ∂nv(x,t)
∂tn=w(x,t)and using (9)and (12), we have:
ˆ
S∂n+1v(x,t)
∂tn+1=ˆ
S∂w(x,t)
∂t=s
uˆ
S(w(x,t))−w(x, 0)
=s
u"sn
unV(x,s,u)−
n−1
∑
k=0s
un−(k+1)∂kv(x, 0)
∂tk#−∂nv(x, 0)
∂tn
=sn+1
un+1V(x,s,u)−
n−1
∑
k=0s
un+1−(k+1)∂kv(x, 0)
∂tk−∂nv(x, 0)
∂tn
=sn+1
un+1V(x,s,u)−
n
∑
k=0s
un+1−(k+1)∂kv(x, 0)
∂tk.
Open J. Math. Anal. 2019,3(2), 81-90 84
3. Main results
3.1. Shehu transform decomposition method
To illustrate the basic idea of this method, we consider a general nonlinear nonhomogeneous partial
differential equation
∂mU(x,t)
∂tm+RU(x,t) + NU(x,t) = g(x,t), (13)
where ∂mU(x,t)
∂tmis the partial derivative of the function U(x,t)of order m(m=1, 2, 3),Ris the linear differential
operator, Nrepresents the general nonlinear differential operator, and g(x,t)is the source term.
Applying the Shehu transform (denoted in this paper by ˆ
S) on both sides of Equation(13), we get
ˆ
S∂mU(x,t)
∂tm+ˆ
S[RU(x,t) + NU(x,t)]=ˆ
S[g(x,t)]. (14)
Using the properties of Shehu transform, we obtain
sm
umˆ
S[U(x,t)]=
m−1
∑
k=0s
um−(k+1)∂kU(x, 0)
∂tk+ˆ
S[g(x,t)]−ˆ
S[RU(x,t) + NU(x,t)], (15)
where m=1, 2, 3.
Thus, we have
ˆ
S[U(x,t)]=
m−1
∑
k=0u
sk+1∂kU(x, 0)
∂tk+um
smˆ
S[g(x,t)]−um
smˆ
S[RU(x,t) + NU(x,t)]. (16)
Operating the inverse transform on both sides of Equation (16), we get
U(x,t) = G(x,t)−ˆ
S−1um
smˆ
S[RU(x,t) + NU(x,t)], (17)
where G(x,t), represents the term arising from the source term and the prescribed initial conditions.
The second step in Shehu transform decomposition method, is that we represent the solution as an infinite
series given below
U(x,t) =
∞
∑
n=0
Un(x,t), (18)
and the nonlinear term can be decomposed as:
NU(x,t) =
∞
∑
n=0
A(U), (19)
where Anare Adomian polynomials [22] of U0,U1,U2, ..., Unand it can be calculated by the formula:
An=1
n!
∂n
∂λn"N ∞
∑
i=0
λiUi!#λ=0
,n=0, 1, 2, · · · . (20)
Substituting (18)and (19)in (17), we have
∞
∑
n=0
Un(x,t) = G(x,t)−ˆ
S−1"um
smˆ
S"R
∞
∑
n=0
Un(x,t) +
∞
∑
n=0
An(U)##. (21)
Open J. Math. Anal. 2019,3(2), 81-90 85
On comparing both sides of the Equation (21), we get
U0(x,t) = G(x,t),
U1(x,t) = −ˆ
S−1hum
smˆ
S[RU0(x,t) + A0(U)]i,
U2(x,t) = −ˆ
S−1hum
smˆ
S[RU1(x,t) + A1(U)]i,
U3(x,t) = −ˆ
S−1hum
smˆ
S[RU2(x,t) + A2(U)]i,
.
.
.
(22)
In general, the recursive relation is given as:
Un+1(x,t) = −ˆ
S−1um
smˆ
S[RUn(x,t) + An(U)], (23)
where m=1, 2, 3, and n>0.
Finally, we approximate the analytical solution U(x,t)by:
U(x,t) = lim
N→∞
N
∑
n=0
Un(x,t), (24)
The above series solutions generally converge very rapidly [23].
3.2. Application
Here, we apply Shehu transform decomposition method to solve some nonlinear partial differential
equations.
Example 1. Consider the nonlinear KdV equation [24]:
Ut+UUx−Uxx =0, (25)
with the initial condition:
U(x, 0) = x. (26)
Applying the Shehu transform on both sides of Equation (25), we get
ˆ
S[Ut]+ˆ
S[UUx]−ˆ
S[Uxx ]=0. (27)
By means of the properties of Shehu transform, we get
ˆ
S[U(x,t)]=x−u
sE[UUx−Uxx]. (28)
Taking the inverse Shehu transform on both sides of Equation (28), we obtain
U(x,t) = x−ˆ
S−1u
sˆ
S[UUx−Uxx]. (29)
By applying the aforesaid decomposition method, we have
∞
∑
n=0
U(x,t) = x−ˆ
S−1"u
sˆ
S ∞
∑
n=0
An(U)−
∞
∑
n=0
(Un)xx !# (30)
Open J. Math. Anal. 2019,3(2), 81-90 86
Figure 1. The graphs of exact solution and approximate solutions of Equation (25)for 3 terms and 4 terms.
On comparing both sides of Equation (30), we get
U0(x,t) = x,
U1(x,t) = −ˆ
S−1u
sˆ
S(A0(U)−U0xx (x,t)),
U2(x,t) = −ˆ
S−1u
sˆ
S(A1(U)−U1xx (x,t)),
U3(x,t) = −ˆ
S−1u
sˆ
S(A2(U)−U2xx (x,t)),
.
.
.
(31)
The first few components of An(U)polynomials [22], for example, are given by:
A0(U) = U0U0,x,
A1(U) = U0U1,x+U1U0,x,
A2(U) = U0U2,x+U2U0,x+U1U1,x,
.
.
.
(32)
Using the iteration formulas (31)and the Adomian polynomials (32), we obtain
U0(x,t) = x,
U1(x,t) = −xt,
U2(x,t) = xt2,
U3(x,t) = −xt3,
.
.
.
(33)
Based on the formula (24), we get
U(x,t) = x−xt +xt2−xt3+· · · , (34)
which gives
U(x,t) = x
1+t,|t|<1, (35)
which is an exact solution to the KdV equation as presented in [25].
The graphs of exact solution and approximate solutions of Equation (25)for 3 terms and 4 terms is given
in Figure 1.
Example 2. Consider the nonlinear gas dynamics equation:
Ut+UUx−U(1−U) = 0, t>0, (36)
with the initial condition:
U(x, 0) = e−x. (37)
Open J. Math. Anal. 2019,3(2), 81-90 87
Applying the Shehu transform and its inverse on both sides of Equation (36), we get
U(x,t) = e−x−ˆ
S−1[u
sˆ
SUUx+U2−U]. (38)
By applying the aforesaid Decomposition Method, we have
∞
∑
n=0
Un(x,y) = e−x−ˆ
S−1"u
sˆ
S ∞
∑
n=0
An(U) +
∞
∑
n=0
Bn(U)−
∞
∑
n=0
Un!#. (39)
On comparing both sides of Equation (39), we obtain
U0(x,t) = e−x,
U1(x,t) = −ˆ
S−1u
sˆ
S(A0(U) + B0(U)−U0(x,t)),
U2(x,t) = −ˆ
S−1u
sˆ
S(A1(U) + B0(U)−U1(x,t)),
U3(x,t) = −ˆ
S−1u
sˆ
S(A2(U) + B0(U)−U2(x,t)),
.
.
.
(40)
The first few components of An(U)polynomials [22] is given by (32), and for Bn(U)for example, given
as follows:
B0(U) = U0U0,
B1(U) = 2U0U1,
B2(U) = 2U0U2+U2
1,
.
.
.
(41)
Using the iteration formulas (40)and the Adomian polynomials (32),(41), we get the first terms of the
solution series that is given by:
U0(x,t) = e−κ,
U1(x,t) = e−κt,
U2(x,t) = e−κt2
2! ,
U3(x,t) = e−κt3
3! ,
.
.
.
(42)
So, the approximate series solution of Equation (36)is given as:
U(x,t) = e−x1+t+t2
2! +t3
3! +· · · . (43)
And in the closed form, is given by:
U(x,t) = e−xet=et−x. (44)
This result is the same as that obtained in [26] using homotopy analysis method. In Figure 2,(a)represents
the graph of exact solution, (b)represents the graph of approximate solutions in 5 terms and (c)represents the
graph of approximate solutions in 4 terms.
Figure 2. (a)Represents the graph of exact solution, (b)represents the graph of approximate solutions in 5
terms, (c)represents the graph of approximate solutions in 4 terms.
Open J. Math. Anal. 2019,3(2), 81-90 88
Example 3. Consider the nonlinear wave-like equation with variable coefficients:
Utt =x2∂
∂x(UxUxx )−x2(Uxx )2−U, 0 <x<1, t>0, (45)
with the initial conditions:
U(x, 0) = 0, Ut(x, 0) = x2. (46)
Applying the Shehu transform and its inverse on both sides of Equation (45), we get
U(x,t) = x2t+ˆ
S−1u2
s2ˆ
Sx2∂
∂x(UxUxx )−x2(Uxx )2−U. (47)
By applying the aforesaid Decomposition Method, we have
∞
∑
n=0
U(x,t) = x2t+ˆ
S−1"u2
s2ˆ
S x2∂
∂x ∞
∑
n=0
Cn(U)!−x2∞
∑
n=0
Dn(U)−
∞
∑
n=0
Un!#. (48)
Comparing both sides of Equation(48), we obtain
U0(x,t) = x2t,
U1(x,t) = ˆ
S−1hu2
s2ˆ
Sx2∂
∂x(C0(U))−x2D0(U)−U0i,
U2(x,t) = ˆ
S−1hu2
s2ˆ
Sx2∂
∂x(C1(U))−x2D1(U)−U1i,
U3(x,t) = ˆ
S−1hu2
s2ˆ
Sx2∂
∂x(C2(U))−x2D2(U)−U2i
.
.
.
(49)
The first few components of Cn(U)and Dn(U)Adomian polynomials [22], for example, are given by:
C0(U) = U0,xU0,xx ,
C1(U) = U0,xU1,xx +U1,xU0,xx,
C2(U) = U0,xU2,xx +U2,xU0,xx +U1,xU1,xx,
.
.
.
(50)
and
D0(U) = (U0,xx )2,
D1(U) = 2U0,xx U1,xx ,
D2(U) = 2U0,xx U2,xx +(U1,xx )2,
.
.
.
(51)
Using the iteration formulas (49)and the Adomian polynomials (50)and (51), we obtain
U0(x,t) = x2t,
U1(x,t) = −x2t3
3! ,
U2(x,t) = x2t5
5! ,
U3(x,t) = −x2t7
7! ,
.
.
.
(52)
The first terms of the approximate solution of Equation (45), is given by
U(x,t) = x2t−t3
3! +t5
5! −t7
7! +· · · . (53)
And in the closed form:
U(x,t) = x2sin(t). (54)
Open J. Math. Anal. 2019,3(2), 81-90 89
Figure 3. The graphs of exact solution and approximate solutions of Equation (45)for 3 terms and 4 terms.
This result represents the exact solution of the Equation (45)as presented in [27] The graphs of exact
solution and approximate solutions of Equation (45)for 3 terms and 4 terms are shown in Figure 3.
4. Conclusion
The coupling of Adomian decomposition method (ADM) and Shehu transform method proved very
effective to solve nonlinear partial differential equations. We can say that this method is easy to implement
and is very effective, as it allows us to know the exact solution after calculate the first three terms only. As a
result, the conclusion that comes through this work is that (STDM) can be applied to other nonlinear partial
differential equations of higher order, due to the efficiency and flexibility.
Author Contributions: All authors contributed equally to the writing of this paper. All authors read and approved the
final manuscript.
Conflicts of Interest: “The authors declare no conflict of interest.”
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