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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.
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) ([14]), homotopy analysis method ([58]), variational iteration
method (VIM) ([912]), homotopy perturbation method (HPM) ([1316]) and DJ iteration method ([1720]).
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
S1[V(s,u)]=v(t), for t>0. (3)
Equivalently
v(t) = ˆ
S1[V(s,u)]=1
2πi
α+i
Z
αi
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
s2a2u2
tnexp(at)
n!un+1
(sau)n+1cosh at us
s2a2u2
Open J. Math. Anal. 2019,3(2), 81-90 83
Proposition 4. If v(x,t)
tand 2v(x,t)
t2exist, then
ˆ
Sv(x,t)
t=s
uV(x,s,u)v(x, 0). (9)
ˆ
S2v(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
Sv(x,t)
t=
Z
0
est
uv(x,t)
tdt =lim
τ
τ
Z
0
est
uv(x,t)
tdt
=lim
τ
hv(x,t)est
uiτ
0+s
u
τ
Z
0
est
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
S2v(x,t)
t2=ˆ
Sw(x,t)
t=s
uˆ
S[w(x,t)]w(x, 0)
=s
uˆ
Sv(x,t)
tv(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
Snv(x,t)
tn=sn
unV(x,s,u)
n1
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
Snv(x,t)
tn=sn
unV(x,s,u)
n1
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
Sn+1v(x,t)
tn+1=ˆ
Sw(x,t)
t=s
uˆ
S(w(x,t))w(x, 0)
=s
u"sn
unV(x,s,u)
n1
k=0s
un(k+1)kv(x, 0)
tk#nv(x, 0)
tn
=sn+1
un+1V(x,s,u)
n1
k=0s
un+1(k+1)kv(x, 0)
tknv(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
SmU(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)]=
m1
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)]=
m1
k=0u
sk+1kU(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
S1um
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
S1"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
S1hum
smˆ
S[RU0(x,t) + A0(U)]i,
U2(x,t) = ˆ
S1hum
smˆ
S[RU1(x,t) + A1(U)]i,
U3(x,t) = ˆ
S1hum
smˆ
S[RU2(x,t) + A2(U)]i,
.
.
.
(22)
In general, the recursive relation is given as:
Un+1(x,t) = ˆ
S1um
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+UUxUxx =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)]=xu
sE[UUxUxx]. (28)
Taking the inverse Shehu transform on both sides of Equation (28), we obtain
U(x,t) = xˆ
S1u
sˆ
S[UUxUxx]. (29)
By applying the aforesaid decomposition method, we have
n=0
U(x,t) = xˆ
S1"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
S1u
sˆ
S(A0(U)U0xx (x,t)),
U2(x,t) = ˆ
S1u
sˆ
S(A1(U)U1xx (x,t)),
U3(x,t) = ˆ
S1u
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) = xxt +xt2xt3+· · · , (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+UUxU(1U) = 0, t>0, (36)
with the initial condition:
U(x, 0) = ex. (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) = exˆ
S1[u
sˆ
SUUx+U2U]. (38)
By applying the aforesaid Decomposition Method, we have
n=0
Un(x,y) = exˆ
S1"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) = ex,
U1(x,t) = ˆ
S1u
sˆ
S(A0(U) + B0(U)U0(x,t)),
U2(x,t) = ˆ
S1u
sˆ
S(A1(U) + B0(U)U1(x,t)),
U3(x,t) = ˆ
S1u
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) = eκ,
U1(x,t) = eκt,
U2(x,t) = eκt2
2! ,
U3(x,t) = eκt3
3! ,
.
.
.
(42)
So, the approximate series solution of Equation (36)is given as:
U(x,t) = ex1+t+t2
2! +t3
3! +· · · . (43)
And in the closed form, is given by:
U(x,t) = exet=etx. (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 )2U, 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
S1u2
s2ˆ
Sx2
x(UxUxx )x2(Uxx )2U. (47)
By applying the aforesaid Decomposition Method, we have
n=0
U(x,t) = x2t+ˆ
S1"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
S1hu2
s2ˆ
Sx2
x(C0(U))x2D0(U)U0i,
U2(x,t) = ˆ
S1hu2
s2ˆ
Sx2
x(C1(U))x2D1(U)U1i,
U3(x,t) = ˆ
S1hu2
s2ˆ
Sx2
x(C2(U))x2D2(U)U2i
.
.
.
(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) = x2tt3
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.”
References
[1] Adomian, G., & Rach, R. (1990). Equality of partial solutions in the decomposition method for linear or nonlinear
partial differential equations. Computers & Mathematics with Applications, 19(12), 9-12.
[2] Adomian, G. (2013). Solving frontier problems of physics: the decomposition method (Vol. 60). Springer Science & Business
Media.
[3] Adomian, G. (1994). Solution of physical problems by decomposition. Computers & Mathematics with Applications,
27(9-10), 145-154.
[4] Adomian, G. (1998). Solutions of nonlinear PDE. Applied Mathematics Letters, 11(3), 121-123.
[5] Liao, S. J. (1992). The proposed homotopy analysis technique for the solution of nonlinear problems (Doctoral dissertation, Ph.
D. Thesis, Shanghai Jiao Tong University).
[6] Raton, B. (2003). Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman and Hall/CRC Press, Boca
Raton.
[7] Liao, S. (2004). On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation,
147(2), 499-513.
[8] Liao, S. (2009). Notes on the homotopy analysis method: some definitions and theorems. Communications in Nonlinear
Science and Numerical Simulation, 14(4), 983-997.
[9] He, J. (1997). A new approach to nonlinear partial differential equations. Communications in Nonlinear Science and
Numerical Simulation, 2(4), 230-235.
[10] He, J. H. (1998). Approximate analytical solution for seepage flow with fractional derivatives in porous media.
Computer Methods in Applied Mechanics and Engineering, 167(1-2), 57-68.
[11] He, J. H. (1998). A variational iteration approach to nonlinear problems and its applications. Mech. Appl, 20(1), 30-31.
[12] He, J. H., & Wu, X. H. (2007). Variational iteration method: new development and applications. Computers &
Mathematics with Applications, 54(7-8), 881-894.
[13] He, J. H. (1999). Homotopy perturbation technique. Computer methods in applied mechanics and engineering, 178(3-4),
257-262.
Open J. Math. Anal. 2019,3(2), 81-90 90
[14] He, J. H. (2005). Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons &
Fractals, 26(3), 695-700.
[15] He, J. H. (2000). A coupling method of homotopy technique and perturbation to VolterraŠs integro-differential
equation. International Journal of Non-Linear Mechanics, 35(1), 37-43.
[16] He, J. H. (2000). A new perturbation technique which is also valid for large parameters. Journal of Sound and Vibration,
5(229), 1257-1263.
[17] Daftardar-Gejji, V., & Jafari, H. (2006). An iterative method for solving nonlinear functional equations. Journal of
Mathematical Analysis and Applications, 316(2), 753-763.
[18] Hemeda, A. A. (2012). New iterative method: application to nth-order integro-differential equations. In International
Mathematical Forum (Vol. 7, No. 47, pp. 2317-2332).
[19] AL-Jawary, M. A. (2014). A reliable iterative method for Cauchy problems. Mathematical Theory and Modeling, 4,
148-153.
[20] Patade, J., & Bhalekar, S. (2015). Approximate analytical solutions of Newell-Whitehead-Segel equation using a new
iterative method. World Journal of Modelling and Simulation, 11(2), 94-103.
[21] Maitama, S., & Zhao, W. (2019). New integral transform: Shehu transform a generalization of Sumudu and Laplace
transform for solving differential equations. International Journal of Nonlinear Analysis and Applications , 17(2), 167-19
[22] Zhu, Y., Chang, Q., & Wu, S. (2005). A new algorithm for calculating Adomian polynomials. Applied Mathematics and
Computation, 169(1), 402-416.
[23] Hosseini, M. M., & Nasabzadeh, H. (2006). On the convergence of Adomian decomposition method. Applied
mathematics and computation, 182(1), 536-543.
[24] Wazwaz, A. M. (2007). The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic
Boussinesq equations. Journal of Computational and Applied Mathematics, 207(1), 18-23.
[25] Ziane, D., & Cherif, M. H. (2015). Resolution of nonlinear partial differential equations by elzaki transform
decomposition method. J. Appr. Theor. App. Math, 5, 17-30.
[26] Jafari, H., Chun, C., Seifi, S., & Saeidy, M. (2009). Analytical solution for nonlinear gas dynamic equation by homotopy
analysis method. Applications and Applied mathematics, 4(1), 149-154.
[27] Gupta, V., & Gupta, S. (2013). Homotopy perturbation transform method for solving nonlinear wave-like equations
of variable coefficients. Journal of Information and Computing Science, 8(3), 163-172.
c
2019 by the authors; licensee PSRP, Lahore, Pakistan. This article is an open access article
distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license
(http://creativecommons.org/licenses/by/4.0/).
... Furthermore, the development of integral transforms have allowed for the efficient computation of PDEs, opening up new possibilities in the production of exact and approximate solutions of the equations among them, PDEs [5,23,32]. Recently, various applications of integral transforms have been found in different areas of engineering, mathematics and physics such as image processing, signal analysis and electric [21,26,30]. This is due to their properties and ability to transform a function from one domain to another domain while preserving important features of the function. ...
... It is worth mentioning that the integral transforms can be used in combination with other methods to address the nonlinear parts of equations. This can provide a much more efficient solution, and can be used to analyse many different systems with greater accuracy [7,29,32]. ...
Article
Full-text available
Our goal in this paper is to generalize the integral transforms and use it with He’s polynomial method to find the solution of the nonlinear partial differential equations. All results of theoretical studies regarding the generalization and its properties are presented. For the He’s polynomial method, it is used to solve the nonlinear part of the partial differential equation. It is shown that the importance of my research is the combination of generalization of integral transforms with He’s polynomial method allows for exact and approximate solutions configurations to be determined. Furthermore, the generalization of integral transforms has been shown to include most, or even all, of the integral transforms and be applicable to a variety of equations, making it a crucial tool in solving them. Finally, the capability of solutions to be obtained quickly and easily through this combined technique provides an invaluable tool for solving problems.
... (2) Definition 3: Inverse Shehu transform is defined as follows [18][19][20]: ...
... Definition 5: Linearity property of Shehu transform [18][19][20]: ...
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Present research deals with the time-fractional Schrödinger equations aiming for the analytical solution via Shehu Transform based Adomian Decomposition Method [STADM]. Three types of time-fractional Schrödinger equations are tackled in the present research. Shehu transform ADM is incorporated to solve the time-fractional PDE along with the fractional derivative in the Caputo sense. The developed technique is easy to implement for fetching an analytical solution. No discretization or numerical program development is demanded. The present scheme will surely help to find the analytical solution to some complex-natured fractional PDEs as well as integro-differential equations. Convergence of the proposed method is also mentioned.
... All these transformations are helpful and efficient methods for solving fractional partial differential equations with initial and boundary conditions. Many recent studies have combined various integral transforms with the Adomian decomposition method and its modifications to accelerate the resolution of these problems see (El-Kalla, 2007;Hussain and Khan, 2010;Kumar et al., 2012;El-Kalla et al., 2019;Khan et al., 2019;Mousa and Elzaki, 2019;Shah et al., 2019;Ziane et al., 2019). This study intends to use a combination of the triple Shehu transform and the Accelerated Adomian decomposition method (El-Kalla, 2007) for solving some fractional non-linear partial differential equations that have the following form Baleanu and Jassim (2020): ...
Conference Paper
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The integral transform methods are powerfully effective for solving fractional partial differential equations. This study employed the composition of the Accelerated Adomian Decomposition Method (AADM) and the triple Shehu transform method (TRHTM) to solve nonlinear fractional partial differential equations. Without the use of estimation or assumptions, the suggested technique yields an approximate-exact solution in the form of a series that converges to the exact solutions. Consequently, the triple Shehu accelerated decomposition method solves fractional partial differential equations quickly and efficiently. Demonstrates the effectiveness of this approach by studying some examples, and the outcomes demonstrate its dependability and effectiveness. Finally, we utilize Mathematica12 to create a graphical solution to the issue.
... Using Eqs. (37) and (38) in (36) yields the following: ...
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There is considerable literature on solutions to the gas-dynamic equation (GDE) and Fokker–Planck equation (FPE), where the fractional derivative is expressed in terms of the Caputo fractional derivative. There is hardly any work on analytical and numerical GDE and FPE solutions involving conformable fractional derivative (CFD). For the reasons stated above, we are required to solve the GDE and FPE in the form of CFD. The main goal of this research is to offer a novel combined method by employing the conformable Shehu transform (CST) and the homotopy perturbation method (HPM) for extracting analytical and numerical solutions of the time-fractional conformable GDE and FPE. The proposed method is called the conformable Shehu homotopy perturbation method (CSHPM). To evaluate its efficiency and consistency, relative and absolute errors among the approximate and exact solutions to three nonlinear problems of GDE and FPE are considered numerically and graphically. Moreover, fifth-term approximate and exact solutions are also compared by 2D and 3D graphs. This method has the benefit of not requiring any minor or major physical parameter assumptions in the problem. As a result, it may be used to solve both weakly and strongly nonlinear problems, overcoming some of the inherent constraints of classic perturbation approaches. Second, while addressing nonlinear problems, the CSHPM does not require Adomian polynomials. Therefore, to solve nonlinear GDE and FPE, just a few computations are necessary. As a consequence, it outperforms homotopy analysis and Adomian decomposition approaches significantly. It does not require discretization or linearization, unlike traditional numerical methods. The convergence and error analysis of the series solutions are also presented.
... Ordinary differential equations (ODEs) that have periodical solutions arise in various fields of applied sciences so that such kinds of problems are widely studied within applied and pure mathematics. In particular, numerical or semi-analytical solutions for these problems have been obtained with several methods such as the FDM [1], FEM [2,3], HAM [4][5][6], ADM [7][8][9][10] and other approaches [11]. In the last few decades, considerable attention has been focused on the periodic solutions of nonlinear oscillators, which are ubiquitous in every area of science related to oscillatory phenomena, not only in the areas of mechanics and physics, but also in other disciplines involving engineering applications [12][13][14][15]. ...
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In the present paper, we investigate the influence of the choice of continuous linear operator for obtaining the approximate periodic solutions of ordinary second-order differential equations. In most of these problems, the periods are unknown, and the determination of these periods and periodic solutions is a difficult issue. So, a new computational method is proposed based on the symmetric operator, namely the reproducing kernel Hilbert space (RKHS) method to obtain the interval of these solutions. This operator, as a consequence of the symmetric inner product, is a symmetric operator and it will be used to show the influence on periodic solutions. The high efficiency of the proposed strategy is presented along with some illustrative examples which demonstrate their periodic interval dealing with the choice of an appropriate continuous linear operator.
... The result (45) is the same as it was presented in [30]. 2 ...
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This article aims to solve Caputo-Fabruzo fractional differential equations using the Aboodh transform together with the Adomian Decomposition method (A-ADM). Since the Aboodh transform can only be applied to linear equations, ADM is an effective technique for approximating solutions of nonlinear differential equations. In nonlinear systems, the Newell-Whitehead-Segel equation plays an important role, explaining the emergence of stripes in 2-dimensional systems. The findings show that the results obtained from the tables provide superior results compared to the existing conformable q-Shehu homotopy analysis transform method (Cq-SHATM) in the literature. With the help of Matlab package program, numerical values were found to depict three-dimensional surfaces and displayed in a table.
Chapter
Foaming arises in many processes of absorption and distillation. The drainage of liquid foams involves the interplay of gravity, surface tension and viscous forces. For the foam density the drainage of liquid can be represented as a nonlinear partial differential equation as a function of time and vertical position. In this paper, Combined Shehu Accelerated Adomian Decomposition method is applied to solve fractional order foam drainage equation. The powerful Shehu Transform(ST) combined with Accelerated Adomian decomposition method (AADM) evaluates the approximate solution in the form of infinite series which converges very rapidly.KeywordsFoam drainage equationShehu transformAccelerated Adomian decomposition methodFractional derivatives
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In this paper, we introduce a Laplace-type integral transform called the Shehu transform which is a generalization of the Laplace and the Sumudu integral transforms for solving differential equations in the time domain. The proposed integral transform is successfully derived from the classical Fourier integral transform and is applied to both ordinary and partial differential equations to show its simplicity, efficiency, and the high accuracy.
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