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Ain Shams Engineering Journal (2016) xxx, xxx–xxx
Ain Shams University
Ain Shams Engineering Journal
www.elsevier.com/locate/asejwww.sciencedirect.com
ENGINEERING PHYSICS AND MATHEMATICS
A computational approach for fractional
convection-diffusion equation via integral
transforms
* Corresponding author. Tel.: +91 9460254205.
E-mail addresses: jagdevsinghrathore@gmail.com (J. Singh),
mathsram09@gmail.com (R. Swroop), devendra.maths@gmail.com
(D. Kumar).
Peer review under responsibility of Ain Shams University.
Production and hosting by Elsevier
http://dx.doi.org/10.1016/j.asej.2016.04.0142090-4479 � 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: Singh J et al., A computational approach for fractional convection-diffusion equation via integral transforms, Ain Sham(2016), http://dx.doi.org/10.1016/j.asej.2016.04.014
Jagdev Singha, Ram Swroop
b,*, Devendra Kumarc
aDepartment of Mathematics, Jagan Nath University, Jaipur 303901, Rajasthan, IndiabDepartment of Mathematics, Arya Institute of Engineering & Technology, RIICO Kukas, Jaipur 303101, Rajasthan, IndiacDepartment of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India
Received 20 April 2015; revised 10 February 2016; accepted 6 April 2016
KEYWORDS
Homotopy analysis trans-
form method;
Homotopy perturbation
method;
Sumudu transform;
Laplace transform;
Fractional convection-
diffusion equation;�h-curve
Abstract In this paper, two efficient analytic techniques namely the homotopy analysis transform
method (HATM) and homotopy perturbation Sumudu transform method (HPSTM) are imple-
mented to give a series solution of fractional convection-diffusion equation which describes the flow
of heat. These proposed techniques introduce significance in the field over the existing techniques
that make them computationally very attractive for applications. Numerical solutions clearly
demonstrate the reliability and efficiency of HATM and HPSTM to solve strongly nonlinear frac-
tional problems.� 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an
open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
In recent years, fractional calculus is playing a very importantrole in all areas of science and engineering. An excellent liter-ature of fractional differentiation and integration operators
was also presented by number of researchers for extensionsof scientific and technical fields, among them were Caputo
[1], Oldham and Spanier [2], Samko et al. [3], Carpinteri and
Mainardi [4], Podlubny [5], Ahmed et al. [6], Kumar et al.[7], Atangana and Alabaraoye [8], Arife et al. [9], Yin et al.[10]. The beauty of this subject is that a fractional derivative
is not a local point property. Fractional calculus considersthe genetic and nonlocal distributed effects. This characteristicmakes it more accurate than the description of the integer-
order derivative.Fractional convection-diffusion equation is solved by num-
ber of methods, such as fractional variational iteration method(FVIM) was used by Merdan [11], which is computationally
complicated procedure because it involved Lagrange multi-plier, correction functional, stationary conditions. The homo-topy perturbation method (HPM) used by Momani and
Yildrim [12] has its own limitation of the small parameterassumption. The Adomian decomposition method (ADM)applied by Momani [13] involved complicated Adomian
s Eng J
Figure 4 4th order HPSTM (HATM, �h ¼ �1) approximate
solutions of yðx; tÞ at x ¼ 0:5 versus time t at different values of afor Example 1.
Figure 2 4th order approximate solution yðx; tÞ at a ¼ 1 for
Example 1.
Figure 1 Exact solution yðx; tÞ versus x and time t for Example 1.
Figure 3 Absolute error E4ðyÞ ¼ jyex � yappj at �h ¼ �1 and a ¼ 1
for Example 1.
2 J. Singh et al.
polynomials which narrow down its applications. Recently, the
flatlet oblique multiwavelets successfully applied and foundnumerical solution for a class of fractional convection-diffusion equations by Irandoust-pakchin et al. [14]. To over-come these disadvantages, we use comparatively two effective
techniques, homotopy analysis transform method (HATM) onthe one hand and on the other hand homotopy perturbationSumudu transform method (HPSTM) to solve fractional
convection-diffusion equation.The homotopy analysis transform method (HATM) basi-
cally shows how the Laplace transform can be used to find
the approximate solutions of time depended fractional
Please cite this article in press as: Singh J et al., A computational approach for frac(2016), http://dx.doi.org/10.1016/j.asej.2016.04.014
convection-diffusion equation by manipulating the homotopyanalysis method (HAM). The advantage of this method is its
Capability of combining two powerful methods, homotopyanalysis method (HAM) and Laplace transform method.Homotopy analysis method (HAM) is an analytic approxima-
tion method was first proposed and applied by Liao [15–18],based on homotopy, a fundamental concept in topology anddifferential geometry [19]. In recent years, HAM attracts the
attention of researchers for solving various nonlinear problemsin science, finance and engineering, such as model for HIV
infection of CD4þT cells [20], finance problems [21,22],
tional convection-diffusion equation via integral transforms, Ain Shams Eng J
Figure 5 �h-curve for different order HATM approximations
represents the valid range of �h is �2:005 6 �h < 0 at
x ¼ 0:5; t ¼ 0:005; a ¼ 1 for Example 1.
Figure 6 Exact solution yðx; tÞ versus x and time t for Example 2.
Figure 8 Absolute error E3ðyÞ ¼ jyex � yappj at �h ¼ �1 and a ¼ 1
for Example 2.
Figure 7 3rd order approximate solution at a ¼ 1 for Example 2.
Fractional convection-diffusion equation 3
Jaulent-Miodek equations [23], nonlinear heat transfer [24],nonlinear heat conduction and convection equations [25],
Burgers-Huxley equation [26], thermal-hydraulic networks[27], and differential–difference equation [28]. The HATM[29–33] is used for handling many nonlinear problems, which
provides us a simple way to ensure the rapid convergence ofthe solution series, valid over larger special and parameterdomains comparatively to VIM, ADM and perturbation
techniques.Homotopy perturbation Sumudu transform method
(HPSTM) is combination of Sumudu transform, homotopy
Please cite this article in press as: Singh J et al., A computational approach for frac(2016), http://dx.doi.org/10.1016/j.asej.2016.04.014
perturbation method and He’s polynomials. Homotopy per-turbation method (HPM) is introduced by Ji-Huan He of
Shanghai University in 1998, used in scientific and technicalproblems [34–38]. Sumudu transform was proposed by Watu-gala [39] in 1993, to solve differential equations and control
engineering problems. Number of researchers paid attentionin various types of problems via Sumudu transform, such asordinary and partial differential equations [40–42], fractional
partial differential equations [43–45], and integral equations[46]. Note that a very interesting fact about Sumudu transformis that the original function and its Sumudu transform have thesame Taylor coefficients except the factor n; see [47]. Thus if
fðtÞ ¼P1n¼0ant
n then fðyÞ ¼P1n¼0Cðnþ 1Þanynn, see [48]. In
the same way, the Sumudu transform sends combinations,
tional convection-diffusion equation via integral transforms, Ain Shams Eng J
Figure 9 2nd order HPSTM (HATM, �h ¼ �1) approximate
solutions of yðx; tÞ versus time t at x ¼ 0:002 at different values of
a for Example 2.
Figure 10 �h-curve for 2nd order HATM approximations repre-
sents the valid range of �h is �2:02 6 �h < 0 at
x ¼ 0:002; t ¼ 0:005; a ¼ 1 for Example 2.
4 J. Singh et al.
Cðm; nÞ; into permutations; Pðm; nÞ and hence it will be veryimportant and useful in the discrete systems. In addition, some
analytical methods use a combination of Sumudu transformfor obtaining series solution which converge to the exact solu-tions, such as homotopy analysis Sumudu transform method
[49], Sumudu decomposition method [50], and variation itera-tion Sumudu transform method [51]. Recently, the homotopyperturbation Sumudu transform method (HPSTM) is fre-
quently used for solving linear and nonlinear equations which
Please cite this article in press as: Singh J et al., A computational approach for frac(2016), http://dx.doi.org/10.1016/j.asej.2016.04.014
are fractional heat-like equations [52], heat and wave-likeequations [53], gas dynamics equation [54], etc.
The convection-diffusion equation models the flow of heat
in situations where there is both diffusion and convection. Inthis paper, we consider the following fractional nonlinearconvection-diffusion problem
@ay
@ta¼ @2y
@x2� c
@y
@xþ UðyÞ þ fðx; tÞ
0 < x 6 1; 0 < a 6 1; t > 0;
ð1Þ
yðx; 0Þ ¼ hðxÞ; 0 < x 6 1; ð2Þwhere UðyÞ is some reasonable nonlinear function of y which ischosen as a potential energy, c is a constant, a is a parameter
describing the order of the time-fractional derivative andyðx; tÞ represents temperature in heat transfer and is causalfunction of time, i.e. vanishing for t < 0. The fractional deriva-
tive is considered in the Caputo sense. The convection-diffusion equations are widely used in science and engineeringas mathematical models for computational simulations, suchas in oil reservoir simulations, transport of mass and energy,
and global weather production, in which an initially discontin-uous profile is propagated by diffusion and convection, the lat-ter with a speed of c [55].
2. Basic definitions
Definition 1. The Laplace transform of continuous (or analmost piecewise continuous) function fðtÞin½0;1Þ is defined
and represented in the following form
FðsÞ ¼ L½fðtÞ� ¼Z t
0
e�stfðtÞdt: ð3Þ
where s denotes a real or complex number.
Definition 2. The fractional derivative of f (t) in the Caputo[56] sense is defined as follows
DafðtÞ ¼ Jn�aDn
¼ 1
Cðn� aÞZ t
0
ðt� sÞn�a�1fnðsÞds;
for n� 1 < a 6 n; n 2 N; x > 0: ð4Þ
Definition 3. The Laplace transform of the Caputo derivative
is given by Caputo see also Kilbas et al. [57] in the form
L½DafðtÞ� ¼ saL½fðtÞ� �Xn�1
r¼0
sa�r�1fðrÞð0þÞ; n� 1 < a 6 n: ð5Þ
Definition 4. The Laplace transform L½fðtÞ� of the Riemann–Liouville fractional order integral is defined as
L½Iat fðtÞ� ¼ s�aFðsÞ: ð6Þ
Definition 5. The Sumudu transform of fractional orderderivative is defined and represented as [54]
S½DaxfðtÞ�¼ u�aS½fðtÞ��
Xn�1
r¼0
u�aþr½fðrÞðtÞ�x¼0;n�1< a6 n;n2N: ð7Þ
tional convection-diffusion equation via integral transforms, Ain Shams Eng J
Fractional convection-diffusion equation 5
Definition 6. The Mittag-Leffler is defined and represented in
the following form [58]:
Ea;bðzÞ ¼X1k¼0
zk
Cðakþ bÞ ; a; b P 0: ð8Þ
3. Basic idea of HATM
To show the basic idea and solution procedure of this method,we take a general time-fractional nonlinear non-homogeneouspartial differential equation of the form:
Dat yðx; tÞ þ Ryðx; tÞ þNyðx; tÞ ¼ gðx; tÞ; n� 1 < a 6 n ð9Þ
where Dat yðx; tÞ represents the Caputo fractional derivative of
the function yðx; tÞ;R is the linear differential operator, N rep-resents the general nonlinear differential operator and gðx; tÞ;is the source term.
By applying the Laplace transform on both sides of Eq. (9),it gives the result
L½Dat y� þ L½Ry� þ L½Ny� ¼ L½gðx; tÞ�: ð10Þ
Making use of the differentiation property of the Laplacetransform, we have
saL½y� �Xn�1
k¼0
sa�k�1yðkÞðx; 0Þ þ L½Ry� þ L½Ny� ¼ L½gðx; tÞ�:
ð11ÞOn simplifying, we have the following equation
L½y�� 1
sa
Xn�1
k¼0
sa�k�1yðkÞðx;0Þþ 1
saL½Ry�þL½Ny��L½gðx; tÞ�½ � ¼ 0: ð12Þ
Next, we define and present the nonlinear operator as
N½/ðx; t; qÞ� ¼ L½/ðx; t; qÞ� � 1
sa
Xn�1
k¼0
sa�k�1/ðkÞðx; t; qÞð0þÞ
þ 1
saL½R/ðx; t; qÞ� þ L½N/ðx; t; qÞ� � L½gðx; tÞ�½ �;
ð13Þwhere q 2 ½0; 1� and /ðx; t; qÞ is a real function of x, t and q.We construct a homotopy as follows:
ð1� qÞL½/ðx; t; qÞ � y0ðx; tÞ� ¼ �hqHðx; tÞN½yðx; tÞ�; ð14Þwhere L denotes the Laplace transform operator, q 2 ½0; 1� isthe embedding parameter, Hðx; tÞ denotes a nonzero auxiliary
function, �h–0 is an auxiliary parameter, y0ðx; tÞ is an initialguess of yðx; tÞ and /ðx; t; qÞ is a unknown function. Obvi-ously, when the embedding parameter q ¼ 0 and q ¼ 1; it
holds
/ðx; t; 0Þ ¼ y0ðx; tÞ; /ðx; t; 1Þ ¼ yðx; tÞ; ð15Þ
respectively. Thus, as q increases form 0 to 1, the solution/ðx; t; qÞ varies from the initial guess y0ðx; tÞ to the solution
yðx; tÞ: Expanding /ðx; t; qÞ in Taylor series with respect toq, we have
/ðx; t; qÞ ¼ y0ðx; tÞ þX1m¼1
ymðx; tÞqm; ð16Þ
Please cite this article in press as: Singh J et al., A computational approach for frac(2016), http://dx.doi.org/10.1016/j.asej.2016.04.014
where
ymðx; tÞ ¼1
m!
@m/ðx; t; qÞ@qm
����q¼0
: ð17Þ
If the auxiliary linear operator, the initial guess, the auxil-
iary parameter �h; and the auxiliary function are properly cho-sen, the series (16) converges at q = 1, then we have
yðx; tÞ ¼ y0ðx; tÞ þX1m¼1
ymðx; tÞ; ð18Þ
which must be one of the solutions of the original nonlinearequations. According to the definition (21), the governingequation can be deduced from the zero-order deformation
Eq. (14).Define the vectors in the following manner
y!m ¼ fy0ðx; tÞ; y1ðx; tÞ; . . . ; ymðx; tÞg: ð19ÞDifferentiating the zeroth-order deformation Eq. (14) m-
times with respect to q and then dividing them by m! andfinally setting q ¼ 0; we get the following mth-order deforma-tion equation:
L ymðx; tÞ � vmym�1ðx; tÞ½ � ¼ �hHðx; tÞRmð y!m�1Þ: ð20ÞApplying the inverse Laplace transform, we have
ymðx; tÞ ¼ vmym�1ðx; tÞ þ �hL�1 Hðx; tÞRmð y!m�1Þ� �
; ð21Þ
where the value of Rmð y!m�1Þ is obtained from the followingformula
Rmð y!m�1Þ ¼ 1
ðm� 1Þ!@m�1N½/ðx; t; qÞ�
@qm�1
����q¼0
; ð22Þ
and the value of vm is given as
vm ¼ 0; m 6 1;
1; m > 1:
�ð23Þ
4. Basic idea of HPSTM
We illustrate the basic idea of HPSTM by considering a gen-
eral time-fractional nonlinear non-homogeneous partial differ-ential equation with the initial condition of the general form:
Dat yðx;tÞ¼Ryðx;tÞþNyðx;tÞþgðx;tÞ; n�1< a6 n; a> 0; ð24Þ
subject to the initial condition
Dm0 yðx; 0Þ ¼ fmðxÞ; where m ¼ 0; . . . n� 1;
Dn0yðx; 0Þ ¼ 0; and n ¼ ½a�;
where Dat yðx; tÞ represents the well-known Caputo fractional
derivative of the function yðx; tÞ;R is the linear differential
operator, N represents the general nonlinear differentialoperator.
By applying the Sumudu transform on both sides of Eq.
(24), we get
S½Dat y� ¼ S½Ry� þ S½Ny� þ S½gðx; tÞ�: ð25Þ
Using the property of the Sumudu transform, we have thefollowing result
S½y� ¼ fðx; tÞ þ uaS½Ry� þ uaS½Ny� þ uaS½gðx; tÞ�: ð26Þ
tional convection-diffusion equation via integral transforms, Ain Shams Eng J
J. Singh et al.
Now applying the Sumudu inverse on both sides of (26) we
get
6
y ¼ Fðx; tÞ þ S�1 uaS½Ry� þ uaS½Ny�½ �; ð27Þwhere F(x, t) represents the term occurring from the knownfunction g(x, t) and the initial conditions.
Now we apply the HPM:
yðx; tÞ ¼X1n¼0
pnynðx; tÞ: ð28Þ
The nonlinear term can be decomposed in the followingway
Nyðx; tÞ ¼X1n¼0
pnHnðyÞ: ð29Þ
Using the He’s polynomial HnðyÞ [59] given as follows:
Hnðy0; . . . ;ynÞ¼1
Cðnþ1Þ@n
@pnNX1r¼0
pryrðx; tÞ !" #
; n¼ 0;1;2; . . . ð30Þ
Substituting (28) and (29) in (27), we have
X1n¼0
pnyn ¼ Fðx; tÞ þ p S�1 uaS RX1n¼0
pnyn
!" #""
þuaS NX1n¼0
pnHn
!" ###: ð31Þ
This is the coupling of the Sumudu transform and the HPMusing He’s polynomials. Comparing the coefficients of likepower of p, the following approximations are obtained.
p0 : y0 ¼ Fðx; tÞ;p1 : y1 ¼ S�1 uaS½Rðy0Þ þH0ðyÞ�b c;p2 : y2 ¼ S�1 uaS½Rðy1Þ þH1ðyÞ�b c;p3 : y3 ¼ S�1 uaS½Rðy2Þ þH2ðyÞ�b c;...
pn : yn ¼ S�1 uaS½Rðyn�1Þ þHn�1ðyÞ�b c;...
ð32Þ
The approximate analytical solution of (24) yðx; tÞ by the
truncated series:
yðx; tÞ ¼ y0ðx; tÞ þX1k¼1
ykðx; tÞ; ð33Þ
5. Numerical examples
In this section, we employ the above reliable approach, in a
realistic and efficient way, to handle nonlinear fractional equa-tion with time-fractional derivative.
Example 1. We consider the following homogeneous frac-tional nonlinear convection-diffusion problem.
@ay
@ta¼ @2y
@x2� @y
@xþ y
@2y
@x2� y2 þ y;
0 < x 6 1; 0 < a 6 1; t > 0: ð34Þsubject to the boundary conditions
Please cite this article in press as: Singh J et al., A computational approach for frac(2016), http://dx.doi.org/10.1016/j.asej.2016.04.014
yð0; tÞ ¼ et; yð1; tÞ ¼ etþ1; ð35Þand the initial condition
yðx; 0Þ ¼ ex: ð36ÞApplying the Laplace transform subject to the initial condi-
tion, we get the following equation
L½yðx; tÞ� � 1
sex � 1
saL
@2y
@x2� @y
@xþ y
@2y
@x2� y2 þ y
� �¼ 0: ð37Þ
The nonlinear operator is
N½/ðx; t;qÞ� ¼L½/ðx; t;qÞ��1
sex� 1
saL
@2/ðx; t;qÞ@x2
�@/ðx; t;qÞ@x
�
þ/ðx; t;qÞ@2/ðx; t;qÞ@x2
�/ðx; t;qÞ2þ/ðx; t;qÞ�:
ð38Þ
And thus
Rð y!m�1Þ ¼ Lðym�1Þ � ð1� vmÞ1
sex � 1
saL
@2ym�1
@x2� @ym�1
@x
�
þXm�1
i¼0
yi@2ym�i�1
@x2�Xm�1
i¼0
yiym�i�1 þ y
#: ð39Þ
The mth-order deformation equation is presented as
L ymðx; tÞ � vmym�1ðx; tÞ½ � ¼ �hRmð y!m�1Þ: ð40ÞBy taking inverse Laplace transform, we arrive at the
following result
ymðx; tÞ ¼ vmym�1ðx; tÞ þ �hL�1½Rmð y!m�1Þ�: ð41ÞSolving the above Eq. (41), for m = 1, 2, 3, . . ., we get
y0ðx;tÞ¼ ex;
y1ðx;tÞ¼��hta
Cðaþ1Þex;
y2ðx;tÞ¼ ð�hþ1Þy1ðx;tÞþ�h2ext2a
Cð2aþ1Þ ;
y3ðx;tÞ¼ ð�hþ1Þy2ðx;tÞþ�h2ðhþ1Þex t2a
Cð2aþ1Þ��h3ext3a
Cð3aþ1Þ ;
..
.
ð42Þ
In the similar manner, the rest of the components ynðx; tÞfor n > 3 can be completely derived and the series solutionsare thus entirely found. Therefore, the approximate solution is
yðx; tÞ ¼ y0ðx; tÞ þX1n¼1
ynðx; tÞ: ð43Þ
When we put a ¼ 1 and �h ¼ �1 in the Eq. (43), then clearly
we can conclude that the obtained solutionPN
n¼0ynðx; tÞ as
N ! 1 converges to the exact solution yðx; tÞ ¼ exþt, whichis same as well-established results obtained with the help of
FVIM [11], HPM [12] and ADM [13].
tional convection-diffusion equation via integral transforms, Ain Shams Eng J
Fractional convection-diffusion equation 7
Now in order to obtain the solution by making use of
HPSTM, we take the Sumudu transform of Eq. (24), subject tothe initial condition (26) its gives
S@ay
@ta
� �¼ S
@2y
@x2� @y
@xþ y
@2y
@x2� y2 þ y
� �;
0 < x 6 1; 0 < a 6 1; t > 0: ð44Þ
S½y� ¼ ex þ uaS@2y
@x2� @y
@xþ y
@2y
@x2� y2 þ y
� �; ð45Þ
Taking inverse Sumudu transform, we arrive at the follow-ing result
y ¼ ex þ S�1 uaS@2y
@x2� @y
@xþ y
@2y
@x2� y2 þ y
� � : ð46Þ
Now using the HPM, we get
X1n¼0
pnyn ¼ ex þ pS�1 ua SX1n¼0
pnyn
!xx
�X1n¼0
pnyn
!x
þX1n¼0
pnHnðyÞ þX1n¼0
pnyn
!!!: ð47Þ
Comparing the coefficient of like powers of p, we get thefollowing components of the series solution
p0 : y0 ¼ ex;
p1 : y1 ¼ ex ta
Cðaþ 1Þ ;
p2 : y2 ¼ ex t2a
Cð2aþ 1Þ ;
p3 : y3 ¼ ex t3a
Cð3aþ 1Þ ;
..
.
Pn : yn ¼ ex tna
Cðnaþ 1Þ ;
..
.
ð48Þ
Proceeding in this manner, the rest of the componentsynðx; tÞ can be found. Therefore, the approximate solution is
given in the following form
yðx; tÞ ¼ ex 1þ ta
Cðaþ 1Þ þt2a
Cð2aþ 1Þ þt3a
Cð3aþ 1Þ�
þ t3a
Cð3aþ 1Þ þ . . .þ tna
Cðnaþ 1Þ þ � � �; ð49Þ
which can be written as
yðx; tÞ ¼ exX1n¼0
tna
Cðnaþ 1Þ : ð50Þ
which converge to the exact solution
yðx; tÞ ¼ exEaðtaÞ: ð51ÞIf we set a ! 1; then
yðx; tÞ ¼ exX1r¼0
tr
Cðrþ 1Þ ¼ exþt: ð52Þ
Please cite this article in press as: Singh J et al., A computational approach for frac(2016), http://dx.doi.org/10.1016/j.asej.2016.04.014
Example 2. Consider the following non-homogeneous frac-
tional nonlinear convection-diffusion problem of the form
@ay
@ta¼ @2y
@x2� c
@y
@xþ @
@t
� hðyÞ þ fðx; tÞ;
0 < x 6 1; 0 < a 6 1; t > 0; ð53ÞThe boundary and initial conditions imposed are
yð0; tÞ ¼ 2t; yð1; tÞ ¼ 1þ 2t; ð54Þsubject to the initial condition
yðx; 0Þ ¼ x2: ð55ÞApplying the Laplace transform subject to the initial condi-
tion, we have
L½yðx; tÞ� � 1
sx2 � 1
saL
@2y
@x2� @y
@xþ @
@ty@y
@x
� � 2x
� �¼ 0:
ð56ÞThe nonlinear operator is
N½/ðx; t;qÞ� ¼L½/ðx;t;qÞ��1
sx2� 1
saL
@2/ðx;t;qÞ@x2
�@/ðx; t;qÞ@x
�
þ @
@t/ðx;t;qÞ@/ðx;t;qÞ
@x
� �2x
�: ð57Þ
And thus
Rð y!m�1Þ ¼ Lðym�1Þ � ð1� vmÞ1
sex � 1
saL
@2ym�1
@x2� @ym�1
@x
�
þ @
@t
Xm�1
i¼0
yi@ym�1�i
@x
!� 2ð1� vmÞx
#: ð58Þ
The mth-order deformation equation is written as
L ymðx; tÞ � vmym�1ðx; tÞ½ � ¼ �hRmð y!m�1Þ: ð59ÞEmploying the inverse Laplace transform, we have
ymðx; tÞ ¼ vmym�1ðx; tÞ þ �hL�1 Rmð y!m�1Þ� �
: ð60ÞSolving the above Eq. (60), for m= 1, 2, 3, . . ., we get the
following components of the series solution
y0ðx;tÞ¼ x2;
y1ðx;tÞ¼��hð�4xþ2Þ ta
Cðaþ1Þ ;
y2ðx;tÞ¼ ð�hþ1Þy1ðx; tÞþ4h2t2a
Cð2aþ1Þþ4�h2ð�3x2þxÞ t2a�1
Cð2aÞ ;
y3ðx;tÞ¼ ð�hþ1Þy2ðx; tÞþ4�h2ð�hþ1Þ t2a
Cð2aþ1Þ�4�h2ð�hþ1Þx2 t2a�1
Cð2aÞþ�h2ð�hþ1Þð�12x2þ4xÞ t2a�1
Cð2aÞ�4�h3ð�12x3þ3x2Þ t3a�2
Cð3a�1Þþ8�h3ð�4xþ2Þ aCð2aÞðCðaþ1ÞÞ2
t3a�1
Cð3aÞ
þ�h3ð�32xþ28Þ t3a�1
Cð3aÞ ;
..
.
ð61Þand so on.
tional convection-diffusion equation via integral transforms, Ain Shams Eng J
8 J. Singh et al.
In this manner the rest of the components can be found.
Therefore, the approximate solution is
yðx; tÞ ¼ y0ðx; tÞ þX1n¼1
ynðx; tÞ: ð62Þ
If we take a ¼ 1; �h ¼ �1 then the solution for the standardconvection-diffusion equation is as follows
yðx; tÞ ¼ Limn!1
ynðx; tÞ¼ x2 þ 2t� 4txþ 2t2 � 12x2tþ 4xt� 12t2
þ 12xt2 � 2t2 � � � : ð63ÞIt is obvious that the self-cancelling ‘noise’ terms appear
between various components. Cancelling the noise terms andkeeping the non-noise terms in Eq. (63), it yields the exact solu-tion presented as
yðx; tÞ ¼ x2 þ 2t; ð64Þwhich is easily verified and formally justified by Yildrim [60].
Now, taking Sumudu transform of Eq. (53), subject to theinitial condition (55), we get
S½y� ¼ x2 þ uaS@2y
@x2� @y
@xþ @
@ty@y
@x
� � 2x
� �¼ 0; ð65Þ
Taking inverse Sumudu transform
y ¼ x2 þ S�1 uaS yxx � yx þ@
@tðyyxÞ � 2x
� � : ð66Þ
Now, applying the homotopy perturbation technique, weget
X1n¼0
pnyn ¼ x2 þ pS�1 ua SX1n¼0
pnyn
!xx
�X1n¼0
pnyn
!x
þ @
@t
X1n¼0
pnHnðyÞ !
� 2x
!!!; ð67Þ
Comparing the coefficient of like of p, we have
p0 : y0 ¼ x2;
p1 : y1 ¼ ð�4xþ 2Þ ta
Cðaþ 1Þ ;
p2 : y2 ¼ 4t2a
Cð2aþ 1Þ þ 4ð�3x2 þ xÞ t2a�1
Cð2aÞ ;
P3 : y3 ¼ �24t3a�1
Cð3aÞ � ð24xþ 4Þ t3a�1
Cð3aÞ
þ ð�24x3 þ 4x2Þ t3a�2
Cð3a� 1Þ
� ð�16xþ 8Þ Cð2aþ 1Þt3a�1
ðCðaþ 1ÞÞ2Cð3aÞ þ 8xt3a�1
Cð3aÞ
þ ð�24x3 þ 8x2Þ t3a�2
Cð3a� 1Þ...
ð68Þ
Please cite this article in press as: Singh J et al., A computational approach for frac(2016), http://dx.doi.org/10.1016/j.asej.2016.04.014
Following the way, the rest of the components ynðx; tÞfor n > 3 can be derived. Therefore, the approximate solutionis
yðx;tÞ¼ x2þð�4xþ2Þ ta
Cðaþ1Þþ4t2a
Cð2aþ1Þþ4ð�3x2þxÞ t2a�1
Cð2aÞ�24t3a�1
Cð3aÞ�ð24xþ4Þ t3a�1
Cð3aÞþð�24x3þ4x2Þ t3a�2
Cð3a�1Þ�ð�16xþ8Þ Cð2aþ1Þt3a�1
ðCðaþ1ÞÞ2Cð3aÞ . . . :
ð69ÞTherefore, the solution for the convection-diffusion Eq.
(53) when a ¼ 1; is
yðx; tÞ ¼ Limn!1
ynðx; tÞ¼ x2 þ 2t� 4txþ 2t2 � 12x2tþ 4xt� 12t2
þ 12xt2 � 2t2 . . . : ð70ÞIt is obvious that the self-cancelling ‘noise’ terms appear
between various components. Cancelling the noise terms andkeeping the non-noise terms in Eq. (70) it yields the exact solu-
tion of Eq. (53) given as
yðx; tÞ ¼ x2 þ 2t; ð71Þwhich is the same as obtained by HATM and FVIM [11],HPM [12] and ADM [13]. It’s clear to see that theHPSTM solution is a special case of the HATM solution when�h ¼ �1:
6. Numerical results and discussions
In this section, we compute the numerical solutions of the tem-perature yðx; tÞ for different time-fractional Brownian motions
a ¼ 0:7; 0:8; 0:9 and also the standard motion a ¼ 1. Figs. 1–5present numerical results obtained by using HATM, HPSTMand exact solutions for fractional nonlinear convection-
diffusion problem (34). From Figs. 1 and 2, we observe thatas the values of time t and space variable x increase, the tem-perature yðx; tÞ increases. It is also observed from Fig. 4 that as
the value of a increases, the temperature yðx; tÞ decreases andconverges to the exact solution at a ¼ 1. Fig. 3 presents abso-lute errors of the results obtained with the help of proposedtechniques. Fig. 5 presents the �h� curve for different order
HATM approximations and shows that the valid range of �his �2:005 6 �h < 0 at x ¼ 0:5; t ¼ 0:005; a ¼ 1.
Figs. 6–10 show the numerical results commutated with the
help of HATM, HPSTM and exact solutions for fractionalnonlinear convection-diffusion problem (53). From Figs. 6and 7, we easily see that as the values of time t and space vari-
able x increase, the temperature yðx; tÞ increases. It is alsoobserved from Fig. 9 that as the value of a increases, the tem-perature yðx; tÞ decreases. Fig. 8 depicts absolute errors of the
results obtained with the help of proposed schemes. Fig. 10presents the �h� curve for 2nd order HATM approximationand shows that the valid range of �h is �2:02 6 �h < 0 atx ¼ 0:002; t ¼ 0:005; a ¼ 1: It is evident that by using more
terms, the accuracy of the results can be dramatically improvedand the errors converge to zero.
tional convection-diffusion equation via integral transforms, Ain Shams Eng J
Fractional convection-diffusion equation 9
7. Conclusions
In this paper, the homotopy analysis transform method(HATM) and homotopy perturbation Sumudu transform
method (HPSTM) have been successfully applied to obtainthe numerical solution of time fractional convection-diffusionequation, which has a good agreement with the exact solution.
Comparing the methodology, we noted that the HATM has aclear advantage over the HPSTM, and it provides the auxiliaryparameter �h, can as be used to adjust and control the conver-gence region of series solutions. These proposed schemes can
be widely utilized to get fast convergent series solution in effi-cient way of different types of equations with strongnonlinearity.
Acknowledgements
The authors are extending their heartfelt thanks to the review-ers for their valuable suggestions for the improvement of thearticle.
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Jagdev Singh is an Associate Professor in the
Department of Mathematics, Jagan Nath
University, Jaipur-303901, Rajasthan, India.
His fields of research include Mathematical
Modelling, Special Functions, Fractional
Calculus, Fluid Dynamics, Nanofluids,
Homotopy Methods, Adomian Decomposi-
tion Method, Laplace Transform Method and
Sumudu Transform Method.
Ram Swroop is an Assistant Professor in the
Arya Institute of Engineering & Technology,
Riico Kukas, Jaipur-303101, Rajasthan,
India. His fields of research include fractional
calculus, homotopy methods, and Laplace
transform method.
Devendra Kumar is an Assistant Professor in
the Department of Mathematics, JECRC
University, Jaipur-303905, Rajasthan, India.
His area of interest is Mathematical Model-
ling, Special Functions, Fractional Calculus,
Fluid Dynamics, Nanofluids, Homotopy
Methods, Adomian Decomposition Method,
Laplace Transform Method and Sumudu
Transform Method.
tional convection-diffusion equation via integral transforms, Ain Shams Eng J