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

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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

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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

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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.

References

[1] Caputo M. Linear models of dissipation whose Q is almost

frequency independent. Part II J R AstronSoc 1967;13:529–39.

[2] Oldham KB, Spanier J. The fractional calculus: integrations and

differentiations of arbitrary order. New York, NY, USA:

Academic Press; 1974.

[3] Samko SG, Kilbas AA, Marichev OI. Fractional integrals’ and

derivatives: theory and applications. Gordon Breach; 1993.

[4] Carpinteri A, Mainardi F. Fractional calculus in continuum

mechanics. New York, NY, USA: Springer; 1997.

[5] Podlubny I. Fractional differential equations, vol. 198. San Diego,

California, USA: Academic Press; 1999. p. 340.

[6] Ahmed E, El-Sayed AMA, El-Saka HAA. Equilibrium points,

stability and numerical solutions of fractional order predator-prey

and rabies models. J Math Anal Appl 2007;325:542–53.

[7] Kumar S, Kocak H, Yildirim A. A fractional model of gas

dynamics equation and its approximate solution by using Laplace

transform. Zeitschrift fur Naturforschung 2012;67:389–96.

[8] Atangana A, Alabaraoye E. Solving a system of fractional partial

differential equations arising in the model of HIV infection of

CD4þT cells and attractor one-dimensional Keller-Segel equa-

tions. Ad Diff Eq 2013;2013. Article 94.

[9] Arife AS, Vanani SK, Soleymani F. The Laplace homotopy

analysis method for solving a general fractional diffusion equation

arising in nano-hydrodynamics. J Comput Theor Nano Sci

2013;10:33–6.

[10] Yin F, Song J, Cao X. A general iteration formula of VIM

fractional heat-and wave-like equations. J Appl Math 2013;2013:9

428079.

[11] Merdan M. Analytical approximate solutions of fractional con-

vection-diffusion equation with modified Riemann–Liouville

derivative by means of fractional variational iteration method.

Iran J Sci Tech 2013;37:83–92.

[12] Momani S, Yildrim A. Analytical approximate solutions of the

fractional convection-diffusion equation with nonlinear source

term by He’s homotopy perturbation method. J Comput Math

2010;87:1057–65.

[13] Momani S. An algorithm for solving the fractional convection-

diffusion equation with nonlinear source term. Commun Nonlin

Sci Simul 2007;12:1283–90.

[14] Irandoust-pakchin S, Dehghan M, Abdi-mazraeh S, Lakestani M.

Numerical solution for a class of fractional convection-diffusion

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

equations using the flatlet oblique multiwavelets. J Vib Contr

2014;20:913–24.

[15] Liao SJ. The proposed homotopy analysis technique for the

solutions of nonlinear problems PhD thesis. Shanghai Jiao Tong

University; 1992.

[16] Liao SJ. Homotopy analysis method a new analytical technique

for nonlinear problems. Commun Nonlin Sci Numer Simul

1997;2:95–100.

[17] Liao SJ. Beyond perturbation: introduction to the homotopy

analysis method. Chaoman and Hall/CRC Pre Boca Raton; 2003.

[18] Liao SJ. On the homotopy analysis method for nonlinear

problems. Appl Math Comput 2004;147:499–513.

[19] Sen S. Topology and geometry for physics. Florida: Academic

Press; 1983.

[20] Ghoreishia M, Ismaila AIBMd, Alomari AK. Application of the

homotopy analysis method for solving a model for HIV infection

of CD4+ T-cells. Math Comput Model 2011;54:3007–15.

[21] Zhu SP. An exact and explicit solution for the valuation of

American put options. Quant Financ 2006;6:229–42.

[22] Zhu SP. A closed-form analytical solution for the valuation of

convertible bonds with constant dividend yield. ANZIAM J

2006;47:477–94.

[23] Rashidi MM, Domairry G, Dinarvand S. The homotopy analysis

method for explicit analytical solutions of Jaulent-Miodek equa-

tions. Numer Methods Part Diff 2009;25:430–9.

[24] Abbasbandy S. Homotopy analysis method to nonlinear

equations arising in heat transfer. Phys Lett A 2006;360:

109–13.

[25] Sajid M, Hayat T. Comparison of HAM and HPM methods for

nonlinear heat conduction and convection equations. Nonlin Anal

Real World Appl 2008;9:2296–301.

[26] Molabahrami A, Khani F. The homotopy analysis method to

solve the Burgers-Huxley equation. Nonlin Anal B: Real World

Appl 2008;13:2091–103.

[27] Cai WH. Nonlinear dynamics of thermal-hydraulic network PhD

thesis. University of Notre Dame; 2006.

[28] Wang Z, Zou L, Zhang H. Applying homotopy analysis method

for solving differential–difference equation. Phys Lett A

2007;369:77–84.

[29] Kumar S. An analytical algorithm for nonlinear fractional

Fornberg-Whitham equation arising in wave breaking based on

a new iterative method. Alex Eng J 2014;53:225–31.

[30] Gondal MA, Arife AS, Khan M, Hussain I. An efficient

numerical method for solving linear and nonlinear partial

differential equation by combining homotopy analysis and trans-

form method. World Appl Sci J 2011;14:1786–91.

[31] Khan M, Gondal MA, Hussain I, Vanani SK. A new comparative

study between homotopy analysis transform method and homo-

topy perturbation transform method on semi in finite domain.

Math Comput Model 2012;55:1143–50.

[32] Khader MM, Kumar S, Abbasbandy S. New Homotopy analysis

transform method for solving the discontinued problems arising in

nanotechnology. Chinese Phys B 2013;22:1059–62.

[33] Kumar D, Singh J, Kumar S, Sushila, Singh BP. Numerical

computation of nonlinear shock wave equation of fractional

order. Ain Shams Eng J 2015;6:605–11.

[34] He JH. Homotopy perturbation technique. Comput Methods

Appl Mech Eng 1999;178:257–62.

[35] He JH. Homotopy perturbation method: a new nonlinear

analytical technique. Appl Math Comput 2003;135:73–9.

[36] He JH. New interpretation of homotopy perturbation method. Int

J Mod Phys B 2006;20:2561–8.

[37] Vanani SK, Yildrim A, Soleymani F, Khan M, Tutkun S.

Solution of the heat equation in the cast-mould heterogeneous

domain using a weighted algorithm based on the homotopy

perturbation method. Int J Numer Methods Heat Fluid Flow

2013;23:451–9.

tional convection-diffusion equation via integral transforms, Ain Shams Eng J

10 J. Singh et al.

[38] Kumar S. A numerical study for solution of time fractional

nonlinear Shallow-water equation in Oceans. Zeitschrift for

Naturforschung A 2013;68:547–53.

[39] Watugala GK. Sumudu transform: a new integral transform to

solve differential equations and control engineering problems. Int

J Math Educ Sci Tech 1993;24:35–43.

[40] Belgacem FBM, Karaballi AA, Kalla SL. Analytical investiga-

tions of the Sumudu transform and applications to integral

production equations. Math Probl Eng 2003;3:103–18.

[41] Kilcman A, Eltayeb H. A note on integral transforms and partial

differential equations. Appl Math Sci 2010;4:109–18.

[42] Eltayeb H, Kılıcman A. On some applications of a new integral

transform. Int J Math Anal 2010;4:123–32.

[43] Darzi R, Mohammadzade B, Mousavi S, Beheshti R. Sumudu

transform method for solving fractional differential equations and

fractional diffusion-wave equation. J Math Comput Sci

2013;6:79–84.

[44] Chaurasia VBL, Singh J. Application of Sumudu transform in

fractional kinetic equations. Gen Math Notes 2011;2:86–95.

[45] Katatbeh D, Belgacem FBM. Applications of the Sumudu

transform to fractional differential equations. Nonlin Stud

2011;18:99–112.

[46] Asiru MA. Sumudu transform and the solution of integral

equations of convolution type. Int J Math Educ Sci Tech

2001;32:906–10.

[47] Zhang J. A Sumudu based algorithm for solving differential

equations. Acad Sci Moldova 2007;15:303–13.

[48] Kilicman A, Eltayeb H, Atan KAM. A note on the comparison

between Laplace and Sumudu transforms. Iran Math Soc

2011;37:131–41.

[49] Rathore S, Kumar D, Singh J, Gupta S. Homotopy analysis

Sumudu transform method for nonlinear equations. Int J Ind

Math 2012;4:301–14.

[50] Kumar D, Singh J, Rathore S. Sumudu decomposition method

for nonlinear equations. Int Math Forum 2012;7:515–21.

[51] Abedl-Rady S, Rida SZ, Arafa AAM, Abedl-Rahim HR. Vari-

ational iteration Sumudu transform method for solving fractional

nonlinear gas dynamics equation. Int J Res Stud Sci Eng Tech

2014;1:82–90.

[52] Atangana A, Kilicman A. The use of Sumudu transform for

solving certain nonlinear fractional heat-like equations. Abstr

Appl Anal 2013;2013:12.

[53] Singh J, Kumar D, Kilicman A. Application of homotopy

perturbation Sumudu transform method for solving heat and

wave-like equations. Malays J Math Sci 2013;7:79–95.

[54] Singh J, Kumar D, Kilicman A. Homotopy perturbation method

for fractional gas dynamics equation using Sumudu transform.

Abstr Appl Anal 2013;2013:8.

[55] Sincovec RF, Madsen NK. Software for non-linear partial

differential equations. ACM Trans Math Softw 1975.

[56] Caputo M. Elasticita e Dissipazione. Zani-Chelli Bologna; 1969.

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

[57] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications

of fractional differential equations. Amsterdam: Elsevier; 2006.

[58] Mittag-Leffler GM. Sur law nouvelle function. C R Acad Sci Paris

(Ser. II) 1903;137:554–8.

[59] Qin YM, Zeng DQ. Homotopy perturbation method for the q-

diffusion equation with a source term. Commun Frac Calc

2012;3:34–7.

[60] Yildrim A. The homotopy perturbation method for approximate

solution of the modified KdV equation. Z Naturforch A, A J Phys

Sci 2008;63:621–6.

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