analytical solutions of convection diffusion problems by combining laplace transform method and homotopy perturbation method

ORIGINAL ARTICLEAnalytical solutions of convection–diffusion problems by combining Laplace transform method and homotopy perturbation method Sumit Gupta a,*, Devendra Kumar b, Jagdev Sing

ORIGINAL ARTICLE

Analytical solutions of convection–diffusion

problems by combining Laplace transform method

and homotopy perturbation method

Sumit Gupta a,*, Devendra Kumar b, Jagdev Singh c

a

Department of Mathematics, Jagan Nath Gupta Institute of Engineering and Technology, Jaipur 302022, Rajasthan, India

b

Department of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India

c

Department of Mathematics, Jagan Nath University, Jaipur 303901, Rajasthan, India

Received 25 February 2014; revised 8 May 2015; accepted 12 May 2015

Available online 9 June 2015

KEYWORDS

Homotopy perturbation

method;

Laplace transform method;

Linear and nonlinear

con-vection–diffusion problems;

He’s polynomials

Abstract The aim of this paper was to present a user friendly numerical algorithm based on homo-topy perturbation transform method for solving various linear and nonlinear convection-diffusion problems arising in physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection The homotopy perturbation transform method is a combined form of the homotopy perturbation method and Laplace transform method The nonlinear terms can be easily obtained by the use of He’s polyno-mials The technique presents an accurate methodology to solve many types of partial differential equations The approximate solutions obtained by proposed scheme in a wide range of the problem’s domain were compared with those results obtained from the actual solutions The comparison shows a precise agreement between the results

ª 2015 Faculty of Engineering, Alexandria 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

The convection–diffusion equation is a combination of the

dif-fusion and convection equations, and describes physical

phe-nomena where particles, energy, or other physical quantities

are transferred inside a physical system due to two processes: diffusion and convection In general form the convection–dif-fusion equation is given as follows:

@u

where u is the variable of interest, D is the diffusivity, such as mass diffusivity for particle motion or thermal diffusivity for heat transport, ~t is the average velocity that the quantity is moving For example, in advection, u might be the concentra-tion of salt in a river, and then ~t would be the velocity of the water flow As another example, u might be the concentration

* Corresponding author Tel.: +91 9929764461.

E-mail addresses: guptasumit.edu@gmail.com (S Gupta), devendra.

maths@gmail.com (D Kumar), jagdevsinghrathore@gmail.com

(J Singh).

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

H O S T E D BY

Alexandria University Alexandria Engineering Journal

www.elsevier.com/locate/aej www.sciencedirect.com

http://dx.doi.org/10.1016/j.aej.2015.05.004

1110-0168 ª 2015 Faculty of Engineering, Alexandria 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/ ).

of small bubbles in a calm lake, and then ~t would be the

aver-age velocity of bubbles rising towards the surface by buoyancy

Rdescribes ‘‘sources’’ or ‘‘sinks’’ of the quantity u For

exam-ple, for a chemical species, R > 0 means that a chemical

reac-tion is creating more of the species, and R < 0 means that a

chemical reaction is destroying the species For heat transport,

R> 0 might occur if thermal energy is being generated by

fric-tion r represents gradient and r represents divergence

Previously various methods have been used to handle these

problems such as variational iteration method (VIM) [1],

Adomian’s decomposition method (ADM)[2], homotopy

per-turbation method (HPM) [3] and Bessel collocation method

[4] Most of these methods have their inbuilt deficiencies like

the calculation of Adomian’s polynomials, the Lagrange

mul-tiplier, divergent results and huge computational work In

recent years numerical methods have also been applied various

physical problems such as Volterra’s population growth model

with fractional order [5], nonlinear Lane-Emden type

equa-tions[6], Hantavirus infection model[7], continuous

popula-tion models for single and interacting species [8] The

homotopy perturbation method (HPM), first proposed by

He[9–15] for solving various linear and nonlinear initial and

boundary value problems The HPM was also studied by many

authors to handle nonlinear equations arising in science and

engineering[16–23] The Laplace transform is totally incapable

of handling nonlinear equations because of the difficulties that

are caused by the nonlinear terms Various ways have been

proposed recently to deal such nonlinearities such as the

Laplace decomposition algorithm[24–27] and the homotopy

perturbation transform method (HPTM)[28–30] to produce

highly effective techniques for solving many nonlinear

problems

The basic motivation of this paper is to apply an effective

modification of HPM to overcome the deficiency We

imple-ment the homotopy perturbation transform method (HPTM)

for solving the convection-diffusion equations Using this

method, all conditions can be satisfied Also very accurate

results are obtained in a wide range via one or two iteration

steps The suggested HPTM provides the solution in a rapid

convergent series which may lead the solution in closed form

The use of He’s polynomials in nonlinear terms first proposed

by Ghorbani[31, 32] Several examples are given to verify the

reliability and efficiency of the HPTM

2 Analysis of the method

The HPTM is a combined form of the HPM and Laplace

transform method We apply the HPTM to the following

gen-eral nonlinear partial differential equation with the initial

con-ditions of the form

where D is the second order linear differential operator

D¼ @2

=@t2, R is the linear differential operator of less order

than D; N represents the general nonlinear differential

operator and gðx; tÞ is the source term Taking the Laplace

transform (denoted in this paper by L) on both sides of

Eq.(2):

L½D uðx; tÞ þ L½R uðx; tÞ þ L½N uðx; tÞ ¼ L ½gðx; tÞ: ð4Þ

Using the differentiation property of the Laplace transform,

we have

L½uðx; tÞ ¼hðxÞ

s þfðxÞ

s2 1

s2L½Ruðx; tÞ þ1

s2L gðx; tÞ½ 

1

Operating with the Laplace inverse on both sides of Eq.(5)

gives uðx; tÞ ¼ Gðx; tÞ  L1 1

s2L½Ruðx; tÞ þ Nuðx; tÞ

where Gðx; tÞ represents the term arising from the source term and the prescribed initial conditions Now we apply the HPM uðx; tÞ ¼X1

n¼0

and the nonlinear term can be decomposed as

N uðx; tÞ ¼X1

n¼0

for some He’s polynomials HnðuÞ that are given by

Hnðu0; u1; ; unÞ ¼1

n!

@n

@pn N X1

i¼0

piui

!

p¼0

; n¼ 0;1; 2; 3

ð9Þ Substituting Eqs.(7) and (8)in Eq.(6), we get

X1 n¼0

pnunðx;tÞ ¼ Gðx;tÞ

 p L1 1

s2L RX1 n¼0

pnunðx;tÞ þX1

n¼0

pnHnðuÞ

:

ð10Þ

which is the coupling of the Laplace transform and the HPM using He’s polynomials Comparing the coefficient of like pow-ers of p, the following approximations are obtained

p0: u0ðx; tÞ ¼ Gðx; tÞ

p1: u1ðx; tÞ ¼ L1 1

s 2L½Ru0ðx; tÞ þ H0ðuÞ

;

p2: u2ðx; tÞ ¼ L1 1

s 2L½Ru1ðx; tÞ þ H1ðuÞ

;

p3: u3ðx; tÞ ¼ L1 1

s 2L½Ru2ðx; tÞ þ H2ðuÞ

; ;

ð11Þ

and so on

3 Numerical examples and error estimation

In this section, we discuss the implementation of our numerical method and investigate its accuracy and stability by applying it

to numerical examples on the convection-diffusion equations Example 3.1 Let us consider the following diffusion-convection problem

@u

@t¼@

2

u

with the initial condition uðx; 0Þ ¼ x þ ex

Taking Laplace transform on both the sides, subject to the

initial condition, we get

L½uðx; tÞ ¼xþ e

x

Taking inverse Laplace transform, we get

uðx; tÞ ¼ ðx þ exÞ þ pL1 1

sL½uxx u

By homotopy perturbation method, we have

uðx; tÞ ¼X1

n¼0

Using(15)in(14), we have

X1

n¼0

pnunðx;tÞ ¼ ðx þ exÞ

þ pL1 1

s L

X1 n¼0

pnunðx; tÞ

!

xx

 L X1 n¼0

pnunðx; tÞ

!

:

ð16Þ

Comparing the coefficients of various powers of p, we get

p0: u0ðx; tÞ ¼ x þ ex;

p1: u1ðx; tÞ ¼ x t;

p2: u2ðx; tÞ ¼ xt 2

2!;

p3: u3ðx; tÞ ¼ xt 3

3!;

ð17Þ

and so on Therefore the series solution is given by uðx; tÞ ¼ exþ x 1  t þt

2

2!t

3

3!þ

uðx; tÞ ¼ exþ xet The numerical results of uðx; tÞ for the approximate solution (18) obtained by using HPTM, the exact solution and the absolute error E7ðuÞ ¼ juex uappj for various values

of t and x are shown by Fig 1(a)–(c) It is observed from

Fig 1(a) and (b) that uðx; tÞ increases with the increase in x and decrease in t Fig 1(a)–(c) clearly show that the

Figure 1 The surface shows the solution uðx; tÞ for Eq.(12): (a) exact solution; (b) approximate solution(18); (c)juex uappj

approximate solution(18) obtained by the present method is

very near to the exact solution It is to be noted that only the

seventh order term of the HPTM was used in evaluating the

approximate solutions for Fig 1 It is evident that the

efficiency of the present method can be dramatically enhanced

by computing further terms of uðx; tÞ when the HPTM is used

Example 3.2 Let us consider the following

diffusion-convection problem

@u

@t¼@

2u

with the initial condition uðx; 0Þ ¼ 1

10ecos x11

By applying aforesaid method subject to the initial

condi-tion, we have

X1

n¼0

pnunðx; tÞ ¼1

10e

cos x11þ pL1 1

sL

X1 n¼0

pnunðx; tÞ

!

xx

"

þ1

sð1 þ cosx  sin2xÞL X

pnunðx;y;tÞ

: ð20Þ

Comparing the coefficients of various powers of p, we get

p0: u0ðx; tÞ ¼ 1

10ecos x11;

p1: u1ðx; tÞ ¼ 1

10ecos x11ðtÞ;

p2: u2ðx; tÞ ¼ 1

10ecos x11 t 2

2!



;

p3: u3ðx; tÞ ¼ 1

10ecos x11 t 3

3!

;

ð21Þ

and so on

Therefore the approximate solution is given by

uðx; tÞ ¼ 1

10e

cos x11 1 t þt

2

2!t

3

3!þ

uðx; tÞ ¼ 1

10ecos x11t The numerical results of uðx; tÞ for the approximate solution

(22)obtained with the help of HPTM, the exact solution and the absolute error E7ðuÞ ¼ juex uappj for various values of t and x are shown by Fig 2(a)–(c) From Fig 2(a)–(c), we

Figure 2 The surface shows the solution uðx; tÞ for Eq.(19): (a) exact solution; (b) approximate solution(22); (c)juex uappj

observed that the approximate solution(22) obtained by the

proposed method is very near to the exact solution

Example 3.3 Let us consider the following

diffusion-convection problem

@u

@t¼@

2

u

@x21

with the initial condition uðx; 0Þ ¼1

2xþ ex=2

By applying aforesaid method subject to the initial

condi-tion, we have

X1

n¼0

pnunðx; tÞ ¼ x

2þ ex=2

þ pL1 1

sL

X1 n¼0

pnunðx; tÞ

!

xx

"

4sL

X1 n¼0

pnunðx; tÞ

!#

p, weget

p0: u0ðx; tÞ ¼x

2þ ex=2;

p1: u1ðx; tÞ ¼x

2 t 4

;

p2: u2ðx; tÞ ¼x

2 ðt=4Þ 2

2! ;

p3: u3ðx; tÞ ¼x

2 ðt=4Þ 3

3! ;

ð25Þ

and so on

Therefore the series solution is given by

uðx; tÞ ¼ ex=2þx

2 1t

4þðt=4Þ

2

2! ðt=4Þ

3

3! þ

!

uðx; tÞ ¼ ex=2þx

2et=4 The numerical results of uðx; tÞ for the approximate solution(26)derived with the application of HPTM, the exact solution and the absolute error E7ðuÞ ¼ juex uappj for various values of t and x are described by Fig 3(a)–(c) It is to be noted from Fig 3(a) and (b) that uðx; tÞ increases with the

Figure 3 The surface shows the solution uðx; tÞ for Eq.(23): (a) exact solution; (b) approximate solution(26); (c)juex uappj

increase in x and decrease in t.Fig 3(a)–(c) clearly show that

the approximate solution (26) obtained by the present

approach is very near to the exact solution

Example 3.4 Let us consider the following nonlinear

diffusion-convection problem

@u

@t¼@

2

u

@x2@u

with the initial equation uðx; 0Þ ¼ ex

By applying aforesaid method subject to the initial

condi-tion, we have

X1

n¼0

pnunðx; tÞ ¼ ex

þ pL1 1

s



L X1 n¼0

pnunðx; tÞ

!

xx

 L X1 n¼0

pnunðx; tÞ

!

x

 L X1

n¼0

pnunðx; tÞ

!

X1 n¼0

pnunðx; tÞ

!

x

 L X1

n¼0

pnunðx;tÞ

!2

þ L X1 n¼0

pnunðx; tÞ

!3 5: ð28Þ

Comparing the coefficients of various powers of p, we get

p0: u0ðx; tÞ ¼ ex;

p1: u1ðx; tÞ ¼ ext;

p2: u2ðx; tÞ ¼ ex t 2

2!;

p3: u3ðx; tÞ ¼ ex t 3

3!;

ð29Þ

and so on

Therefore the series solution is given as

uðx; tÞ ¼ ex 1þ t þt

2

2!þt

3

3!þ

uðx; tÞ ¼ exþt The numerical results of uðx; tÞ for the approximate solution(30)find by applying HPTM, the exact solution and the absolute error E7ðuÞ ¼ juex uappj for various values of t and x are depicted by Fig 4(a)–(c) It is to be noted from

Figure 4 The surface shows the solution uðx; tÞ for Eq.(27): (a) exact solution; (b) approximate solution(30); (c)juex uappj

Fig 4(a) and (b) that uðx; tÞ increases with the increase in both

xand t FromFig 4(a)–(c), we can see that the approximate

solution(26)obtained by the present scheme is very near to the

exact solution

4 Conclusions

In this paper, we have applied the homotopy perturbation

transform method (HPTM) for solving convection-diffusion

equations The proposed method is applied without using

lin-earization, discretization or restrictive assumptions It may be

concluding that the HPTM using He’s polynomials is very

powerful and efficient in finding the analytic solutions for a

wide class of problems The solution procedure using He’s

polynomials is simple, but the calculation of Adomian’s

poly-nomials is complex The method gives more realistic series

solutions that converge very rapidly in physical problems

The negligible relative errors have been observed even with just

the first two terms of the HPTM solution, which indicate that

the HPTM needs much less computational work compared

with the other semi-analytical methods such as HPM, VIM

and ADM The fact that HPTM solves nonlinear problems

without using Adomian’s polynomials is a clear advantage of

this algorithm over the decomposition method and might find

wide applications

Acknowledgements

The authors are extending their heartfelt thanks to the

reviewer for his valuable suggestions for the improvement of

the article

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