Physics Letters A 374 (2010) 2921–2924

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Physics Letters A

www.elsevier.com/locate/pla

Exact solutions of variable coefficient nonlinear diffusion–reaction equationswith a nonlinear convective term

Ajay Mishra, Ranjit Kumar ∗

Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 December 2009Received in revised form 5 March 2010Accepted 17 March 2010Available online 21 March 2010Communicated by A.R. Bishop

Keywords:Nonlinear diffusion–reaction equationAuxiliary equation methodSolitary wave solutions

Attempts have been made to look for the exact solutions of certain types of nonlinear diffusion–reactionequations which involve not only the quadratic and quartic nonlinearities but also a time-dependentnonlinear convective flux term. In particular, the solitary wave solutions are found. Such equations arisein a variety of contexts in physical and biological problems.

© 2010 Published by Elsevier B.V.

1. Introduction

Many phenomena in physics, chemistry, biology, ecology andsocial science have been modeled in recent times in terms ofa variety of nonlinear partial differential equations (NLPDEs). Asmathematical models of the phenomena, the investigation of exactsolutions of NLPDEs will help one to understand the underlyingmechanism that governs these phenomena or to provide betterknowledge of its physical content and possible applications. In par-ticular, the diffusion process and the associated phenomenon issuch a common concept that it occurs in a variety of disciplinesand in different contexts in the same discipline. A large numberof applications of nonlinear diffusion–reaction (DR) equations havealready been known in the literature [1].

On the other hand, as per demand of several physical situationsit has become of considerable interest [2–11] to find the exactsolutions of a nonlinear partial differential equation when the pa-rameters depend explicitly on time. In particular, time-dependentnonlinear DR equations with polynomial nonlinearity in their anal-ogous forms have turned out to be of great interest in recent years[11,12]. Time dependence in convection v arises in the situationwhere wind affects the population near a hot spot of favorablegrowth rates (an oasis) surrounded by a less favorable desert re-gion [12]. An exact solution of these equations, if become available,will further add to their scope of applications in various studies.

The purpose of this Letter is to find the exact solutions of cer-tain types of nonlinear DR equations which involve quadratic and

* Corresponding author. Tel.: +91 11 27666796; fax: +91 11 27667061.E-mail address: ranjit@srb.org.in (R. Kumar).

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quartic nonlinearities with a nonlinear time-dependent ‘convectiveflux term’ using auxiliary equation method [2]. In particular, weinvestigate the exact solutions of the nonlinear DR equations,

Ct + k(t)CCx = DCxx + αC − βC2, (1)

and

Ct + k(t)C2Cx = DCxx + αC − βC4, (2)

where C = C(x, t), is the concentration or the density variable de-pending on the phenomenon under study; D is the diffusion co-efficient, and k,α,β are real constants. Eqs. (1) and (2) describea transport phenomenon in which both diffusion and convectionprocesses are of equal importance, i.e., the nonlinear diffusioncould be thought of as equivalent to the nonlinear convection ef-fects. In particular, the second term on the left-hand side, kCCx (orkC2Cx) is the replacement of the conventional vCx-term [1]. Eq. (1)has been studied [1] in a variety of situations like in the studies ofion-exchange columns, population dynamics of insects, chromatog-raphy, etc., but Eq. (2), which we are investigating here perhaps forthe first time, could also be a potential candidate from this pointof view for such studies.

2. Exact solutions of Eq. (1)

2.1. Time-independent k

Based on the method of auxiliary equation, we first transformthe partial differential equation in to a total differential equationby defining a variable ξ = x− wt . Using this transformation, Eq. (1)can be expressed as

2922 A. Mishra, R. Kumar / Physics Letters A 374 (2010) 2921–2924

DC ′′ + wC ′ − kCC ′ + αC − βC2 = 0. (3)

For the solution of Eq. (3) we make an ansatz [13]

C(ξ) =l∑

i=0

ai zi(ξ), (4)

where ai are all real constants to be determined, l is a positiveinteger which can be determined by balancing the highest orderderivative term with the highest order nonlinear term in theseequations, and z(ξ) satisfies the following auxiliary ordinary dif-ferential equation [13]

dz

dξ= b + z2(ξ), (5)

where b will be determined later. Eq. (5) has the following generalsolutions:

(i) If b < 0, then

z(ξ) = −√

−b tanh(√

−bξ), or

z(ξ) = −√

−b coth(√

−bξ).

(ii) If b > 0, then

z(ξ) = √b tan(

√bξ), or

z(ξ) = −√bcot(

√bξ).

(iii) If b = 0, then

z(ξ) = −1

ξ.

Using the balancing procedure we get l = 1 for Eq. (3). Thissuggests the choice of C(ξ) in Eq. (4) as

C(ξ) = a0 + a1z(ξ). (6)

Substituting (6) along with (5) in Eq. (3) and then setting thecoefficients of z j(ξ) ( j = 0,1, . . . ,3), to zero in the resultant ex-pression, one obtains a set of algebraic equations involving a0, a1and w as

−ka21 + 2Da1 = 0,

wa1 − ka0a1 − βa21 = 0,

−ka21b + 2Da1b + αa1 − 2βa0a1 = 0,

wa1b − ka0a1b − βa20 + αa0 = 0, (7)

which can be solved for the four unknowns a0, a1, b and w to give

a0 = α

2β, a1 = 2D

k,

b = −α2k2

16β2 D2, w = k2α + 4Dβ2

2βk, (8)

and finally, the solution C(ξ) of Eq. (3) turns out to be

C(ξ) = α

2β

[1 − tanh

(αk

4βDξ

)], (9)

which is a solitary wave solution of Eq. (3). Note that the ampli-tude of C(ξ) is independent of both the parameters D and k butdecreases with nonlinear parameter β . On the other hand its veloc-ity w , depends on both the parameters D and k. Thus, the solutionobtained here can be used to explain the population dynamics ofsome insects [1] and small rodents [14] as well as variety of phe-nomena like ion-exchange columns, chromatography, etc., [1].

2.2. Time-dependent k

For time-dependent k, we define the variable ξ = p(t)x +q(t) [2]. For the solution of Eq. (1) we make an ansatz C(ξ) =∑l

i=0 ai(t)zi(ξ), where all the coefficients ai in Eq. (4) now be-comes time-dependent. Using the balancing procedure we getl = 1. This suggests the choice of C(ξ) as

C(ξ) = a0(t) + a1(t)z(ξ). (10)

Substituting (10) along with (5) in to Eq. (1) and then setting thecoefficients of z j(ξ) ( j = 0,1, . . . ,3), to zero in the resultant ex-pression, one obtains a set of equations involving a0, a1, p and qas

ka21 p − 2Da1 p2 = 0,

a1 pt x + a1qt + ka0a1 p + βa21 = 0,

a1t + ka21 pb + 2βa0a1 − αa1 − 2Da1 p2b = 0,

a0t + a1bpt x + a1bqt + ka0a1 pb − αa0 + βa20 = 0, (11)

which is a set of coupled nonlinear ordinary differential equations.To solve Eqs. (11), we will use the symbolic computation techniqueof Mathematica. In general it is not possible to solve Eqs. (11) forall parametric values. To solve these equations we impose the extracondition i.e., pt x + qt + ka0 p = −αa0/a1b. Using this condition,Eqs. (11) can be solved for the unknowns a0(t), a1(t), p(t), q(t)and b to give

a0(t) = 2αe2αt

e2αc1 + βe2αt, (12)

a1(t) = ± αeαt√e2α2c2 + 2β2be2αt

, (13)

p(t) = k(t)a1(t)

2D; b = − 2

β

[e2α2c2

3e2αtβ − e2αc1

], (14)

q(t) = −∫

αa0(t)

ba1(t)dt −

∫xpt(t)dt −

∫k(t)a0(t)p(t)dt, (15)

where c1 and c2 are constants of integration. Finally, the solutionC(ξ) of Eq. (3) turns out to be

C(ξ) = a0(t) − a1(t)√

−b tanh(√

−bξ)

= 2αe2αt

e2αc1 + βe2αt

∓ αeαt√e2α2c2 + 2β2be2αt

√2

β

(e2α2c2

3e2αtβ − e2αc1

)

× tanh

(√2

β

(e2α2c2

3e2αtβ − e2αc1

)ξ

). (16)

One can see from above equations that amplitude as well as veloc-ity of the wave for the time-dependent k becomes time-dependent.Also, the parametric dependence of the final solution for both thecases are similar, e.g., compare Eqs. (9) and (16).

3. Exact solutions of Eq. (2)

3.1. Time-independent k

By using the transformation ξ = x − wt , we write Eq. (2) as

DC ′′ + wC ′ − kC2C ′ + αC − βC4 = 0. (17)

A. Mishra, R. Kumar / Physics Letters A 374 (2010) 2921–2924 2923

Again, using the balancing procedure we get, l = 1. SubstitutingEqs. (5) and (6) in Eq. (17) we get a set of algebraic equations

−βa41 − ka3

1 = 0,

2Da1 − 2ka0a21 − 4βa0a3

1 = 0,

wa1 − ka20a1 − ka3

1b − 6βa20a2

1 = 0,

2Da1b + αa1 − 2ka0a21b − 4βa3

0a1 = 0,

wa1b − ka20a1b + αa0 − βa4

0 = 0. (18)

After solving set of algebraic equations, one obtains

a0 = βD

k2, a1 = −k

β,

w = −k6α + 16β4 D3

4β2 Dk3, b = β4 D2

4k6− α

16D, (19)

along with a constraining relation, α = ± 8β4 D3

k6 , among the con-stant coefficients in (17). Using the above constraining relation weget two values of b. Finally the solution C(ξ) of Eq. (17) becomes

C(ξ) = βD

k2

[1 + 1

2tanh

(β2 D

2k3ξ

)], (20)

for α = 8β4 D3

k6 , and

C(ξ) = βD

k2

[1 −

√3

4tan

(√3β2 D

2k3ξ

)], (21)

for α = − 8β4 D3

k6 . Note that Eq. (20) is a solitary wave solutionof Eq. (17) while solution (21) is physically not acceptable. Inthis case, amplitude as well as velocity of the wave depends onnonlinear parameter β , diffusion coefficient D as well as density-dependent diffusion coefficient k. Here, amplitude and velocity ofthe wave increases with D and β but, decreases with k.

3.2. Time-dependent k

In this case by using the balancing procedure one obtains l = 1.This suggests the choice of C(ξ) in the form of Eq. (10). Afterdefining the variable ξ = p(t)x + q(t) and using the above pro-cedure as for Eq. (1), one obtains a set of coupled nonlinear differ-ential equations as

βa41 + ka3

1 p = 0,

2ka0a21 p − 2Da1 p2 + 4βa0a3

1 = 0,

a1qt + a1 pt x + ka20a1 p + ka3

1 pb + 6βa20a2

1 = 0,

a1t + 2ka0a21 pb − 2Da1 p2b − αa1 + 4βa3

0a1 = 0,

a0t + a1bqt + a1bpt x + ka20a1 pb − αa0 + βa4

0 = 0. (22)

Again to solve the set of Eqs. (22) we have to impose the condi-tion qt + pt x + ka2

0 p = −αa0/a1b. After solving the set of couplednonlinear differential equations, one obtains

a0(t) = 2α)1/3e2αt

(e6αc1 + βe6αt)1/3, (23)

a1(t) = − e∫ t

t′αk(t′)6−4bD3β4

k(t′)6 dt′√∫ tt′′

8β2bDe2∫ t′′t′

αk(t′)6−4bD3β4

k(t′)6 dt′

k(t′′)2 dt′′ − c2

, (24)

p(t) = −βa1(t), b = 6D2β4 ± √

36β8 D4 − 4k6β4 Dα2 6

, (25)

k(t) 2p k

q(t) = −∫

αa0(t)

ba1(t)dt −

∫xpt(t)dt −

∫k(t)

(a0(t)

)2p(t)dt,

(26)

where c1 and c2 are constants of integration. Finally, the solutionC(ξ) of Eq. (2) becomes

C(ξ) = (2α)1/3e2αt

(e6αc1 + βe6αt)1/3

+ e∫ t

t′αk(t′)6−4bD3β4

k(t′)6 dt′√∫ tt′′

8β2bDe2∫ t′′t′

αk(t′)6−4bD3β4

k(t′)6 dt′

k(t′′)2 dt′′ − c2

×√

−6D2β4 ± √36β8 D4 − 4k6β4 Dα

2p2k6

× tanh

(√−6D2β4 ± √

36β8 D4 − 4k6β4 Dα

2p2k6ξ

), (27)

for b < 0, and

C(ξ) = (2α)1/3e2αt

(e6αc1 + βe6αt)1/3

− e∫ t

t′αk(t′)6−4bD3β4

k(t′)6 dt′√∫ tt′′

8β2bDe2∫ t′′t′

αk(t′)6−4bD3β4

k(t′)6 dt′

k(t′′)2 dt′′ − c2

×√

6D2β4 ± √36β8 D4 − 4k6β4 Dα

2p2k6

× tan

(√6D2β4 ± √

36β8 D4 − 4k6β4 Dα

2p2k6ξ

), (28)

for b > 0. From Eq. (25) one can see that b is real only for

α <9β4 D3

k6 . Again, one can see from above equations that the am-plitude as well as velocity of the corresponding wave becomestime-dependent.

4. Concluding remarks

With a view to extending the scope of applications of nonlinearDR equations with quadratic and quartic nonlinearities, the roleof certain types of nonlinear convective terms in these equationsis investigated. Further, the presence of time-dependent nonlinearconvection term k(t) in the DR equations lead to an interestingeffect on their solutions.

The existence of the kink and antikink shaped soliton solutionsis demonstrated in certain parametric domain. Certain observationsfrom the solution (9) of Eq. (3) and solutions (20) and (21) ofEq. (17) are in order: (i) It can be seen that amplitude of C(ξ)

in Eqs. (20) or (21) decreases with k—a measure of contribution ofnonlinear convective flux term in Eqs. (3) and (17). On the otherhand, amplitude of C(ξ) remains unaffected with respect to k incase of solution (9) of Eq. (3). The wave velocity w in both thecases depends on k (see Eqs. (8) and (19)). In many biological andphysical systems, dispersal is dominated by density-dependent dif-fusion coefficient k [1]. Thus, the solutions obtained here can beused to explain such biological and physical phenomena. Similarly,the parameter responsible for the nonlinearity in Eqs. (3) and (17)plays just opposite roles in these solutions. (ii) For time-dependentnonlinear convective flux term k, the amplitude as well as velocityof the resulting solution becomes time-dependent (see Eqs. (16),(27) and (28)).

2924 A. Mishra, R. Kumar / Physics Letters A 374 (2010) 2921–2924

In view of the fact that NL DR equations are used in ex-plaining a variety of physical phenomenon and the choice ofnonlinearity in these equations is a part of the modeling pro-cess of the phenomenon under study, it is expected that the re-sults obtained in this work can offer some clue in making suchchoices.

Acknowledgements

One of the authors (AM) would like to thank CSIR, New Delhi,Gov. of India for Junior Research Fellowship during the course ofthis work. We would also like to thank the referee for many usefulsuggestions that help us to improve this Letter.

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