Physics Letters A 376 (2012) 1833–1835

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Physics Letters A

www.elsevier.com/locate/pla

Memory effects in Fisher equation with nonlinear convection term

Ajay Mishra a, Ranjit Kumar b,∗a Department of Physics and Astrophysics, University of Delhi, Delhi 110007, Indiab Department of Physics, Dyal Singh College, University of Delhi, Delhi 110003, India

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

Article history:Received 1 January 2012Received in revised form 17 March 2012Accepted 17 April 2012Available online 20 April 2012Communicated by A.R. Bishop

Keywords:Fisher equationFinite memory effectAuxiliary equation methodSolitary wave solution

Memory effect in diffusion–reaction equation plays important role in physical, biological and chemicalsciences. An exact solutions of Fisher equation in the presence of nonlinear convection term with finitememory transport is obtained. Solutions of corresponding diffusion–reaction equation without memoryeffect is also obtained and a comparison is made between obtained solutions. In particular, the solitarywave solutions are found.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Nonlinear diffusion–reaction (DR) equation plays important rolein physical, biological, chemical and social sciences [1]. In partic-ular, DR equation with finite memory transport has played impor-tant role in recent years [2–15]. Nonlinear DR equation with finitememory have been applied to the population growth models [7–9],forest fire [10], neolithic transitions [11] and in several other areas[12–15]. When memory effect is taken into account then parabolicDR equation get converted into hyperbolic DR equation. On theother hand nonlinear convection term in DR equation also becomesimportant in certain biological processes [1,16]. An exact solutionof these equations, if become available, will further add to theirscope of applications in various studies. The purpose of this Letteris to find the exact solutions of Fisher equation in the presence ofnonlinear convection term with finite memory transport.

Diffusion equation without finite memory is derived from thecontinuity equation, ∂u(x,t)

∂t = − ∂ J (x,t)∂x , where u(x, t) is the con-

centration of the particle and J (x, t) is the flux of the diffusingparticle. When combined with the Fick’s law, which states that theflux of the diffusing particle in any part of the system is propor-tional to the density gradient: J (x, t) = −D ∂u(x,t)

∂x , one obtain the

one dimensional diffusion equation ∂u(x,t)∂t = D ∂2u(x,t)

∂x2 . Here D isthe diffusion coefficient. Memory effect in diffusion equation ariseswhen dispersal of the particle is mutually not independent [2–5].When memory effect is taken into account then we have the fol-

* Corresponding author. Tel.: +91 11 27666796; fax: +91 11 27667061.E-mail address: du.ranjit@gmail.com (R. Kumar).

0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2012.04.037

lowing modification of Fick’s law [2] in the presence of nonlinearconvection term

J (x, t + τ ) = −D∂u

∂x+ vu2, (1)

where ∂u∂t is the time rate of change of concentration at time t and

J (x, t + τ ) is the flux of the particles at a later time t + τ . Here τis the delay time and v is the coefficient of nonlinear convectiveflux term. Now in the presence of source term f (u), conservationequation gets modified to

∂u

∂t= −∂ J

∂x+ f (u). (2)

By expanding J in Eq. (1) up to first order in τ , one obtain

J (x, t) + τ∂ J (x, t)

∂t= −D

∂u

∂x+ vu2. (3)

Now, differentiate Eq. (3) with respect to x, Eq. (2) with respect tot and eliminate J (x, t) from the resulting expression, we obtain

utt − βDuxx − f ′(u)ut + β(ut − f (u)

) + kβuux = 0. (4)

Here β ≡ 1τ , k ≡ 2v and f ′(u) = df

du . Note that Eq. (4) reduces toBurgers equation [5] in the absence of source term ( f (u) = 0). Inparticular we are considering f (u) = αu − γ u2 and for this formof f (u) we get f ′(u) = α − 2γ u. For this particular choice of f (u),Eq. (4) can also be interperated as the Burgers–Fisher equation.Note that Eq. (4) describe a transport phenomenon in which bothdiffusion and convection processes are of equal importance. Afterusing this form of f (u) and f ′(u) in Eq. (4) and using the trans-formation ξ = x − wt , we get

1834 A. Mishra, R. Kumar / Physics Letters A 376 (2012) 1833–1835

(w2 − βD

)u′′ + w(α − β)u′ + (kβ − 2γ w)uu′ − βαu + βγ u2

= 0. (5)

By taking τ = 1/β = 0 in Eqs. (4) and (5) one obtain the corre-sponding DR equation without finite memory transport

ut + kuux = Duxx + f (u), (6)

and

Du′′ + wu′ − kuu′ + αu − γ u2 = 0. (7)

One can see from above that Eq. (4) is a hyperbolic DR equationwhile Eq. (6) is a parabolic DR equation. Again, note that in theabsence of source term ( f (u) = 0), Eq. (6) reduces to Burgers equa-tion. In next section we will discuss auxiliary equation method tofind the solutions of above nonlinear DR equations.

2. Method

For the solutions of (5) and (7) we make an ansatz [17]

u(ξ) =l∑

i=0

ai zi(ξ), (8)

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 [17]

dz

dξ= b + z2(ξ), (9)

where b will be determined later. Eq. (9) has the following generalsolutions:

(i) If b < 0, then

z(ξ) = −√

−b tanh(√

−bξ), or

z(ξ) = −√

−b coth(√

−bξ). (10)

(ii) If b > 0, then

z(ξ) = √b tan(

√bξ), or z(ξ) = −√

b cot(√

bξ). (11)

(iii) If b = 0, then

z(ξ) = −1

ξ. (12)

3. Solutions of Eq. (5)

Using the balancing procedure i.e., by using the ansatz (8) inEq. (5) and balancing the highest order derivative term u′′ with thehighest order nonlinear term uu′ , one obtains l = 1. This suggeststhe choice of u(ξ) in Eq. (8) as

u(ξ) = a0 + a1z(ξ). (13)

Substituting (13) along with (9) in Eq. (5) 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, a1,b and w as

2a1(

w2 − βD) + (βk − 2γ w)a2

1 = 0, (14)

wa1(α − β) + a0a1(βk − 2γ w) + βγ a21 = 0, (15)

2a1b(

w2 −βD) + a2

1b(βk −2γ w)−βαa1 + 2a0a1βγ = 0, (16)

wa1b(α − β) + a0a1b(βk − 2γ w) − βαa0 + βγ a2 = 0, (17)

0

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

a0 = α

2γ, a1 = 2(βD − w2)

(βk − 2γ w),

w = β(αk2 + 4Dγ 2)

2γ k(α + β), b = α2

2a1(αk − 2γ w). (18)

From Eq. (18) one can see that b < 0 for αk − 2γ w < 0. Substitut-ing Eq. (18) in (13) and using Eq. (10) we get the following resultfor u(ξ)

u(ξ) = α

2γ

[1 − tanh

(γ kα(α + β)

(4Dβγ 2 − α2k2)(x − wt)

)], (19)

and

u(ξ) = α

2γ

[1 − coth

(γ kα(α + β)

(4Dβγ 2 − α2k2)(x − wt)

)]. (20)

Eq. (19) is a solitary wave solution of Eq. (5) whereas solution (20)diverge. Since u(ξ) represent the concentration of certain specieswhich cannot go to infinity hence solution (20) is physically notacceptable. When b > 0 then the corresponding solutions are inthe form of tan and cot (see Eq. (11)) both of which diverge andtherefore they are physically not acceptable.

4. Solutions of Eq. (7)

Solutions of Eq. (7) is already obtained by us in Ref. [16]. Forsake of completeness we rewrite here the solutions with slightchange in notation (parameter β in Ref. [16] is replaced by γ andC(ξ) by u(ξ) in the present work).

u(ξ) = α

2γ

[1 − tanh

(αk

4γ Dξ

)], (21)

and

u(ξ) = α

2γ

[1 − coth

(αk

4γ Dξ

)], (22)

and wave speed w as

w = αk2 + 4Dγ 2

2γ k. (23)

Eq. (21) is again a solitary wave solution of Eq. (7) while Eq. (22) isphysically not acceptable. Note that we can get solutions (21) and(22) by taking 1/β = 0 in Eqs. (18), (19) and (20).

In the case of Fisher equation with finite memory transport andwithout convection (k = 0), the wave velocity of the correspond-

ing nonlinear DR equation [5] is given by w =√

βD1+ 6

25 (y−1/y)2 with

y =√

βα (parameter k in Ref. [5] is represented by α in our work).

In the case of Fisher equation without convection (k = 0) and fi-nite memory transport, i.e. 1

β= 0, the wave velocity is given by

w = 5√

αD6 . From here one can see that wave speed w is indepen-

dent of nonlinear parameter γ . When nonlinear convection termis included in the DR equation then one can see from Eqs. (18)and (23) that wave speed w in both cases depends on nonlinearparameter γ as well as coefficient k of nonlinear convection term.

5. Concluding remarks

Variety of phenomena in physical, chemical and biological sci-ence are described by the interaction of diffusion and reaction or

A. Mishra, R. Kumar / Physics Letters A 376 (2012) 1833–1835 1835

by the interaction between convection and diffusion. In this Let-ter we have obtained the exact solutions of Burgers–Fisher equa-tion in which both convection as well as diffusion play importantrole.

The existence of the kink and antikink shaped soliton solutionsis demonstrated in certain parametric domain. Certain observa-tions from the solutions (19) and (20) of Eq. (5) and solutions(21) and (22) of Eq. (7) are in order: (i) It can be seen thatwave speed w of Eq. (5) depends on time delay τ = 1/β . FromEq. (18) one can see that wave speed w increases with β . In thelimit τ = 1/β = 0, one obtains the wave speed of correspond-ing DR equation without finite memory transport (see Eq. (23)).(ii) Wave speed in both cases depends on α, γ , D and k. Thuswave speed depends on all three factors i.e., reaction, diffusionand convection coefficients. In many biological and physical sys-tems, dispersal is dominated by both diffusion coefficient D aswell as convection coefficient k [1,16]. Thus, the solutions obtainedhere can be used to explain such biological and physical phenom-ena.

In view of the fact that nonlinear DR equations are used inexplaining 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

We would like to thank R.S. Kaushal and Awadhesh Prasad forhelpful discussions. A.M. would like to thank CSIR, New Delhi, Gov.of India for Senior Research Fellowship during the course of thiswork. We would like to thank the referee for many useful sugges-tions that help us to improve this Letter.

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