A Numerical Study of the Nonlinear Reaction-Diffusion Equation withDifferent Type of Absorbent Term by Homotopy Analysis MethodPraveen Kumar Gupta and Swati Verma

Department of Mathematics and Statistics, Center for Mathematical Sciences, BanasthaliUniversity, Banasthali – 304 022, India

Reprint requests to P. K. G. ; E-mail: praveen.rs.apm@itbhu.ac.in

Z. Naturforsch. 67a, 621 – 627 (2012) / DOI: 10.5560/ZNA.2012-0066Received April 17, 2012 / revised June 28, 2012 / published online September 19, 2012

In this paper, based on the homotopy analysis method (HAM), a new powerful algorithm is usedfor the solution of the nonlinear reaction-diffusion equation. The algorithm presents the procedure ofconstructing a set of base functions and gives the high-order deformation equation in a simple form.Different from all other analytic methods, it provides us with a simple way to adjust and control theconvergence region of the solution series by introducing an auxiliary parameter h. The solutions ofthe problem of presence and absence of absorbent term and external force for different particularcases are presented graphically.

Key words: Homotopy Analysis Method; Nonlinear Reaction-Diffusion Equation; PartialDifferential Equation; External Force; Reaction Term.

Mathematics Subject Classification 2000: 14F35, 35K57, 45K05

1. Introduction

The nonlinear reaction diffusion equation as well asa system of nonlinear reaction-diffusion equations arevery attractive in recent years, since they have practicalapplications in many fields of science and engineer-ing. The general form of the governing equations in-volves temporal and spatial derivatives of the field plusa source/sink or forcing term. In underwater acoustics,for example, the source term explains the effects ofwave refraction and spreading due to a discrepancy inthe ocean refractive index. In the prediction of shore-line evolution this term stands for the supply or re-moval of sediment along the shoreline through cross-shore movement.

In the early eighties, Rose [1, 2] discussed a non-linear advection-diffusion problem for flow in porousmedia, where he defined a continuous, piecewise lin-ear finite element Galerkin method and derived ratesof convergence based on assumed asymptotic rates ofdegeneracy. Magenes et al. [3] considered a class ofproblems including the Stephan problem in the porousmedium. During the calculation, they rest the strictequality by using the asymptotically correct Chernoffformulation. In 1988, Nochetto and Verdi [4] devel-

oped a parallel degenerate parabolic equation and de-scribed a continuous, piecewise linear finite elementGalerkin method and proved its convergence; in ad-dition, they removed error estimates in measure forthe free boundaries that emerge in the solutions. Inthis continuation, Barrett and Knabner [5] and Arbo-gastr et al. [6] considered the problem of solute trans-port. Here, they also defined a similar linear finite el-ement Galerkin method, and the authors used a regu-larization of the problem to obtain their results. Pao [7]has solved a reaction-diffusion equation with nonlinearboundary conditions.

In 1996, Tang et al. [8] developed the model forEl Nino events with a nonlinear convection-reaction-diffusion equation. In this model, the effects of con-vection, turbulent diffusion, linear feedback, and non-linear radiation on the anomaly of sea surface tempera-ture are considered. After some time, Wang [9] defineda new technique (monotone technique) for the diffu-sion equation with nonlinear diffusion coefficients andfind the solution.

In 2010, Yıldırım and Pınar [10] discussed the ap-plication of the exp-function method for solving thenonlinear reaction diffusion equation which is arisingin the mathematical biology. Mishra and Kumar [11]

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622 P. K. Gupta and S. Verma · A Numerical Study of Nonlinear Reaction-Diffusion Equation

introduced the exact solutions for a variable coefficientnonlinear diffusion-reaction equation with a nonlin-ear convective term. In this model, the authors foundthe solitary wave equation and this type of equation isarisen in a variety of contexts in physical and biolog-ical problems. At the same time, Rida [12] found theapproximate analytical solutions of generalized linearand nonlinear reaction-diffusion equations in an infi-nite domain. The solutions are obtained in compact andelegant forms in terms of generalized Mittag–Lefflerfunctions, which are suitable for numerical computa-tion.

In this paper, the authors present an application ofa new iterative scheme based on the homotopy analysismethod (HAM) for the reaction diffusion-equation

∂u(x, t)∂ t

+(

u(x, t)∂u(x, t)

∂x

)= v

∂ 2u(x, t)∂x2

− ∂

∂x(F(x)·u(x, t))−

∫ t

0

(a(t −ξ )

∂u(x,ξ )∂x

)dξ

(1)

with initial condition

u(x,0) = g(x), (2)

where u(x, t) is the displacement, v the constant thatdefines the kinematic viscosity, F(x) the external force,and a(t) the absorbent term.

The homotopy analysis method which provides ananalytical approximation solution is applied to variousnonlinear problems [13 – 15]. The motivation of thispaper is to extend the application of HAM proposedby Liao [16] and applied by several authors [17 – 21]in different mathematical/physical/biological prob-lems to solve nonlinear differential equations. In thenext section, we discuss a new reliable approach ofHAM. This new modification can be implementedfor integer/fractional-order nonlinear differential equa-tions [22].

2. Solution of the Problem

Consider the reaction-diffusion equation with exter-nal force and absorbent term a(t) = α

tβ−1

Γ (β ) , where α isthe absorbent coefficient:

∂u∂ t

+u∂u∂x

= v∂ 2u∂x2 −

∂

∂x[F(x)·u]

− α

Γ (β )

∫ t

0(t −ξ )β−1 ∂u

∂xdξ , t > 0,

(3)

subject to the initial condition

u(x,0) = g(x). (4)

In view of the algorithm presented in the previous sec-tion, if we select the auxiliary linear operator as L = ∂

∂ t ,we can construct the homotopy

(1−q)L[u(x, t;q)−u0(x, t)]

= qhH(t)[

∂u(x, t;q)∂ t

+u(x, t;q)∂u(x, t;q)

∂x− v

∂ 2u(x, t;q)∂x2

+∂

∂x[F(x)·u(x, t;q)]

+α

Γ (β )

∫ t

0(t −ξ )β−1 ∂u(x,ξ ;q)

∂xdξ

](5)

According to HAM, we have the initial guess

u0(x, t) = g(x). (6)

Taking H(t) = 1 with the initial guess (6) into thehomotopy equation (5), then equating the terms withidentical powers of q, we obtain the following set oflinear differential equation:

q1 : Dtu1 = h

(Dtu0 +u0

∂u0

∂x− v

∂ 2u0

∂x2 +∂

∂x[F(x)·u0]

+α

Γ (β )

∫ t

0(t −ξ )β−1 ∂u0

∂xdξ

),

q2 : Dtu2 = Dtu1 + h

(Dtu1 +u0

∂u1

∂x+u1

∂u0

∂x

− v∂ 2u1

∂x2 +∂

∂x[F(x)·u1]

+α

Γ (β )

∫ t

0(t −ξ )β−1 ∂u1

∂xdξ

),

q3 : Dtu3 = Dtu2 + h

(Dtu2 +u0

∂u2

∂x+u1

∂u1

∂x

+u2∂u0

∂x− v

∂ 2u2

∂x2 +∂

∂x[F(x)·u2]

+α

Γ (β )

∫ t

0(t −ξ )β−1 ∂u2

∂xdξ

),

...

P. K. Gupta and S. Verma · A Numerical Study of Nonlinear Reaction-Diffusion Equation 623

qn : Dtun = Dtun−1 + h

(Dtun−1 +

n−1

∑i=0

ui∂un−1−i

∂x

− v∂ 2un−1

∂x2 +∂

∂x[F(x)·un−1]

+α

Γ (β )

∫ t

0(t −ξ )β−1 ∂un−1

∂xdξ

). (7)

Now, we use its initial solution u(x,0) = g(x) as aninitial approximation by the iteration formula (7), andwe get the values of u1,u2,u3, . . . and so on.

3. Particular Cases

In this section, we are considering some cases of theequations (1) and (2) to demonstrate the reliability ofthe method and its wider applicability for solving non-linear diffusion equations with external force and ab-sorbent term.

Case 1. Here the expressions of the displacementu(x, t) are deduced for different particular cases for ini-tial condition g(x) = x.

A. For k = 1, α = 1, i.e. the one-dimensional nonlin-ear diffusion equation in the presence of an externalforce and a reaction term:

u(x, t) =

[x− h(3+3h+ h2)tx+ h2x

t3

3

+h(3+3h+ h2)t1+β

Γ (2+β )− h2(3+2h)

t2+β

Γ (3+β )

−h3 t3+β

(3+β )Γ (2+β )+ · · ·

]. (8)

B. For k = 0, α = 1, i.e. the one-dimensional non-linear diffusion equation in the presence of an ab-sorbent term and the absence of an external force:

u(x, t) =

[x+ h(3+3h+ h2)xt + h2(3+2h)xt2

+ h2xt3 + h(3+3h+ h2)t1+β

Γ (2+β )

+2h2(3+2h)t2+β

Γ (3+β )+4h3 t3+β

Γ (4+β )

+h3 t3+β

(3+β )Γ (2+β )+ · · ·

]. (9)

C. For k = 1, α = 0, i.e. the one-dimensional non-linear diffusion equation in the absence of an ab-sorbent term and the presence of an external force:

u(x, t) =[

x− h(3+3h+ h2)xt + h2xt3

3+ · · ·

]. (10)

Case 2. Here the expressions of the displacementu(x, t) are deduced for different particular cases for ini-tial condition g(x) = x2.

A. For k = 1, α = 1, i.e. the one-dimensional nonlin-ear diffusion equation in the presence of an externalforce and a reaction term:

u(x, t) =

[x2 + h(3+3h+ h2)(−2−3x2 +2x3)t

+ h2(3(4−8x+9x2 −20x3 +10x4)

+2h(2−4x+9x2 −20x3 +10x4))t2

2!+ h2(−26+192x−219x2 +152x3 −210x4

+84x5)t3

3!+2h(3+3h+ h2)x

t1+β

Γ (2+β )

+2h2(3+2h)(−5+6x)xt2+β

Γ (3+β )

+2h3(−20+19x−63x2 +44x3)t3+β

Γ (4+β )

+2h3(−2−9x2 +8x3)t3+β

(3+β )Γ (2+β )

+2h2(3+2h)t2+2β

Γ (3+2β )

+4h3(−3+7x)t3+2β

Γ (4+2β )

+h3xt3+2β

(3+2β )Γ (2+β )2 + · · ·

]. (11)

B. For k = 0, α = 1, i.e. the one-dimensional non-linear diffusion equation in the presence of an ab-sorbent term and the absence of an external force:

u(x, t) =

[x2 +2h(3+3h+ h2)(−1+ x3)t

+ h2(10hx3 +15x3 −4h−12)xt2 +2h2x2

· (−16+7x3)t3 +2h(3+3h+ h2)xt1+β

Γ (2+β )

624 P. K. Gupta and S. Verma · A Numerical Study of Nonlinear Reaction-Diffusion Equation

+12h2(3+2h)x2 t2+β

Γ (3+β )+ h3(−40+88x3)

· t3+β

Γ (4+β )+ h3(−4+16x3)

t3+β

(3+β )Γ (2+β )

+2h2(3+2h)t2+2β

Γ (3+2β )+28h3x

t3+2β

Γ (4+2β )

+4h3xt3+2β

(3+2β )Γ (2+β )2 + · · ·

]. (12)

C. For k = 1, α = 0, i.e. the one-dimensional non-linear diffusion equation in the absence of an ab-sorbent term and the presence of an external force:

u(x, t) =

[x2 +h(3+3h+ h2)(−2−3x2 +2x3)t

+ h2(3(4−8x+9x2 −20x3 +10x4)

+2h(2−4x+9x2 +−20x3 +10x4))t2

2!+ h2(−26+192x+−219x2 +152x3 −210x4

+84x5)t3

3!+ · · ·

]. (13)

Case 3. Here the expressions of the displacementu(x, t) are deduced for different particular cases for ini-tial condition g(x) = ex.

A. For k = 1, α = 1, i.e. the one-dimensional nonlin-ear diffusion equation in the presence of an externalforce and a reaction term:

u(x, t) = ex

[1+ h(3+3h+ h3)(−3+ ex −2x)t

+ h2(25+ e2x(9+6h)+44x+12x2 +4h(3+7x

+2x2)−2ex(17+12x+2h(5+4x)))t2

2!+ h2(−79+16e3x −3e2x(35+18x)−122x

−60x2 −8x3 +4ex(47+48x+12x2))t3

3!

+ h(3+3h+ h3)t1+β

Γ (2+β )

+4h2(3+2h)(−2+ ex − x)t2+β

Γ (3+β )

+ h2(3+2h)t2+2β

Γ (3+2β )+ h3(−15+10ex −6x)

· t3+2β

Γ (4+2β )+ h3 t3+3β

Γ (4+3β )+ exh3

· t3+2β

(3+2β )Γ (2+β )2 + · · ·

]. (14)

B. For k = 0, α = 1, i.e. the one-dimensional non-linear diffusion equation in the presence of an ab-sorbent term and the absence of an external force:

u(x, t) = ex

[1+(−1+ ex)h(3+3h+ h3)t

+ h2(1−2ex(5+2h)+ e2x(9+6h))t2

2!

+(−1+28ex −51e2x +16e3x)h2 t3

3!+ h(3+3h

+ h3)t1+β

Γ (2+β )+2h2(−1+2ex)h2(3+2h)

· t2+β

Γ (3+β )+(3+ ex(−32+21ex))h3 t3+β

Γ (4+β )

+ ex(−2+3ex)h3 t3+β

(3+β )Γ (2+β )+ h2(3+2h)

· t2+2β

Γ (3+2β )+(−3+10ex)h3 t3+2β

Γ (4+2β )

+ exh3 t3+2β

(3+2β )Γ (2+β )2

+h3 t3+3β

Γ (4+3β )+ · · ·

]. (15)

C. For k = 1, α = 0, i.e. the one-dimensional non-linear diffusion equation in the presence of an ab-sorbent term and the absence of an external force:

u(x, t) = ex

[1+ h(3+3h+ h2)(−3+ ex −2x)t

+ h2(25+ e2x(9+6h)+44x+12x2 +4h(3+7x

+2x2)−2ex(17+12x+2h(5+4x)))t2

2!+ h2(−79+16e3x −122x−60x2 −8x3 −3e2x(35

+18x)+4ex(47+48x+12x2))t3

3!+ · · ·

]. (16)

4. Numerical Results and Discussion

In this section, the numerical values of the displace-ment u(x, t) at the initial condition g(x) = ex for vari-ous values of t and different ranges of x with the properchoice of h at v = 1 are obtained, and the results are de-

P. K. Gupta and S. Verma · A Numerical Study of Nonlinear Reaction-Diffusion Equation 625

Fig. 1. Plot of u(x, t) vs. h at α = 1, k = 1, v = 1, β = 1, x = 1,and t = 1 for g(x) = ex.

Fig. 2 (colour online). Plot of u(x, t) vs. t at α = 1, k = 1, v =1, x = 1, h = −0.4, and g(x) = ex for different values of β .

Fig. 3. Plot of u(x, t) vs. h at α = 1, k = 0, v = 1, β = 1, x = 1,and t = 1 for g(x) = ex.

picted through Figures 1 – 7. Here, the authors considerfourth-order term approximation of the series solutionthroughout the numerical computation. As stated byLiao [16], it is seen that when h = −1 the result re-

Fig. 4 (colour online). Plot of u(x, t) vs. t at α = 1, k = 0,v = 1, x = 1, h = −0.4, and g(x) = ex for different values ofβ .

Fig. 5. Plot of u(x, t) vs. h at α = 0, k = 1, v = 1, β = 1, x = 1,and t = 1 for g(x) = ex.

Fig. 6. Plot of u(x, t) vs. t at α = 0, k = 1, v = 1, x = 1,h = −0.4, and g(x) = ex for different values of β .

sounds with the result obtained by the other mathemat-ical tool homotopy perturbation method (HPM) but wecannot say that our solution is always converging ath = −1.

626 P. K. Gupta and S. Verma · A Numerical Study of Nonlinear Reaction-Diffusion Equation

Fig. 7 (colour online). Behaviour of u(x, t) with respect to x and t are obtained when g(x) = ex, h = −0.4 ; (a) α = 1, k = 1;(b) α = 1, k = 0; (c) α = 0, k = 1.

Figure 1 represents the plot of displacementu(x, t; h) against h for k = 1, α = 1, β = 1, x = 1, v = 1,and t = 1. Since u(x, t; h) converges to the exact valuesfor different values of h, there exist horizontal line seg-ments shown in the figures, which are usually calledh-curves and show the validity of the region of con-vergence of the series solutions of equations (8) – (16).The results justify the statement of Liao [16, 17] thatby means of HAM, the convergence region and therate of series solution can be adjusted and controlledby plotting h-curves.

In Section 3, Particular Cases, we have calculatedthe values of u(x, t) with presence and/or absence ofabsorbent and external forces and showed that if bothterms, i.e. absorbent and external forces (α = 1 andk = 1), are present the normal solution increases at in-crease of time but slower than with the presence of onlyan external force for various values of β = 1/4, 1/2,3/4, and β = 1 (Fig. 2). If the absorbent term (α = 1and k = 0) is present, the regular solution is rapidly de-creasing with the increase of time for various values ofβ = 1/4, 1/2, 3/4, and β = 1 (Fig. 4), and if the ex-ternal force (α = 0 and k = 1) is present, the regularsolution increases at the increase of time (Fig. 6) and itis also physically justified.

It is also observed from Figures 1, 3, and 5, that forsmaller time the region of convergence will be betterfor every value of β . The numerical results u(x, t) forvarious values of x, t, and h are depicted through Fig-ures 7a, b, and c.

5. Conclusion

The numerical study of this article demonstrates theeffect of external force and absorbent term in the non-linear reaction-diffusion equation. Here the presenceof an external force increases the rate of diffusion inthe considered nonlinear system. But the rate becomesslower in the presence of an absorbent term. Again inthe absence of the external force, the reaction termfacilitates the diffusivity of the system much slowerwhich shows that the reaction term controls the systemof becoming stable. Therefore, the dynamic responseand stability scopes are improved in the presence of anabsorbent term which provides a damping force.

Also all the Figures 1 – 7 evidently demonstrate thestatement of Liao [16] that the method provides greaterfreedom to choose an initial approximation, the auxil-iary linear operator, and the auxiliary parameter h toensure the convergence of the series solution. And thisarticle also clearly defined the effect of external forceand absorbent term in linear, bilinear, and exponentialtype of initial conditions. Hence, we have concludedthat the diffusion process can be controlled and re-stricted by the use of the external/absorbent term, re-spectively.

Acknowledgement

The authors are extending their heartfelt thanks tothe reviewers for their valuable suggestions for theimprovement of the article.

P. K. Gupta and S. Verma · A Numerical Study of Nonlinear Reaction-Diffusion Equation 627

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