International Journal of Applied Mathematics and Computation

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International Journal of Applied Mathematics and ComputationVolume 3(4),pp 238–241, 2011http://ijamc.psit.in

The modified extended tanh method with the Riccatiequation for solving (3 + 1)-dimensionalKadomtsev-Petviashvili (KP ) equation

Nasir Taghizadeh, Mohammad Ali MirzazadehDepartment of Mathematics, Faculty of Mathematical Sciences,University of Guilan, P.O.Box 1914, Rasht, IranEmail: [email protected],[email protected]

Abstract:

In this paper, the modified extended tanh method is used to construct new exact travelingwave solutions of the (3+1)-dimensional Kadomtsev-Petviashvili equation. The modified ex-tended tanh method is one of most direct and effective algebraic method for obtaining exactsolutions of nonlinear partial differential equations. The method can be applied to noninte-grable equations as well as to integrable ones.

Key words: Modified extended tanh method; Riccati equation; (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation.

1 Introduction

The investigation of the exact traveling wave solutions of nonlinear evolutions equations paly animportant role in the study of nonlinear physical phenomena. For example, the wave phenomenaobserved in fluid dynamics, elastic media, optical fibers, etc.In recent years, there was interest in obtaining exact solutions of nonlinear partial differentialequation by the extended tanh method. The standard tanh method is developed by Malfliet [1].Recently, Wazwaz investigated exact solutions of nonlinear partial differential equation by theextended tanh method [2-4]. Fan in [5] presented the generalized tanh method for constructingthe exact solutions of nonlinear partial differential equation, such as, the (2+1)-dimensional sine-Gordon equation and the double sine-Gordon equation.The aim of this paper is to find exact solutions of the (3+1)-dimensional Kadomtsev-Petviashviliequation by modified extended tanh method with the Riccati equation.

2 Modified extended tanh method

Let us describe the modified extended tanh method. For given a nonlinear equation

F (u, ux, uy, ut, uxx, uxy, uxt, ...) = 0, (2.1)

when we look for its traveling wave solutions, the first step is to introduce the wave transforma-tion u(x, y, t) = u(ξ), ξ = sx + ly + nt + d, and change Eq. (2.1) to an ordinary differentialequation(ODE)

H(u, u′, u′′, u′′′, ...) = 0. (2.2)

The next crucial step is to introduce a new variable φ = φ(ξ), which is a solution of the Riccatiequation

dφ

dξ= k + φ2. (2.3)

239

The modified extended tanh method admits the use of the finite expansion:

u(x, y, t) = u(ξ) =

N∑i=0

aiφi(ξ) +

N∑i=1

biφ−i(ξ), (2.4)

where the positive integer N is usually obtained by balancing the highest-order linear term withthe nonlinear terms in Eq. (2.2). Expansion (2.4) reduces to the generalized tanh method [5]for bi = 0, i = 1, ..., N. Substituting (2.3) and (2.4) into Eq. (2.2) and then setting zero allcoefficients of φi(ξ), we can obtain a system of algebraic equations with respect to the constantsk, s, l, n, a0, ..., aN , b1, ..., bN . Then we can determine the constants k, s, l, n, a0, ..., aN , b1, ..., bN .The Riccati equation (2.3) has the general solutions:If k < 0 then

φ(ξ) = −√−k tanh(

√−kξ), (2.5)

φ(ξ) = −√−k coth(

√−kξ).

If k = 0 then

φ(ξ) = −1

ξ. (2.6)

If k > 0 thenφ(ξ) =

√k tan(

√kξ), (2.7)

φ(ξ) = −√k cot(

√kξ).

Therefore, by the sign test of k, we can obtain exact solutions of Eq. (2.1).

3 The (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation

For the (3 + 1)-dimensional KP [6]

(ut + 6uux + uxxx)x − 3uyy − 3uzz = 0. (3.1)

Let us consider the traveling wave solutions

u(x, y, z, t) = u(ξ), ξ = sx+ ly +mz + nt+ d, (3.2)

where s, l,m, n and d are constants.Substituting (3.2) into (3.1), then (3.1) is reduced to the following nonlinear ordinary differentialequation

s4u′′′′ + (sn− 3(m2 + l2))u′′ + 6s2(uu′)′ = 0. (3.3)

Integrating twice of Eq. (3.3), setting the constants of integrating to zero, we obtain

s4u′′(ξ) + (sn− 3(m2 + l2))u(ξ) + 3s2u2(ξ) = 0. (3.4)

It is easy to show that N = 2, if balancing u′′ with u2. Therefore, the modified extended tanhmethod (2.4) admits the use of the finite expansion:

u(ξ) = a0 + a1φ(ξ) + a2φ2(ξ) +

b1φ(ξ)

+b2

φ2(ξ). (3.5)

Thus, by Eq. (2.3), we get

u′′(ξ) = 6a2φ4(ξ) + 2a1φ

3(ξ) + 8ka2φ2(ξ) + 2ka1φ(ξ) + 2b2 + 2k2a2

+2kb1φ(ξ)

+8kb2φ2(ξ)

+2k2b1φ3(ξ)

+6k2b2φ4(ξ)

. (3.6)

240

u2(ξ) = a22φ4(ξ) + 2a1a2φ

3(ξ) + (2a0a2 + a21)φ2(ξ) + (2b1a2 + 2a0a1)φ(ξ) + 2b2a2

+ 2b1a1 + a20 +2b2a1 + 2b1a0

φ(ξ)+

2a0b2 + b21φ2(ξ)

+2b1b2φ3(ξ)

+b22

φ4(ξ). (3.7)

Substituting Eqs. (3.5) and (3.6) and (3.7) into Eq. (3.4), and equating the coefficients of likepowers of φi(i = −4,−3,−2,−1, 0, 1, 2, 3, 4) to zero yields the system of algebraic equations toa0, a1, a2, b1, b2, s, l,m, n and k

φ4 : 6s4a2 + 3s2a22 = 0,

φ3 : 6s2a1a2 + 2s4a1 = 0,

φ2 : 3s2a21 + 6s2a0a2 + (sn− 3(m2 + l2))a2 + 8s4ka2 = 0,

φ1 : (sn− 3(m2 + l2))a1 + 6s2a0a1 + 2s4ka1 + 6s2b1a2 = 0,

φ0 : 6s2b1a1 + 6s2b2a2 + 3s2a20 + (sn− 3(m2 + l2))a0 + 2s4b2 + 2s4k2a2 = 0,

φ−1 : 6s2b2a1 + 6s2b1a0 + 2s4kb1 + (sn− 3(m2 + l2))b1 = 0,

φ−2 : 3s2b21 + 6s2b2a0 + (sn− 3(m2 + l2))b2 + 8s4kb2 = 0,

φ−3 : 6s2b1b2 + 2s4k2b1 = 0,

φ−4 : 6s4k2b2 + 3s2b22 = 0.

Solving the resulting system, by using Maple, we find the following solutions

k =−sn+ 3l2 + 3m2

4s4, a0 = −−sn+ 3l2 + 3m2

6s2, a1 = 0, a2 = −2s2, b1 = b2 = 0. (3.8)

k = −−sn+ 3l2 + 3m2

4s4, a0 =

−sn+ 3l2 + 3m2

2s2, a1 = 0, a2 = −2s2, b1 = b2 = 0. (3.9)

k = −−sn+ 3l2 + 3m2

16s4, a0 =

−sn+ 3l2 + 3m2

4s2, a1 = 0, a2 = −2s2, b1 = 0, (3.10)

b2 = −s2n2 − 6snl2 − 6snm2 + 9l4 + 18m2l2 + 9m4

128s6.

k =−sn+ 3l2 + 3m2

16s4, a0 =

−sn+ 3l2 + 3m2

12s2, a1 = 0, a2 = −2s2, b1 = 0, (3.11)

b2 = −s2n2 − 6snl2 − 6snm2 + 9l4 + 18m2l2 + 9m4

128s6.

k =−sn+ 3l2 + 3m2

4s4, a0 = −−sn+ 3l2 + 3m2

6s2, a1 = a2 = b1 = 0, (3.12)

b2 = − (−sn+ 3l2 + 3m2)2

8s6.

k = −−sn+ 3l2 + 3m2

4s4, a0 =

−sn+ 3l2 + 3m2

2s2, a1 = a2 = b1 = 0, (3.13)

b2 = − (−sn+ 3l2 + 3m2)2

8s6.

By using (2.5) and (2.7), the sets (3.8) -(3.13) give the following solitons solutions:

u1(x, y, z, t) =sn− 3l2 − 3m2

6s2(1 + 3 tan2(

√−sn+ 3m2 + 3l2

4s4(sx+ ly +mz + nt+ d))).

u2(x, y, z, t) =sn− 3l2 − 3m2

6s2(1 + 3 cot2(

√−sn+ 3m2 + 3l2

4s4(sx+ ly +mz + nt+ d))).

241

u3(x, y, z, t) =−sn+ 3l2 + 3m2

2s2sech2(

√−sn+ 3m2 + 3l2

4s4(sx+ ly +mz + nt+ d)).

u4(x, y, z, t) = −−sn+ 3l2 + 3m2

2s2csch2(

√−sn+ 3m2 + 3l2

4s4(sx+ ly +mz + nt+ d)).

u5(x, y, z, t) =−sn+ 3l2 + 3m2

4s2− −sn+ 3l2 + 3m2

8s2

× tanh2(

√−sn+ 3m2 + 3l2

16s4(sx+ ly +mz + nt+ d))

− s2n2 − 6snl2 − 6snm2 + 9l4 + 18m2l2 + 9m4

8s2(−sn+ 3m2 + 3l2)

× coth2(

√−sn+ 3m2 + 3l2

16s4(sx+ ly +mz + nt+ d)).

u6(x, y, z, t) =−sn+ 3l2 + 3m2

12s2− −sn+ 3l2 + 3m2

8s2

× tan2(

√−sn+ 3m2 + 3l2

16s4(sx+ ly +mz + nt+ d))

− s2n2 − 6snl2 − 6snm2 + 9l4 + 18m2l2 + 9m4

8s2(−sn+ 3m2 + 3l2)

× cot2(

√−sn+ 3m2 + 3l2

16s4(sx+ ly +mz + nt+ d)).

4 Conclusion

In this paper, the modified extended tanh method has been successfully applied to find the solutionsof the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation. The modified extended tanhmethod is used to find new exact traveling wave solutions. Thus, we can say that the proposedmethod can be extended to solve the problems of nonlinear partial differential equations whicharising in the theory of solitons and other areas.

References

[1] Malfliet W. Solitary wave solutions of nonlinear wave equations. Am J Phys 1992; 60(7) : 650 − 654.

[2] Wazwaz AM. The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and theTzitzeica-Dodd-Bullough equations. Chaos Solitons Fractals 2005; 25(1) : 55 − 63.

[3] Wazwaz AM. The extended tanh method for new soliton solutions for many forms of the fifth-order-KdVequation. Appl Math Comput 2007 : 184(2); 1002 − 1014.

[4] Wazwaz AM. The extended tanh method: exact solutions of the Sine-Gordon and Sinh-Gordon equations. ApplMath Comput 2005; 167 : 1196 − 1210.

[5] Fan E, Hon YC. Generalized tanh method extended to special types of nonlinear equations. Z.Naturforsch2002; 57a : 692 − 700.

[6] Alagesan T, Uthayakumar A, Porsezian K. Painlev analysis and Backlund transformation for a three-dimensional Kadomtsev-Petviashvili equation. Chaos Solitons Fractals 1997; 8 : 893 − 895.



A mathematical approach on field equations with a case forhigher dimensional cosmological models

R K Dubey1 , Abhijeet Mitra2 and Bijendra Kumar Singh3

1,2,3Department of Mathematics, Govt. Science P G College(Affiliated to A P S University Rewa), Rewa (MP) IndiaEmail: [email protected]

Abstract:

A mathematical solution to Einstein’s field equations with a perfect fluid source, with vari-able gravitational constant G and Cosmological constant Λ for FRW space-time in higher di-mensions is obtained and case study has also been done where the values of ρ(t), G(t),Λ(t), q(t),and dH(t) has been obtained and their nature is also analyzed.

Key words: Cosmology; Higher dimension; Variable gravitational coupling (G) and Cos-mological Constant term (Λ).

1 Introduction

The formation of our universe i.e. study of modern cosmology took off in 1917 with a paper byAlbert Einstein that attempted to describe the universe by means of a simplified mathematicalmodel. Five years later Alexander Friedmann constructed models of the expanding universe thathad their origin in big-bang theory [1].

he idea of a static universe or ’Einstein Universe’ is one which demands that space is not expand-ing nor contracting but rather is dynamically stable. Albert Einstein proposed such a model as hispreferred cosmology by adding a cosmological constant (Λ) to his equations of general relativity tocounteract the dynamical effects of gravity which in a universe of matter would cause the universeto collapse. ’Einstein Universe’ is one of Friedman’s solutions of Einstein’s field equations for thevalue of cosmological constant (Λ).This is only stationary solution of all Friedman’s solutions, andbecause it is stationary, it is thought to be non-physical by majority of astronomers [2, 3]. Thoseastronomers think that universe is expanding because there is observed a phenomena of Hubbleredshift and it is interpreted by those astronomers as a Doppler’s shift caused by galaxies mov-ing away from our own Galaxy. Therefore, it is thought that the real solution of Einstein’s fieldequations cannot be stationary. As discussed earlier by many researchers [4] that a constant (Λ)cannot explain the huge difference between the cosmological constant inferred from observationsand energy density resulting from quantum field theories. In the year 1930 and onwards eminentcosmologists such as A.S. Eddington and Abbe Lemaitree [5, 6] felt that the -term introducedcertain attractive features into cosmology and that models based on it should also be discussed.In modern cosmology the reception given to the Λ-term has varied from hostile to the ecstatic.The term is quietly forgotten if the observational situation does not depend on models based onit. It is resurrected if it is found that the standard Friedmann models without this term are beingseverely constrained by observations.

Corresponding author: R K Dubey

243

To solve the above discussed problem, variable Λ was introduced such that Λ was large in theearly universe and then decayed with evolution.

A number of models with different decay laws for the variation of cosmological term wereinvestigated during last two decades (Chen and Wu [7] ,Pavon [8] Carvalho, Lima and Waga[9], Lima and Maia [10] Lima and Trodden [11] Arbab and Abdel- Rahaman [12] Vishwakarma[13] Cunha and Santos [14] Carneiro and Lima [15]. It was Bertolami [16, 17] who obtainedCosmological Models with time dependent G and Λ terms and suggested Λ ∼ R−2 ∼ t−2.

Similarly the gravitational constant G couples geometry to matter in the Einstein field equa-tions, and in expanding universe it is considered as a function of time. A possible time variableG was suggested by Dirac [18]. Many authors [19, 20] have proposed linking of the variation of Gwith that of Λ within the rules of general relativity theory.

This new thought leaves Einstein’s equations nearly unchanged as a variation in Λ is accompa-nied by the variation in G.

In this paper we have discussed a case with the above considerations to get some interestingresults.

2 The Field equations

Assuming a homogeneous and an isotropic (perfect or neutral) higher dimensional universe givenby Friedmann- Liematrie Robertson- Walker (FLRW) space-time metric

ds2 = c2dt2 − a2(t)

[dr2

1− kr2+ r2(dx2m)

](2.1)

where dx2m = da21 + sin2a1da22 + . . .+ sin2a1sin

2a2 . . . sin2am−1da

2m.

Here a(t), k = 0,±1 and D = m + 2 stand for scale factor, curvature parameter and totalnumber of space time dimensions respectively.

The Einstein field equations with time varying cosmological and gravitational ’constants’,

aik −1

2agik = 8πG [(ρ+ p)uiuk − ρgik] + Λgik (2.2)

For the metric (1) yields two independent equations,

m(m+ 1)

2

[(a

a

)2

+k

a2

]= 8πGρ+ Λ (2.3)

ma

a+m(m− 1)

2

[(a

a

)2

+k

a2

]= −8πGP + Λ (2.4)

Here equation (3) is time-time component and equation (4) the space- space component of the fieldequation (2). The over-dot denote derivative with respect to time. Solving equations (3) & (4),we get the continuity equation

From eqn. (3) we get (a

a

)2

+k

a2=

2(8πGρ+ Λ)

m(m+ 1)(2.5)

Differentiating w.r.t. t (time co-ordinate)

2

(a

a

)(aa− (a)2

a2

)− 2k

a3a =

2

m(m+ 1)(8πGρ+ 8πGρ+ Λ)

(1

a3

)[aaa− a3 − ka

]=

(8πGρ+ 8πGρ+ Λ)

m(m+ 1)

244

aaa = a3(8πGρ+ 8πGρ+ Λ)

m(m+ 1)+ a3 + ka

a =a3

aa

(8πGρ+ 8πGρ+ Λ)

m(m+ 1)+a3

aa+ka

aa(2.6)

Substituting the value of a obtained in (6) to (4) we get

m

a

[a2(8πGρ+ 8πGρ+ Λ)

m(m+ 1)a+a2

a+k

a

]+m(m− 1)

2

[(a

a

)2

+k

a2

]= −8πGP + Λ

a(8πGρ+ 8πGρ+ Λ)

(m+ 1)a+

(m+

m(m− 1)

2

)[(a

a

)2

+k

a2

]= −8πGP + Λ

Substituting the value of(aa

)2+ k

a2 from (5)

a(8πGρ+ 8πGρ+ Λ)

(m+ 1)a+

(m(m+ 1)

2

)2 (8πGρ+ Λ)

m(m+ 1)= −8πGP + Λ

a(

8πGρ+ 8πGρ+ Λ)

(m+ 1)a+ 8πGρ+ 8πGP = 0

Dividing the entire equation above by 8πG we get

aρ

a(m+ 1)+G

G

ρa

(m+ 1)a+

aΛ

8πG(m+ 1)a+ P + ρ = 0

Multiply the above equation by a(m+1)a we get

ρ+ ρG

G+

Λ

8πG+ (P + ρ)(m+ 1)

a

a= 0 (2.7)

From equation (7) we observe that energy density is not conserved for matter field due to varyingnature of scalars G & Λ [34, 35].

Since the Principle of equivalence requires only gik, Λ & G should not involve. So in this caselaw of Conservation of energy-momentum holds and it shows from equation (7)

ρ+ (m+ 1)(ρ+ P )a

a= 0 (2.8)

Using equation (8), equation (7) gives

ρG

G=−Λ

8πG⇒ G = − Λ

8πρ(2.9)

Using equation of state P = (γ − 1)ρ. On putting P = (γ − 1)ρ in (8)

ρ

ρ= −(m+ 1)γ

a

a(2.10)

Integrating we getlog ρ = −(m+ 1)γ log a+ log b2

ρ = b2a−(m+1)γ (2.11)

Where b2 = ρ0a+(m+1)γ0 and suffix 0 represents the present value of the parameters.

245

From equation (10) we get aa = − 1

(m+1)γρP , Substituting this value of a

a in eqn. (3) we get

m(m+ 1)

2× 1

(m+ 1)2γ2ρ2

ρ2= 8πGρ+ Λ− k

a2

(m(m+ 1)

2

)ρ2

ρ3= (m+ 1)2γ2

[16πG

m(m+ 1)+

2Λ

m(m+ 1)ρ− k

a2ρ

](2.12)

Again differentiating (12) and using equation (9)

ρ32ρρ− ρ2.3ρ2ρ(ρ3)2

= (m+ 1)2γ2

[16πG

m(m+ 1)+

2

m(m+ 1)

(ρΛ− Λρ

ρ2

)− k

(2aaρ+ a2ρ

(a2ρ)2

)]

2ρρ

ρ3− 3ρ2ρ

ρ4= (m+ 1)2γ2

[16πG

m(m+ 1)+

2Λ

m(m+ 1)ρ− 2ρΛ

m(m+ 1)ρ2− k2

a2p

(a

a

)− kρ

a2ρ2

]2ρρ

ρ3− 3

(ρ

ρ

)2ρ

ρ2= (m+ 1)2γ2

[−2Λρ

m(m+ 1)ρ2+

2k.1

(m+ 1)γ

ρ

ρ2a2− kρ

a2ρ2

]Above equation is obtained using equation 9. So, we have

2ρ

ρ− 3

(ρ

ρ

)2

= (m+ 1)γ2[

2k − (m+ 1)γk

γa2− 2Λ

m

]2ρ

ρ− 3

(ρ

ρ

)2

= (m+ 1)γ2[k (2− (m+ 1)γ)

γa2− 2Λ

m

](2.13)

Since from eqn. (10) aa = −1

(m+1)γρρ

H =−1

(m+ 1)γ

ρ

ρ

H =−1

(m+ 1)γ

(ρρ− ρρ.

ρ2

)

=−1

(m+ 1)γ

ρ

ρ+

1

(m+ 1)γ

(ρ

ρ

)2

2H =−2ρ

(m+ 1)γρ+

2

(m+ 1)γ

(ρ

ρ

)2

(2.14)

From eqn. (13) we get

−2ρ

(m+ 1)γρ+

3

(m+ 1)γ

(ρ

ρ

)2

=((m+ 1)γ − 2)

a2k +

2Λγ

m

Substituting eqn. (14) in eqn. (13) we get

2H + (m+ 1)γH2 +(2− (m+ 1)γ)k

a2− 2Λγ

m= 0 (2.15)

where H = aa is the Hubble parameter.

246

3 Solution and their analysis

Many authors have suggested that Λ ∼ a−2 by making different assumptions. Chen and Wu havesuggested that in the relation Λ = αa−2, the constant is related to the curvature parameter k andhence we may assume

Λ = b1a−2 (3.1)

In order to have

2H + (m+ 1)γH2 = 0 (3.2)

we make specific assumption of Λ given by eqn. (16) so that the last two terms of eqn. (15) getcanceled. i.e,

(2− (m− 1)γ)k

a2− 2Λγ

m= 0

or,

Λ =m

2γ(2− (m+ 1)γ)

k

a2⇒ m

2γ(2− (m+ 1)γ)k = b1 (3.3)

Now from equation (17) we obtain

2H

H2= −(m+ 1)γ

which on integration provides

− 2

H= −(m+ 1)γt⇒ 1

H=

(m+ 1)γ

2t (3.4)

Let (m+1)γ2 t = u and H = a

a(m+ 1)γ

2dt = du

Multiplying both sides by H and integrating we get

H(m+ 1)γ

2dt = H.du

Since H = aa and H = 1

u . Therefore,

(m+ 1)γ

2

a

adt =

1

udu

Integrating,

a =

((m+ 1)γ

2b3t

) 2(m+1)γ

(3.5)

where b3 is an integrating constant. By using eqn. (20, 11, 16), we get

ρ = b2

((m+ 1)γ

2b3t

) −2(m+1)γ

(m+1)γ

=b2 × 4

((m+ 1)γb3)2t2(3.6)

Λ = b1

[(m+ 1)γ

2b3t

] −4(m+1)γ

(3.7)

247

Substituting the value of ρ and Λfrom eqn. (21) and (22) in equation (9) and integrating, weobtain

G =2b1b38πb2

[(m+ 1)γ

2b3t

]1− 4(m+1)γ

On integrating, we obtain,

G = b4 +2b1

8πb2

1

((m+ 1)γ − 2)

[(m+ 1)y

2b3t

]2− 4(m+1)γ

Using equation (18) we get

G = b4 +mk

8γπb2

[(m+ 1)γ

2b3t

]2− 4(m+1)γ

(3.8)

where b4 is the constant of integration. In order to satisfy all the field equations by new values ofa, ρ,Λ, G we obtain relation between constants as

b4 =m(m+ 1)b23

16πb2,

which is obtained by putting equations (20, 21, 22, 23) in equation (12). From eqn. (19) we cancalculate the age of universe in terms of H as

t =2

(m+ 1)γH

The deceleration parameter q for the present model takes the form

q = − aa

(a)2

Since a =(

(m+1)γ2 b3t

) 2(m+1)γ

. Differentiating with respect to t, we get

a =2

(m+ 1)γb3

2

(m+ 1)γ

((m+ 1)γ

2γb3

) 2(m+1)γ

−1

Again differentiating with respect to t,

a =

(2

(m+ 1)γb3

)22

(m+ 1)γ[

2

(m+ 1)γ− 1][

((m+ 1)γ

2γb3

) 2(m+1)γ

−2

]

Putting value of a, a, ain equation q = − aa(a)2. , we get

q = −

(2

(m+1)γ − 1)

2(m+1)γ

=1

2[(m+ 1)γ − 1] (3.9)

Equation (24) shows that q is constant in this model and depends on the dimensionality of the

space-time. The relation between temperature and energy density as T ∼ ργ−1γ has been widely

accepted in literature. With the help of eqn. (21) we obtain

ρα1

t2

and

248

T ∼ ργ−1γ ∼= ρ1−

1γ ⇒ T ∼ t−2+

2γ (3.10)

The horizon distance dH(t) at time t is the proper distance travelled by light emitted at t = te

dH(t) = a(t) lim te→0

[∫ t

te

dt,

a(t,)

](3.11)

dH(t) =

((m+ 1)γ

2b3t

) 2(m+1)γ

lim te → 0

∫ t

te

dt′((m+1)γ

2 b3t′) 2

(m+1)γ

=

((m+ 1)γ

2b3t

) 2(m+1)γ

((m+1)γ

2 b3t)1− 2

(m+1)γ(1− 2

(m+1)γ

) 2

(m+ 1)γb3

=

(m+1)γb3t(m+1)γb3

(m+1)γ−2(m+1)γ

=

((m+ 1)γ

(m+ 1)γ − 2

)t (3.12)

From equation (27) it is clear that distance between two observers depend on the dimensionalityof space-time.

4 CONCLUSION

In this paper we have tried to present a case of cosmological model with varying gravitationalcoupling G and cosmological term Λ in higher dimensions.

It is clear from the values obtained in the case for ρ(t), G(t),Λ(t), T (t), q(t) and dH(t) that thesequantities depend on the dimensionality of space-time. The results obtained in this paper are infavour of the views of astronomical observations [36, 37].

The cosmological term Λ decides the behaviour of the universe in the model. In this paperwe have discussed a case, in which we obtain a positive value of Λwhich will correspond to anegative effective mass density. So in this case we can expect that expansion of universe will tendto accelerate.

The observations on magnitude and red shift of type Ia Supernova [38, 39, 40] suggest that ouruniverse may be an accelerating one or otherwise with induced cosmological density, through thecosmological Λ-term. Thus our models are consistent with the results of the observations made inrecent times.

5 Acknowledgement

I acknowledge my grateful thanks to CRO, Bhopal of University Grant Commission New Delhi forproviding financial assistance under Minor Research Project.

References

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249

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[12] Lima, J.A.S. and Trodden, M.; Phys. Rev. D 53, 4280 (1996)

[13] Arbab, A.I. and Abdel Rahaman, A.M.M.; Phys. Rev. D. 50, 7725 (1994).

[14] Vishwakarma, R.G.; Gen. Relativ. Gravit. 33, 1973 (2001).

[15] Cunha, J.V. and Santos, R.C.; Int. J. Mod. Phys. D 13, 1321 (2004).

[16] Carneiro, S. and Lima, J.a.S.; Int. J. Mod. Phys. A 20, 2465 (2005).

[17] Bertolami, O.; Nuovo Cimento 1393 (36 (1986).

[18] Bertolami, O.; Fortschr. Phys. 34, 829 (1986)

[19] P.A.M. Dirac.; Nature, 61, 323 (1937).

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[21] Berman, M.S.; Gen. Rel. Grav. 23, 465 (1991).

[22] Berman, M.S.; Phys. Rev., D 43, 1075 (1991).

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[24] Kalligas, D., Wesson, P., Everitt, C.W.f.; Gen., Rel. Grav., 24, 315 (1992).

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[31] Pradhan, A., Pandey, P., Singh, G.P.; Deshpandey, R.V., Spacetime & Sulstance, 6 (116) (2003)

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Quadrature method for cylindrical wire antenna

M.P.RamachandranMission Development Group,ISRO Satellite Centre, Bangalore 560 017, INDIAEmail: [email protected]

Abstract:

The distributed current in the straight cylindrical antenna can be obtained by solving theHallen equation with certain unknown constants. In this paper the Hallen equation is reducedto a Cauchy singular integral equation (CSIE). Quadrature method is then applied to theCSIE to obtain a linear system of equations. This approach enables to resolve the unknownconstants with the condition that the current vanishes at the ends. This alternative approachis now well posed. A couple of examples are worked out and distributed current is computed.

Key words: Quadrature; cylindrical antenna; integral equation

1 Introduction

Consider the cylindrical antenna integral equation for a perfectly conducting tube of length 2h,with a radius a and described by Hallen equation [1]

A(z) =−jωεk

∫ z

0

E(z′) sin k(z − z′) dz′ + C1 cos kz + C2 sin kz |z| ≤ h (1.1)

where

A(z) =1

4π

∫ +h

−hK(z, z′)I(z′)dz′ (1.2)

We need to solve for the total axial current I(z’) in (1.1). The kernel in (1.2) is

K(z, z′) =1

2π

∫ +π

−π

e−jkP

Pdφ′ (1.3)

where P = P (z − z′, φ′) =√

[(z − z′)2 + 4a2 sin2(φ′

2 )]. Here k is the wave number, ω is the

angular frequency and ε characterises the medium. The antenna’s length is aligned along the z-axis and the current flows along its length. Literature has extensively discussed the determinationof the distributed current either from (1.1) or from the Pocklington equation (see [2, 3]). In (1.1),C1 = A1(0) and C2k = A1(0) (1 denotes the derivative). These constants however depend on theunknown function. Rynne [2] has observed that the Hallen approach becomes well posed and isequivalent to the solution of the Pocklington equation provided the constants have certain specific

Corresponding author: M.P.Ramachandran

251

values and that the current vanishes at the ends. This suggestion is satisfied here while solvingthe CSIE using quadrature methods. This is the proposal in the paper.

In Section 2 we first briefly discuss the kernel and obtain equivalent CSIE from (1.1). Tomake I(z) vanish at the ends of the interval, an appropriate condition is obtained, which dependson current, the incident field and the kernel. This condition helps to determine the unknownconstants appearing in the Hallen equation. Quadrature method is then applied to the CSIE inSection 3. The integral equation is finally replaced by a linear system of equations and the solutionis the distributed current. In Section 4 of the paper we suggest separating the incident field E(z)into odd and even parts. Subsequently the unknown constants in the Hallen equation are eithereliminated or determined by using the boundary condition. The computed current is illustratedfor convergence against various dimensions of the linear system.

2 Cauchy Singular Integral Equation

The kernel in equation (1.2) can replaced as K = Kr = e−jkr

r here r =√

[(z − z′)2 + a2] forcylindrical antenna while a << h and a << λ. The equation (1.1) then becomes ill-posed [4]. Thekernel in (1.3) has a logarithmic singularity as suggested by Schelkunoff [1] and derived by Pearson[5]. This singularity should be incorporated to obtain sensible solution. We extend the approachof Jones [6] to decompose K as:

K = K1 +K2 +K3

where

K1 =1

π

∫ π

0

(u2 + a2φ′2)−1/2 dφ′ (2.1)

K2 =1

π

∫ π

0

[(u2 + 4a2 sin2(φ′

2))−1/2 − (u2 + a2φ′2)−1/2] dφ′ (2.2)

and

K3 =−1

2π

∫ π

−π

1− ejkP

Pdφ′ (2.3)

The integral in (2.1) has a closed form expression:

K1 = Ks +K1b (2.4)

where

Ks =−1

aπln |z − z′| (2.5)

K1b =1

aπln a+ ln |π + [(

u

a)2 + π2]1/2| (2.6)

The kernel can thus be expressed as a sum of singular and non-singular part:

K = Ks +KB (2.7)

Here,KB = K1b +K2 +K3 (2.8)

The term Ks has the logarithmic singularity. In absence of any closed form expression for K2,it is replaced by applying trapezoidal rule and the expression in [7] to evaluate K3. Alternativeapproximation for KB is given in [4]. Note that the first derivative of the bounded part KB can beshown to be bounded and continuous, while its second derivative can be shown to have logarithmicsingularity (see [3, 6]).

252

The equation (1.1) defined over [−h, h] is normalized to [−1, 1] in order to facilitate the applica-tion of the quadrature formulae. Differentiating the equation (1.1) with respect to z, and denoting∂K(z,z′)

∂z as K1(z, z′), we have the CSIE

(1/4π)

∫ +1

−1K1(z, z′)I(z′)dz′ = (−jεω)

∫ z

0

E(z′) cos k(z − z′)dz′ − C1k sin kz + C2k cos kz (2.9)

The analytic theory of CSIE is discussed in detail in [8] as a Riemann-Hilbert problem. Weshall use a more general notation instead of (2.9) for simplicit

(−1/πa)

∫ +1

−1I(z′)/(z − z′)dz′ +

∫ +1

−1R(z, z′)I(z′)dz′ = F (z)jzj = 1 (2.10)

where by using (2.7) and (2.8) we haveR = KB

1 (2.11)

F (z) = (−4πjεω)

∫ z

0

E(z′) cos k(z − z′)dz′ − C1k sin kz + C2k cos kz (2.12)

The solution of (2.10) has integrable singularity at the ends of the interval.

I(z) = (1− z2)−1/2c− (a/π)

∫ +1

−1χ(t)(1− t2)1/2/(z − t)dt (2.13)

where

c = (1/π)

∫ +1

−1I(z′)dz′ (2.14)

χ(t) = F (t)−∫ +1

−1R(t, t′)I(t′)dt′ (2.15)

In physical problems, ‘c’ in (2.14) is usually known and hence closed form solution can bederived when R(z, z′) is zero. The numerical solution of CSIE has a prolific literature (see [9]) eversince the work of Erdogan [10] appeared. The current that is confined to the wire is bounded andvanishes in the neighborhood of the ends. This physical condition implies that,

I(±1) = 0 (2.16)

To enable the current I(z′) satisfy this relation we write [8]:

I(z′) = (1− z′2)1/2Ψ(z′), (2.17)

The following condition needs to be satisfied by the solution of (2.13) (with (2.14) and (2.15))in order to satisfy (2.16). This equation is given below which contains the constants C1 and C2

and specifies them. ∫ +1

−1(1− z2)−1/2χ(z)dz = 0 (2.18)

The numerical solution of CSIE wherein the solution vanishes at the ends of the interval and inthe absence of any unknown constants has been discussed in [11] - [14]. It needs to be mentionedthat Tan [15] has also considered the Cauchy conversion (2.3) and has approximated the solutionby Chebychev polynomials as in [10] while recently Bruno [16] has used them while solving (1.1).

While differentiating (1.1) and deriving the CSIE (2.9), the derivatives of the kernel in (2.7)and (2.8) are obtained as follows.

K1s = (−1//πa)(1/u) (2.19)

253

K11b = [π + ((u/a)2 + π2)−1/2(u/a2)(u/a)2 + π2−1/2](1/πa) (2.20)

K2 is approximated using trapezoidal rule on the variable ϕ’ over [-1,1] and then its derivative isobtained. Also, the derivative of K3 (composed of real and imaginary parts) is first obtained andapproximated to a known accuracy by applying Mclaurin expansion (see [7]).

As mentioned earlier other alternative approximate form for KB1 as derived from [4] can also

be used.

3 Quadrature and numerical solution

The Cauchy integral in (2.10) is approximated as follows [11] :

∫ +1

−1(1− z′2)1/2Ψ(z′)/(z′ − z)dz′ =

∫ +1

−1(1− z′2)1/2(Ψ(z′)−Ψ(z))/(z′ − z)dz′ − zΨ(z)

≈ (π/(n+ 1))

n∑k=1

(1− tk2)Ψ(tk)/(tk −−z)−Ψ(z)−z + Tn+1(z)/Un(z) − zΨ(z) (3.1)

hereUn(tk) = 0; tk = cos(kπ/(n+ 1)).; k = 1, 2, . . . n (3.2)

When we select z such that,

z = xr, Tn+1(xr) = 0;xr = cos((2r − 1)π/2(n+ 1)); r = 1, 2 . . . , (n+ 1) (3.3)

We notice the terms trailing the summation in (3.1) vanishes. The Gauss-Chebychev quadratureis applied to the regular integral in (2.10) for z = xr to obtain∫ +1

−1(1− z′2)1/2R(xr, z

′)Ψ(z′)dz′ = π/(n+ 1)

n∑k=1

(1− tk2)R(xr, tk)Ψ(tk) (3.4)

Thus effectively we have reduced (2.10) as

(1/a(n+ 1))

n∑k=1

(1− tk2)Ψ(tk)/(xr − tk) + π(n+ 1)

n∑k=1

(1− tk2)R(xr, tk)Ψ(tk) = F (xr) (3.5)

where r = 1, 2..., (n + 1). The above linear system has (n + 1) equations in (n + 2) unknowns,namely Ψ(tk) , k = 1, 2, . . . n, C1 and C2. The current I(z′), at z = tk can be obtained using(2.17). Quadrature methods have the advantage of being simpler and eliminate the need of theevaluation of integrals though the collocation points are restricted while arriving at the linearsystem of equations.

Next we use ∫ +1

−1(1− z′2)−1/2F (z′)dz′ = π/(n+ 1)

n+1∑r=1

F (xr) (3.6)

along with the approximation (3.4) in (2.18) to get

π/(n+ 1)

n+1∑r=1

F (xr)− (π/(n+ 1))

n∑k=1

(1− tk2)R(xr, tk)Ψ(tk) = 0 (3.7)

Interestingly while summing all the (n+1) equations in (3.5) and using the formula

n+1∑r=1

1/(tk − xr) = 0 (3.8)

254

Figure 1: Quadrature Method : 2h = 1.2 λ and a= 0.001λ

The first summation term in (3.5) vanishes and we arrive at (3.7). Thus, we have an importantobservation that the solution of (3.5) for Ψ (tk) , k = 1,2,. . . .,n , C1 and C2 obtained by discretising(2.10) also satisfies the equation (3.7) which is nothing but the discretisation of the boundarycondition (2.18). Thus unknown constants in (3.5) satisfy boundary condition in (2.18) and attainspecific values as suggested by Rynne in being well posed. Next, as any constant term (from theeven component of the right side terms) appearing in (1.1) vanishes on differentiation to (2.9), wepropose the following to establish the equivalence of CSIE to (1.1). An additional condition isobtained after multiplying either side of (1.1) by (1− z′2)−1/2 and then integrating over [−1, 1] inorder to remove the singularity by using the formula

(1/π)

∫ +1

−1lnjz − z′j(1− z2)−1/2dz = − ln 2, (3.9)

We have

(1/4π)

∫ +1

−1−ln2/a+

∫ +1

−1KB(1− z2)−1/2dzI(z′)dz′ =

∫ +1

−1g(z) + C1 cos kz(1− z2)−1/2dz

(3.10)where g(z) denotes the first right side term in (1.1). This equivalence approach has been suggestedin [17] where the logarithmic integral equation has a solution that possesses integrable singularity.Notice that the term containing C2 vanishes. Then applying the quadrature (3.6) for the integralsin (3.10) we get

n∑k=1

(1/4(n+1))− ln2/a+π/(n+1)

n+1∑r=1

KB(xr, tk)(1−tk2)Ψ(tk) = π(n+1)n+1∑r=1

g(xr)+C1 cos kxr

(3.11)The linear system in (3.5) along with that in (3.11) has (n+2) equations to solve for as many

unknowns. Alternatively the value of C1 from (3.11) can be substituted in (3.5) to solve for theremaining (n+1) unknowns in (n+1) equations. This shall become clear in the next section.

4 Case Study

The external source field in (1.1) could in general be separated as a sum of even and odd parts.The incident field is set to be a plane wave and that a constant say C0 (an even function). In(2.10), it can be shown that R(v) = −R(u) , where v = −(u). Whenever the input function in(2.10) is odd, the solution Ψ(z) is then an even function. In (1.1) the constant C2 is absent andhence can be written as

255

Figure 2: Quadrature Method : 2h = λ /2 and a= λ/5000 Figure 3: Convergence I(0)

A(z) = (C0jωε

k2(1− cos kz) + C1 cos kz (4.1)

The differentiation leads to a CSIE which is

1

4π

∫ +1

−1K1(z, z′)I(z′)dz′ = C0

jωε

k2(k sin kz)− C1k sin kz (4.2)

The additional equation in (3.11) is

∫ +1

−1I(z′) dz′−ln2/a+

∫ +1

−1KB(1− z2)−1/2dz = (4πCojωε/k

2)(π − d) + C1d (4.3)

where d =∫ +1

−1 (cos kz)(1−z2)−1/2dz. Equation (4.3) enables to eliminate C1 in (4.2). The equation(4.2) is reduced to a linear system as in (3.11). It contains (n+1) equations with only n unknowns;Ψ(tk), k = 1, 2, . . . , n. It is interesting to note that if n is an even number, then when z = 0 (thatis xr = 0) the particular linear equation in (3.11) is trivial as Ψ(z) is an even function. Ignoringthat equation, the remaining n equations enable to solve uniquely for Ψ(tk), else if n is an oddnumber then we end up having an over determined system of equations and adopt the suggestionin [13] to obtain an optimal solution.

We give the outline of the convergence of this solution. After eliminating C1, the system in(3.5) is obtained. We find in (4.2) that R = K1

B is continuous and that the input function is alsocontinuous. Then the theorem in Elliot [18] (p142) assures that the numerical solution of (3.5)converges to the exact solution at the discrete points tk.

We considered an example where the antenna length, 2h is 1.2λ. We choose λ = 21.2 to make

h = 1. The radius is 0.001λ and the incident plane wave is constant. Behaviour of the absolutevalue of the current (in milli Amps) is depicted in Fig 1 for various values of nodes, n. For valuesof n higher than 94, there was no change in the second decimal.

Next, when we change λ = 4 and set radius equal to ( 5λ1000 ). This is to solve (1.1) as in [19, 20]

and is a case of a cylinder whose length is half wave length. The result when n is 18 or 72 is givenin Fig 2 (a) and is normalized to λ = 1 and this agrees with that in [19, 20]. The value of I(0),obtained by interpolation, is seen in Fig 2 (b) converging for n greater than 70 to 3.486. This valueis 3.485 when n is 120 or 150.

Before concluding we notice that whenever the source field is an odd function, the additionalcondition (4.1) is trivial and C1 = 0. Equations in (3.5) directly determine the current and constantC2. However, the convergence analysis mentioned earlier is not applicable because of the presenceof the constant C2 and is beyond the scope of this paper (see [14]).

256

5 Conclusion

The Cauchy singular integral equation (CSIE) with the pair of unknown constants and an additionalequation are deduced from the Hallen equation having a kernel containing logarithmic singularity.The quadrature method based method to solve the CSIE is formulated. This approach allowsin resolving the unknown constants besides satisfying the boundary condition while computingthe distributed current. The proposed alternative approach is thus well posed [2]. Results arepresented for a couple of applications.

References

[1] S.A.Schelkunoff, Advanced Antenna Theory, John Wiley placeStateNew York, 1952.

[2] B.P.Rynne , ‘ The well-posedness of the integral equations for the thin wire antennas’, IMA J Appl Math 49:35-44 (1992).

[3] T.K.Sarkar, ‘ A study of various methods for computing electromagnetic fields utilizing thin wire integralequation, Radio Science 18 : 29-38 (1983) .

[4] P.J.Davies, D.B.Duncan and S.A.Funken,’ Accurate and efficient algorithms for frequency domain scatteringfrom thin wire’, Jnl of Compt Phys 168 : 155-183 (2001).

[5] L.W.Pearson , A separation of logarithmic singularity in the exact kernel of the cylindrical antenna integralequation , IEEE Trans Antennas and Propagation, AP 23: 256-258 (1975).

[6] D.S.Jones, ‘Note on the integral equation for a straight wire antenna’, IEE Proc Microwave, Antennas andPropagation, 128 :114-116 (1981)

[7] M.P.Ramachandran, ’On the bounded part of the kernel in the cylindrical antenna integral equation’, AppliedComputational Electromagnetics Society Journal, 13 : 71- 77 (1998).

[8] N.I.Muskelishvilli, Singular Integral Equations, P.Nordhoff, placeCityGroningen, (1953)

[9] M.A.Golberg, Introduction to numerical solution of Cauchy-type singular integral equation, in ‘NumericalSolution of Integral equations’ , Ed M.A. Golberg, (1990) Plenum Press, New York.

[10] F.Erdogan ‘Approximate solution of systems of singular integral equation ‘, SIAM J Appl Math 17 :1041 -1059(1969).

[11] F.Erdogan and G.D.Gupta ,’ On numerical solution of singular integral equation ‘, Quart Appl Math 30 :525-534 (1972).

[12] N.I.Iokimidis, ‘ Some remarks on the numerical solution of Cauchy type singular integral equations with indexequal to -1’, Computers and Structures , 14: 403-407 (1981).

[13] E.Jen and R.P . Srivastav, ‘ Solving singular integral equations using Gaussian quadrature and over determinedsystems’, Comp and Math with applics, 9: 625-632 (1983).

[14] J.A.Cuminato, ‘Numerical solution of cauchy-type integral equations of index -1 by collocation methods,’Advances in Computational Mathematics, 6 : 47-64 (1996).

[15] S.H.Tan , ‘Modelling of electrically thick cylindrical antennas’, International Jnl of Numerical Modelling: Elec-tronic Networks, Devices and Fields, 3 : 195-206 (1990) .

[16] O.P.Bruno and M.C.Haslam ,’Regularity theory and superalgebraic solvers for wire antenna problems’, SIAMJ Sci Compt, 29 :1375-1402 (2007).

[17] M.P.Ramachandran,’ Numerical solution of an integral equation with logarithmic singularity, ‘ Comp and MathApplics, 26 :51-57 (1993) .

[18] D.Elliot,’ Rates of convergence for the method of classical collocation for solving singular integral equations’,SIAM J Numer Analy , 21 : 136-148 (1984).

[19] S.J.Orfanidis , Electromagnetics waves and Antennas , [Online] www.ece.rutgers.edu/˜orfanidi/ewa/ch21.pdf

[20] M.C.van Beurden and A.Tijhuis,’ Analysis and regularization of the thin-wire integral equation with reducedkernel’, IEEE Trans Antennas and Propogation, AP 55: 120-129 (2007) .



On finite π - directable automata

Milena BogdanovicTeacher Training Faculty in Vranje, SerbiaEmail: [email protected] [email protected]

Abstract:

Directable automata, known also as synchronizable, cofinal and reset automata, are a sig-nificant type of automata with very interesting algebraic properties and important applicationsin various branches of Computer Science. The central concept that we introduce and discuss inthis paper is the concept of π-directable automata as a concomitant specialization concept ofthe directable automata and generalization of the concept definite automata. Are introducedand a new class of automata, such as trapπ-directable automata, the local trap π-directableautomata, uniformly locally trap π-directable automata, finite π-directable automata.

Key words: π-directable automata; trapπ-directable automata; the local trap π-directableautomata; uniformly locally trap π-directable automata; finite π-directable automata.

1 Indtroduction and basic concepts

Directable automata, known also as synchronizable, cofinal and reset automata, are a significanttype of automata with very interesting algebraic properties and important applications in variousbranches of Computer Science (synchronization of binary messages, symbolic dynamics, verificationof software, etc.). They have been a subject of interest of many eminent authors since 1964, whenthey were introduced by J. Cerny in [3], although some of their special types were investigatedeven several years earlier. Various specializations and generalizations of directable automata haveappeared recently. T. Petkovic, M. Ciric and S. Bogdanovic in [4] introduced and studied trap-directable, trapped, monogenically, locally and generalized directable automata, as well as otherrelated kinds of automata. These automata have been also studied by Z. Popovic, S. Bogdanovic,T. Petkovic and M. Ciric in [5] and [6]. We also refer to the survey paper by S. Bogdanovic, B.Imreh, M. Ciric and T. Petkovic [2], devoted to directable automata, their generalizations andspecializations.

The purpose of this paper is to study the concept of π-directable automata as a concomi-tant specialization concept of the directable automata and generalization of the concept definiteautomata.

Automata be referred the π-directable automata if for each input word u ∈ X+there is k ∈ Nso that there is uk ∈ DW (A). Similarly, if for each input word u ∈ X+ there is k ∈ N so thatuk ∈ LDW (A) , uk ∈ GDW (A) , uk ∈ TDW (A) , uk ∈ LTDW (A) , uk ∈ TW (A) , then wecall the automata A the locally-π-directable automata, the general π-directable automata, thetrap-π-directable automata, the uniformly locally trap-π-directable and the π-trapped automata,respectively.

To mark a class consisting of these automata will use tags that are being introduced followingtable.

Corresponding author: M. Bogdanovic

258

Table 1

Mark Class of automata Mark Class of automataπDir π−directable TπDir trap-π-directableULπDir uniformly locally π-directable ULTπDir uniformly locally trap π-directableGπDir general π-directable πTrap π-trapped

2 Characterization of the π-directable automata

The automata A is colled general π-directable automata if for each word u ∈ X+, there is k ∈ N ,so that uk is the general directing word, ie. uk ∈ GDW (A) [1].Lemma 2.1. For an arbitrary automaton A, sets of TDW (A), LTDW (A), TW (A), DW (A),LDW (A) and GDW (A) are the ideals of free monoids X∗ and holds conditions:(i) TDW (A) 6= ∅ =⇒ TDW (A) = LTW (A) = TW (A) = DW (A) = LDW (A) = GDW (A);(ii) LTDW (A) 6= ∅ =⇒ LTDW (A) = LDW (A) = TW (A) = GDW (A);(iii) TW (A) 6= ∅ =⇒ TW (A) = GDW (A);(iv) DW (A) 6= ∅ =⇒ DW (A) = LDW (A) = GDW (A);(v) LDW (A) 6= ∅ LDW =⇒ (A) = GDW (A).Theorems that follow provide a variety of characteristics of π-directable automata.Theorem 2.1. For the automata A the following conditions are equivalent:1) A is generally π-directable automata;2) A is an extension of local π-directable automata using trap π-directable automata;3) S is a nil-extensions of rectangular bands.Proof: 1) → 2). Let the automaton A is generally π-directable. Then, it is the extension ofthe locally directable automata B using trap-directable automata C. As the automata B is locallydirectable, then LDW (A) 6= ∅, and based on Lemma 2.1, we have LDW (B) = GDW (B).Thus, for every word u ∈ X∗ there is k ∈ N such that uk ∈ GDW (B) = LDW (B), and B isuniformly locally π-directable automata.On the other hand, since C is the trap directable automata, it is TDW (A) 6= ∅, and by theLemma 2.1 it follows that TDW (C) = GDW (C). Therefore, for every word u ∈ X+, thereexists k ∈ N , so that uk ∈ TDW (C) = GDW (C), and the automata C is an trap directableautomata.This we have proved the implication 1) → 2).2) → 1). Let A is the extension of uniformly locally π-directable automata B using the trapπ-directable automata C. Consider an arbitrary word u ∈ X+. Then exist k, l ∈ N such thatuk ∈ LDW (B) and ul ∈ TDW (C) . Based on the feature of the sets of directing words, appliesuk+l = ukul ∈ GDW (A) . Thus we have proved that A is the generally π-directable automata.1)→ 3). Notice, an arbitrary word u ∈ X+. Then there exists k ∈ N such that uk ∈ GDW (A),which means that ηku = ηuk is an bi-zero in S(A). Let E be the set of all-zero of S(A). Thenwe have that E is a rectangular bar and the ideal of S(A), and how we proved that for everyηu ∈ S(A) there exists k ∈ N such that ηku ∈ E, this conclude that the S(A) is the nil-extensionsof rectangular bands E.3) → 1). Let S(A) is the nil-extensions of rectangular bands E. Consider arbitrary word u ∈X+ . By assumption, there exists k ∈ N such that ηku ∈ E, ie. ηuk ∈ E. This means that ηuk isthe bi-zero of S(A), which implies that the uk ∈ GDW (A). Thus we have proved that A is thegenerally π-directable automata.This is proof of the theorem is complete.Let A is the π-directable automata. Then, for each word u ∈ X+ is there n ∈ N so that isun ∈ DW (A). The smallest number n ∈ N such that un ∈ DW (A) , is the level of directingword u. Clearly, the directing words of the automata A has the same level of guidance 1.Following theorems are fully proven in [1].Now we describe the uniformly locally trap-π-directable automata.

259

Theorem 2.3. For the automata A the following conditions are equivalent:1) A is uniformly locally π-directable automata;2) A is the direct sum of π-directable automata, Aα, α ∈ Y and every word u ∈ X+ has a limitedlevel of guidance in automata Aα, α ∈ Y ;3) S(A) is a nil-extension right zero bands.Automata A is the π-trapped, if for each word u ∈ X+, for which there is k ∈ N, hold uk ∈ TW (A).The next theorem describes, among other things, the structure of the transition semigroup of theπ-trapped automata.Theorem 2.4. For the automata A the following conditions are equivalent:1) A is the π-trapped automata;2) A is an extension of discrete automata with trap π-directable automata;3)A is a nil-extension of left zero bands.The following theorem gives a complete characterization of uniformly locally trap π-directableautomata.Theorem 2.6. For the automata A the following conditions are equivalent:1) A is uniformly locally trap π-directable automata;2) A is the extension retractiveof the discrete automata with the trap πdirectable automata;3) A is the direct sum of trap π-directable automata, Aα, α ∈ Y and every word u ∈ X+ has alimited level of guidance in automata Aα, α ∈ Y ;4) A is the product subdirect of a discrete automata and a trap π-directable automata;5) A is the parallel composition of a discrete automata, and a trap π-directable automata;6) S(A) is a nil-semigroup.

3 Finite π-directable automata

In this section we prove that in the case of finite automata is no difference between the π-directableautomata and the definite automata. Similar features will be demonstrated for other types ofautomata that are discussed in the previous sections.Theorem 3.1. The finite automata A is a trap π-directable automata if and only if it is nilpotentautomata.Proof: Let the automata A is the trap π-directable automata. This means that there is a statea0 ∈ A, so that for all a ∈ A and for every word u ∈ X+, for which there is n ∈ N suchthat un ∈ DW (A), holds au = a0. The transition semigroup S(A) of the automata A is anil-semigroup. On the other hand, A is a finite automata, so the transition semigroup S(A) isfinite. Any finte nil-semigroup is nilpotent, then S(A) is nilpotent semigroup. The automata A isthe direct sum of the automata nilpotent Aα, α ∈ Y . However, A is the trap-directable automata,and it is the indecomposable of the direct sum. This means that |Y | = 1, ie. Y = α , andA = Aα. So, A is the nilpotent automata.The reversal of the theorem is clear.The proof that a finite automata who is the π-directable, it is also definite, given the followingtheorem.Theorem 3.2. The finite automata A is a π-directable automata if and only if it is definiteautomata.Proof: Let the automata A is the π-directable automata. Then, A is a locally π-directableautomata and the transition semigroup of the automata A is nil-extension of the right zero bands.However, as A is a finite automata, then S(A) is finite semigroup, so it must be a nilpotentextension of right zero bands. From this fact it follows that A is a direct sum of the automatadefinite, with the same degree of definiteness. Furthermore, the automata A is indecomposable indirect sum, because it is the π-directable, so it must be an automata definite.The reversal of the theorem is clear.We can prove the following two theorems with analogous considerations.

260

Theorem 3.3. The finite automata A is generally π-directable if and only if it is the generaldefinite.Theorem 3.4. The finite automata A is a uniformly locally π-directable if and only if it isuniformly locally definite.Last theorem shows that, for finite automata, the concept of uniform local trap π-directability isthe same concept as uniform local nilpotentility.Theorem 3.5. The finite automata A is a uniformly locally trap π-directable if and only if it isuniformly locally nilpotent.Proof : Let A be a uniformly locally nilpotent automata. Then each monogenic subautomataof A is nilpotent, and how this monogenic subautomata can be finite, according to Theorem 3.2.we have that every monogenic subautomata of A is the trap π-directable. Therefore, A is locallytrap π-directable automata, and as a finite automata, that it is uniformly locally trap π-directableautomata.

4 Conclusion

In the second half of the twentieth century, the concepts of information and the processing andtransfer of information, have become central in many areas of modern science. The very math-ematical abstraction of these concepts plays an important role in ensuring that their the studyapply exact mathematical models. As one of those mathematical abstraction, forties, fifties yearsof this century, came the notion of automata. The automata are viewed as systems that can beused for processing and transmission of certain kinds of information. Sixties and later, there is aconsiderable number of books on the Theory of Automata, which resulted in the development ofthis area as one of the most important in the field of Computre Science.

Here we introduce the notion of the π-directable automata, which is also the concept of spe-cialization and generalization of idea automata directable and definite automata. Also, we intro-duce and describe the concepts of the trap π-directable automata, locally and uniformly locallyπ-directable automata, locally and uniformly locally trap π-directable automata etc..

References

[1] Milena Bogdanovic, Directable Automata, Their Generalizations and Specializations (Direktabilni automati,njihova uoptenja i specijalizacije), (in Serbian), MSc thesis, University of Ni, Faculty of Sciences and Mathe-matics, 2001.

[2] S. Bogdanovic, B. Imreh, M. Ciric and T. Petkovic, Directable automata and their generalizations – A survey,in: S. Crvenkovic and I. Dolinka (eds.), Proc. VIII Int. Conf. ”Algebra and Logic” (Novi Sad, 1998), Novi SadJournal of Mathematics 29 (2) (1999), 31-74. [3] J. ernı, Poznmka k homognym experimentom s konecinımiautomatami, Mat.-fyz. cas. SAV 14 (1964), 208–215.

[3] T. Petkovic, M. Ciric and S. Bogdanovic, Decompositions of automata and transition semigroups, Acta Cy-bernetica (Szeged) 13 (1998), 385-403.

[4] Z. Popovic, S. Bogdanovic, T. Petkovic, and M. Ciric, Trapped automata, Publicationes Mathematicae Debrecen60 (3-4) (2002), 661-677.

[5] Z. Popovic, S. Bogdanovic, T. Petkovic, and M. Ciric, Generalized directable automata, in: Words, languagesand combinatorics, III, Proceedings of the Third International Colloquium in Kyoto, Japan, (M. Ito and T.Imaoka, eds.), World Scientific, 2003, pp. 378-395.



New approach for convergence of the series solution to aclass of Hammerstein integral equations

M. A. Abdoua, I. L. El-Kallab, A. M. Al-BugamicaDepartment of Mathematics Faculty of Education,Alexandria University, Egypt.bPhysics and Engineering Mathematics Department,Faculty of Engineering, Mansoura University, PO 35516, Mansoura, Egypt.Email: al [email protected] of Mathematics,Faculty of Applied Sciences, Taif University, KSA.

Abstract:

The proof of convergence of the series solution to a class of nonlinear two-dimensionalHammerstein integral equation (NTHIE), including the necessary and sufficient conditionsthat guarantee a unique solution, is introduced. Adomian Decomposition Method (ADM) andHomotopy Analysis Method (HAM) are used to solve the NTHIE. It was found that, whenusing the traditional Adomian polynomials (1.4), ADM and HAM are exactly the same. Butwhen using the proposed accelerated Adomian polynomials formula (1.5), ADM convergesfaster than HAM. The proposed accelerated Adomian polynomials formula is used directlyto prove the convergence of the series solution. Convergence approach is reliable enough toestimate the maximum absolute truncated error.

Key words: Hammerestein integral equation; Cauchy-Schwarz inequality; Fixed pointtheorem; Adomian Method; Homotopy Analysis Method.

1 Introduction

Integral equations provide an important tool for modeling a numerous phenomena and processesand also for solving boundary value problems for both ordinary and partial differential equations.Their historical development is closely related to the solution of boundary value problems in poten-tial theory. Progress in the theory of integral equations also had a great impact on the developmentof functional analysis. Reciprocally, the main results of the theory of compact operators have takenthe leading part to the foundation of the existence theory for integral equations of the second kind[1]-[4]. Therefore, many different methods are used to obtain the solution of the linear and non-linear integral equations. During the last years, significant progress has been made in numericalanalysis of one-dimensional version. However, the numerical methods for two-dimensional integralequations seem to have been discussed in only a few places. Brunner and Kauthen [5] introducedcollocation and iterated collocation methods for solving the two-dimensional Volterra integral equa-tion (T-DVIE). In [6] authors proposed a class of explicit Runge-Kutta-type methods of order 3for solving nonlinear T-DVIE. In [7] authors studied the approximate solution of T-DVIEs by thetwo-dimensional differential transform method. Abdou, in [8]-[11], used different methods to ob-tain the solution of Fredholm-Volterra integral equation of the first and second kinds in which the

Corresponding author: I. L. El-Kalla

262

Fredholm integral term is considered in position while the Volterra integral term is considered intime. In this paper, we use ADM and HAM for solving NTHIE

µu (x, t) = f(x, y) + λ

∫ b

a

∫ d

c

k (x, y, t, s) γ (t, s, u(t, s)) dtds, (1.1)

of the second kind where, the free term f(x, y) ∈ J = [a, b] × [c, d], k ∈ J × J is the given kerneland γ (x, y, u(x, y)) is the nonlinear term containing the unknown function u which represents thesolution of the NTHIE (1.1). In general µ defines the kind of the integral equation, µ = 0 for thefirst kind, µ = const 6= 0 for the second kind and µ = µ(x, y) for the third kind. Also, λ is aconstant, may be complex, that has a physical meaning.

1.1 Adomian Decomposition Method (ADM)

ADM has been known as a powerful device for solving many functional equations as algebraicequations, ordinary and partial differential equations and integral equations [12]-[15]. In manypapers for example [16]-[18] ADM were used to solve some classes of integral equations in onedimension. In this work, the two dimensions NTHIE (1.1) of the second kind (µ = const 6= 0) willbe solved using ADM. Without loss of generality (1.1) can be rewritten in the form

u(x, y) = f(x, y) + λ

∫ b

a

∫ d

c

k(x, y, t, s)γ(t, s, u(t, s))dtds. (1.2)

ADM assume the solution in the series form u(x, y) =∞∑n=0

un(x, y) and the nonlinear term γ(t, s, u)

is decomposed into an infinite series of Adomian polynomials

γ(t, s, u) =

∞∑n=0

An, (1.3)

which can be determined using the traditional formula

An =1

n!

dn

dλn[γ(t, s,

∞∑i=0

λiui)]λ=0. (1.4)

Another formula of Adomian polynomials, called accelerated Adomian polynomials, was deducedby El-Kalla in [19]-[20] in the recursive form

An = γ(Sn)−n−1∑i=0

Ai, (1.5)

where the partial sum Sn =n∑i=0

ui(x, y) and A0 = γ(u0). Application of ADM to Eq. (1.2) yields

u(x, y) =

∞∑i=0

un(x, y), (1.6)

where,

u0(x, y) = f(x, y), ui(x, y) = λ

∫ b

a

∫ d

c

k(x, y, t, s)Ai−1dtds, i ≥ 1 (1.7)

263

1.2 Homotopy Analysis Method (HAM)

Since 1992, Liao in [21] employed the basic ideas of the homotopy in topology to propose a generalanalytic method for nonlinear problems, namely the homotopy analysis method and then modifiedit step by step [22, 23]. This method has been successfully applied to solve different types of non-linear problems [24, 25]. HAM is in principle based on Taylor series with respect to an embeddingparameter. More importantly, different from all perturbation and traditional non-perturbationmethods, HAM provides us a simple way to ensure the convergence of solution series, and there-fore, the HAM is valid even for strongly nonlinear problems. In this subsection NTHIE (1.2) willbe solved using the HAM.

N [u] = u(x, y)− f(x, y)− λ∫ b

a

∫ d

c

k(x, y, t, s)γ(t, s, u(t, s))dtds = 0, (1.8)

where, N is an operator and u = (x, y) is the unknown function. Let u0(x, y) denote an initialguess of the exact solution u(x, y), h 6= 0 an auxiliary parameter, H(x, y) an auxiliary function andL an auxiliary linear operator with the property L[g(x, y)] = 0 when g(x, y) = 0. Using r ∈ [0, 1],as an embedding parameter, we construct such a homotopy

(1− r)L[φ(x, y; r)− u0(x, y)]− rhH(x, y)N [φ(x, y; r)] = H[φ(x, y; r);u0(x, y), H(x, y), h, r], (1.9)

where H is a second auxiliary function . It should be emphasized that we have a great freedom tochoose the initial guess u0(x, y), the auxiliary linear operator L, the non-zero auxiliary parameterh and the auxiliary function H(x, y). Assume the homotopy (1.9) to be zero, i.e.

H[φ(x, y; r);u0(x, y), H(x, y), h, r] = 0.

We have the so called zero-order deformation equation

(1− r)L[φ(x, y; r)− u0(x, y) = rhH(x, y)N [φ(x, y; r)]. (1.10)

When r = 0, the zero-order deformation (1.10) becomes

φ(x, y; 0) = u0(x, y), (1.11)

also, when r = 1, h 6= 0 and H(x, y) 6= 0, the zero–order deformation (1.10) is equivalent to

φ(x, y; 1) = u(x, y). (1.12)

By Taylor’s theorem φ(x, y; r) can be represents in a power series form of r as follows

φ(x, y; r) = u0(x, y) +

∞∑m=1

um(x, y)rm (1.13)

where,

um(x, y) =1

m!

∂mφ(x, y; r)

∂rm|r=0. (1.14)

The nonzero auxiliary parameter h, and the auxiliary function H(x, y)are properly chosen, so thatthe power series (1.13) of φ(x, y; r) converges at r = 1. Under these assumptions we have the seriessolution

u(x, y) = φ(x, y; 1) = u0(x, y) +

∞∑m=1

um(x, y), (1.15)

therefore, we can define the vector

−→u n(x, y) = φ(x, y; 1) = u0(x, y), u1(x, y), ..., un(x, y). (1.16)

264

According to the definition (1.13), the governing equation of um(x, y) can be derived from thezero-order deformation equation (1.10). Differentiating (1.10) m times with respective to r, thendividing by m! and setting r = 0,we have the so called mth-order deformation equation

L[um(x, y)− ηmum−1(x, y)] = hH(x, y)<m(−→u m−1(x, y)),

um(0, 0) = 0, (1.17)

where,

<m(−→u m−1(x, y)) =1

(m− 1)!

∂m−1N [φ(x, y; r)]

∂rm−1|r=0, (1.18)

and ηm = 0 (for m ≤ 1) or = 1 (for m > 1). Note that the high-order deformation of Eq. (1.17)is governing by the linear operator L, and the term <m(−→u m−1(x, y)) can be expressed simplyby (1.18). To obtain a simple iteration formula for um(x, y), choose Lu = u as an auxiliarylinear operator, the zero–order approximation u0(x, y) = f(x, y) is taken and the nonzero auxiliaryparameter h and the auxiliary function H(x, y) can be taken as h = −1, H(x, y) = 1. Applicationof HAM on (1.2) yields

u0(x, y) = f(x, y)

um(x, y) = λ

∫ b

a

∫ d

c

k(x, y, t, s)<m−1(φp)dtds,m ≥ 1,

and the corresponding homotopy series solution is given by

u(x, y) =

∞∑m=0

um(x, y) (1.19)

2 Existence of a unique solution

To prove the existence of a unique solution of (1.1), we use the principle of contraction mappingon a complete metric space. Rewrite (1.1) in the operator form

Wu(x, y) =1

µf(x, y) +Wu(x, y), (µ 6= 0), (2.1)

where,

Wu(x, y) =λ

µ

∫ b

a

∫ d

c

k(x, y, t, s)γ(t, s, u(t, s))dtds, (2.2)

and assuming the following conditions:1- The kernel k(x, y, t, s) satisfies the continuity condition |k(x, y, t, s)| ≤ N , N is a constant2- The given function f(x, y) is continuous in the Banach space C[J ] and satisfy(∫ b

a

∫ d

c

|f(x, y)|2 dxdy

) 12

= δ

where δ is a constant.3- For constants A > A1 and A > P, the function γ(x, y, u(x, y)) satisfies:

i− ∫ b

a

∫ d

c

|γ(x, y, u(x, y))|2 dxdy 12 ≤ A1 ‖u(x, y)‖

ii− |γ(x, y, u1(x, y))− γ(x, y, u2(x, y))| ≤ M(x, y) |u1(x, y)− u2(x, y)|

265

where ‖M(x, y)‖ = P

4- The unknown function u(x, y) ∈ C[J ] with the norm ‖u(x, y)‖ =∣∣∣∫ ba ∫ dc |u(x, y)|2 dxdy

∣∣∣ 12Lemma 1. Under the conditions (1) – (3-i), the operator W, defined by (1.2), maps the spaceC[J ] into itself.

Proof. Using equations (1.2), (1.3) and condition 2 with Cauchy-Schwarz inequality, we have

∥∥Wu(x, y)∥∥ ≤ δ

|µ|+

∣∣∣∣λµ∣∣∣∣ |k(x, y, t, s)|

∫ b

a

∫ d

c

|γ(x, y, u(x, y))|2 dxdy 12 .

Using conditions (1) and (3-i), the above inequality takes the form∥∥Wu(x, y)∥∥ ≤ δ

|µ|+ σ ‖u(x, y)‖ , σ =

∣∣∣∣λµ∣∣∣∣ NA. (2.3)

The inequality (1.4) shows that the operator W maps the space C[J ] into itself. Also, inequality(1.4) shows that the operators W and W are bounded where,

‖Wu(x, y)‖ = σ ‖u(x, y)‖ (2.4)

Lemma 2. If the conditions 1) and 3-ii) are satisfied, then the operator W contractive in C[J ].

Proof. For any two functions u1(x, y), u2(x, y) ∈ C[J ], formulas (1.2) and (1.3) lead to

∥∥(Wu1 −Wu2)(x, y)∥∥ ≤ ∣∣∣∣λµ

∣∣∣∣∥∥∥∥∥∫ b

a

∫ d

c

|k(x, y, t, s)| |γ(t, s, u1(t, s))− γ(t, s, u2(t, s))| dtds

∥∥∥∥∥ .Using the conditions 1) and 3-ii) and applying Cauchy-Schwarz inequality we have∥∥(Wu1 −Wu2)(x, y)

∥∥ ≤ σ ‖u1(x, y)− u2(x, y)‖ . (2.5)

Under the condition σ < 1, the operator W contractive and the proof is complete.

Theorem 2.1. If the conditions (1)–(3) are satisfied, then the integral equation (1.1) has a uniquesolution in the space C[J ] whenever |µ| > |λ| NA.

Proof. From Lemma1 and Lemma 2 since σ =∣∣∣λµ ∣∣∣ NA < 1 this leads to |µ| > |λ| NA and the proof

is complete.

3 Convergence Analysis

Convergence of the Adomian series solution was studied for different problems and by many au-thors. In [26, 27], convergence was investigated when the method applied to a general functionalequations and to specific type of equations in [28, 29]. In convergence analysis, Adomian’s poly-nomials play a very important role however, these polynomials cannot utilize all the informationconcerning the obtained successive terms of the series solution, which could affect directly theaccuracy as well as the convergence region and the convergence rate. In the present analysis wesuggest an alternative approach for proving the convergence. This approach depends mainly onEl-Kalla accelerated Adomian polynomial formula (1.5). As a result to this approach, the rate ofconvergence for the series solution is accelerated and the maximum absolute truncated error of theseries solution is estimated. Define a mapping F : E → E where, E = (C[J ], ‖.‖) is the Banachspace of all continuous functions on J with the norm ‖u(x, y)‖ = max

∀x,y∈J|u(x, y)| .

266

Theorem 3.1. The series (1.6) converges to a unique continuous solution of Eq. (1.2) if f(x, y)and k(x, y, t, s) are continuous and bounded functions and γ(t, s, u) satisfies Lipschitz condition.

Proof. Define a sequence Sn of partial sum such that Sn = u0 + u1 + ... + un we are going toprove that Sn is a Cauchy sequence in Banach space E. Let Sn and Sm are two arbitrary distinctpartial sums in the sequence Sn and n > m we have

‖Sn − Sm‖ = maxx,y∈J

|Sn − Sm| .

Using El-Kalla formula (5) we have

‖Sn − Sm‖ = maxx,y∈J

∣∣∣∣∣n∑

i=m+1

ui (x, y)

∣∣∣∣∣= max

x,y∈J

n−1∑i=m

λ

∫ b

a

∫ d

c

k(x, y, t, s)Aidtds

= |λ| maxx,y∈J

∣∣∣∣∣∫ b

a

∫ d

c

k(x, y, t, s)[γ(Sn−1)− γ(Sm−1)]dtds

∣∣∣∣∣≤ |λ|N(b− a)(d− c) max

x,y∈J|γ(Sn−1)− γ(Sm−1)| .

Since γ(u) satisfies Lipschitz condition so ∃ a constant L such that |γ(u)− γ(h)| ≤ L |u− h|. andwe can write

‖Sn − Sm‖ ≤ α ‖Sn−1 − Sm−1‖ ,

where, α = |λ|NL(b− a)(d− c). Substituting n = m+ 1 we have

‖Sm+1 − Sm‖ ≤ α ‖Sm − Sm−1‖ ≤ α2 ‖Sm−1 − Sm−2‖ ≤ · · · ≤ αm ‖S1 − S0‖ .

Using the triangle inequality we have

‖Sn − Sm‖ ≤ ‖Sm+1 − Sm‖+ ‖Sm+2 − Sm+1‖+ · · ·+ ‖Sn − Sn−1‖≤

[αm + αm+1 + · · ·+ αn−1

]‖S1 − S0‖

≤ αm[1 + α+ α2 + · · ·+ αn−m−1

]‖S1 − S0‖

≤ αm(

1− αn−m

1− α

)‖u1 (x, y)‖ .

If 0 < α < 1, i.e. 1− αn−m ≤ 1 then

‖Sn − Sm‖ ≤αm

1− αmax∀x,y∈J

|u1 (x, y)| , (3.1)

but max∀x,t∈J

|u1 (x, y)| <∞ (since f (x, y) is bounded) then ‖Sn − Sm‖ → 0 as m→∞, from which

we conclude that Sn is a Cauchy sequence in E so, the series∑∞i=0 ui(x, y) converges and the

proof is complete.

3.1 Error Estimate

Upon the final result of theorem 2, we can estimate the maximum absolute truncated error of theseries solution in the next theorem.

Theorem 3.2. The maximum absolute truncation error of the series (1.6) to problem (1.2) is

estimated to be: max∀x,y∈J

|u (x, t)−∑mi=0 ui (x, y)| ≤ αm+1

1−α max∀x,y∈J

|γ (u0)|.

267

Proof. From Theorem 2 inequality (3.1) we have

‖Sn − Sm‖ ≤αm


|u1 (x, t)| .

As n→∞ then Sn → u (x, y) and we have

max∀x,y∈J

|u1 (x, y)| ≤ |λ1|NL(b− a)(c− d) max∀x,y∈J

|γ (u0)| ≤ α max∀x,y∈J

|γ (u0)| ,

so,

‖u (x, t)− Sm‖ ≤αm+1


|γ (u0)| .

Finally the maximum absolute truncation error in the interval J is

max∀x,y∈J

∣∣∣∣∣u (x, y)−m∑i=0

ui (x, y)

∣∣∣∣∣ ≤ αm+1


|γ (u0)| . (3.2)

4 Numerical Experiments and discussions

Example1. Consider the NTHIE

u(x, y) = f(x, y) +

∫ 1

0

∫ 1

0

(exys2)(u(t, s))kdtds, (4.1)

Using Maple 10, this example will be solved by ADM and HAM when k = 1 (as a linear case) andwhen k = 2 (as a nonlinear case). In linear case, the free term f(x, y) = x2y2−0.06666666667e(xy)

while in the nonlinear case f(x, y) = x2y2−0.02857142857e(xy). The exact solution in both cases isu(x, y) = x2y2. Using 10 term approximation when k = 1, table 1 presents a comparison betweenerrors resulted from ADM (ErrorADM ) and HAM (ErrorHAM ) at some points of x, y, 0 ≤ x, y ≤ 1.

Table 1 (Linear case, k=1)

x y ErrorADM ErrorHAM0.0 0.0 6.36100E − 08 6.36100E − 080.2 0.2 6.62060E − 08 6.62060E − 080.4 0.4 7.46500E − 08 7.46500E − 080.6 0.6 9.12000E − 08 9.12000E − 080.8 0.8 1.20600E − 07 1.20600E − 071.0 1.0 1.72900E − 07 1.72900E − 07

In the nonlinear case (k = 2) we solve using the traditional Adomian polynomials (1.4) andusing the accelerated Adomian polynomials (5). Table 2 presents a comparison between errorsresulted from ADM with traditional formula (1.4) (ErrorADM1), with accelerated formula (1.5)(ErrorADM2) and HAM (ErrorHAM ) at some points of x, y, 0 ≤ x, y ≤ 1.

Table 2 (Nonlinear case, k=2)

x y ErrorADM1 ErrorADM2 ErrorHAM0.0 0.0 2.20000E − 10 2.00000E − 11 2.20000E − 100.2 0.2 2.28978E − 10 2.08162E − 11 2.28978E − 100.4 0.4 2.58172E − 10 2.34702E − 11 2.58172E − 100.6 0.6 3.15332E − 10 2.86665E − 11 3.15332E − 100.8 0.8 4.17335E − 10 3.79296E − 11 4.17335E − 101.0 1.0 5.98022E − 10 5.43656E − 11 5.98022E − 10

268

Example2. Consider the NTHIE

u(x, y) = f(x, y) +

∫ 1

0

∫ 1

0

cos(xy)(u(t, s))kdtds (4.2)

Also, Maple 10 is used to solve this example by ADM and HAM when k = 1 (as a linear case) andwhen k = 2 (as a nonlinear case). In linear case, the free term f(x, y) = sin(xy)− 0.2118455042xywhile in the nonlinear case f(x, y) = sin(xy)−0.08246639025xy. The exact solution in both cases isu(x, y) = sin(xy). Using 10 term approximation when k = 1, table 3 presents a comparison betweenerrors resulted from ADM (ErrorADM ) and HAM (ErrorHAM ) at some points of x, y, 0 ≤ x, y ≤ 1.

Table 3 (Linear case, k=1)

x y ErrorADM ErrorHAM0.0 0.0 0.00000E + 00 0.00000E + 000.2 0.2 4.00000E − 12 4.00000E − 120.4 0.4 1.60000E − 11 1.60000E − 110.6 0.6 3.60000E − 11 3.60000E − 110.8 0.8 6.40000E − 11 6.40000E − 111.0 1.0 1.00000E − 10 1.00000E − 10

In the nonlinear case (k = 2) we solve using the traditional Adomian polynomials (1.4) andusing the accelerated Adomian polynomials (1.5). Table 2 presents a comparison between errorsresulted from ADM with traditional formula (1.4) (ErrorADM1), with accelerated formula (1.5)(ErrorADM2) and HAM (ErrorHAM ) at some points of x, y, 0 ≤ x, y ≤ 1.

Table 4 (Nonlinear case, k=2)

x y ErrorADM1 ErrorADM2 ErrorHAM0.0 0.0 0.00000E + 00 0.00000E + 00 0.00000E + 000.2 0.2 4.00000E − 13 0.00000E + 00 4.00000E − 130.4 0.4 1.60000E − 12 0.00000E + 00 1.60000E − 120.6 0.6 3.60000E − 12 0.00000E + 00 3.60000E − 120.8 0.8 6.40000E − 12 0.00000E + 00 6.40000E − 121.0 1.0 1.00000E − 11 0.00000E + 00 1.00000E − 11

It is clear that, in linear case both ADM and HAM give the same approximate solution. In thenonlinear case, It was found that, both HAM and ADM with traditional formula (1.4) are exactlythe same, but ADM with accelerated formula (1.5) converges faster than HAM.

5 Conclusion

The necessary and sufficient conditions that guarantee a unique solution to NTHIE is introduced.ADM and HAM are exactly the same for solving linear two-dimensional Hammerstein integralequation. In nonlinear case, both HAM and ADM with traditional formula (1.4) are exactly thesame, but ADM with accelerated formula (1.5) converges faster than HAM. Moreover, acceleratedAdomian polynomials formula (1.5) is used directly in convergence analysis. Convergence analysisis reliable enough to estimate the maximum absolute truncated error of the series solution.

References

[1] K.E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the SecondKind, SIAM, Philadelphia, 1976.

[2] F.G. Tricomi, Integral Equations, New York (1985).

[3] L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Philadelphia, New York, 1985.

[4] M.A. Golberg, Numerical Solution of Integral Equations, Plenum press, New York, 1990.

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[5] H. Brunner, J. P. Kauthen, The numerical solution of two-dimensional Volterra integral equations by collocationand iterated collocation, IMA J. Numer. Anal. 9 (1989) 47-59.

[6] B. A. Beltyukov, L.N. Kuznechikhina, A Runge-Kutta method for the solution of two-dimensional nonlinearVolterra integral equations, Differential Equations, 12 (1976) 1169-1173.

[7] P. Darania and A. Ebadian, Numerical solutions of the nonlinear two-dimensional Volterra integral equations,NJOM, Volume 36 (2007), 163-174.

[8] M. A. Abdou, Fredholm-Volterra integral equation of the first kind and the contact problem, Apple. Math.Comput, 125, (2002), 177-193.

[9] M .A. Abdou, Fredholm-Volterra integral equation and generalized potential kernel, Apple. Math. and Com-puting, 13, (2002), 81-94.

[10] M. A. Abdou, On asymptotic method for Fredholm-Volterra integral equation of the second kind in contactproblems, J .Comp. Appl . Math. 154, (2003), 431-446.

[11] M. A. Abdou, Fredholm-Volterra integral equation with singular kernel, Apple. Math. Comput, 137, (2003),231-243.

[12] G. Adomian, Solving Frontier problems of Physics: The Decomposition method, Kluwer, 1995.

[13] I. L. El-Kalla, New results on the analytic summation of Adomian series for some classes of differential andintegral equations, Appl. Math. Comp., 217, (2010), 3756–3763.

[14] A.M. El-Sayed, I. L. El-Kalla, E. A. Ziada, Analytical and numerical solutions of multi-term nonlinear fractionalorders differential equations, Applied Numerical Mathematics, 60, (2010), 788–797.

[15] I. L. El-Kalla, Error estimate of the series solution to a class of nonlinear fractional differential equations,Commun Nonlinear Sci Numer Simulat, 16, (2011), 1408–1413.

[16] I. L. El-Kalla, Convergence of the Adomian method applied to a class of nonlinear integral equations, Appl.Math. Lett., 21, (2008), 372-376.

[17] A. Wazwaz, The combined Laplace transform–Adomian decomposition method for handling nonlinear Volterraintegro–differential equations, Appl. Math. and Comp., 216, (2010), 1304-1309.

[18] A. Wazwaz, S. Khuri, Two methods for solving integral equations, Appl. Math. and Comp, Volume 77, (1996),79-89.

[19] I. L. El-Kalla, Convergence of Adomian’s Method Applied to A Class of Volterra Type Integro-DifferentialEquations, Inter. J. of Differential Equations and Appl., 10, No.2, (2005), 225-234.

[20] I. L. El-Kalla, Error analysis of Adomian series solution to a class of nonlinear differential equations, Appliedmath. E-Notes, 7(2007), 214-221.

[21] S.J. Liao, The proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis,Shanghai Jiao Tong University, 1992.

[22] S.J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall/CRC Press,Boca Raton, 2003.

[23] S.J. Liao, On the homotopy analysis method for nonlinear problems. Appl. Math. Comput., 147, (2004), 499-513.

[24] S.J. Liao Notes on the homotopy analysis method: some definitions and theorems, Communications in NonlinearScience and Numerical Simulation, (2008), 4-13.

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Application of homotopy perturbation method for highorder nonlinear partial differential equations

Esmail Alibeigi, Mostafa Eslami, Ahmad Neirameh1

Department of Mathematics Islamic Azad University of Gonbad Kavos, IranEmail: 1neirameh [email protected]

Abstract:

In the paper, we extend the homotopy perturbation method to solve nonlinear fifth orderKdV equation. As a result, we successfully obtain some available approximate solutions ofthem. Numerical solutions obtained by the homotopy perturbation method are compared withthe exact solutions. The results reveal that the proposed method is very effective and simplefor obtaining approximate solutions of nonlinear partial differential equations.

Key words: Homotopy perturbation method; fifth order KdV equation.

1 Introduction

The application of the homotopy perturbation method (HPM) in nonlinear problems has beendevoted by scientists and engineers, because this method is to continuously deform a simple problemwhich is easy to solve into the under study problem which is difficult to solve. The homotopyperturbation method [5], proposed first by He in 1998 and was further developed and improvedby He [1-8]. It can be said that He’s homotopy perturbation method is a universal one, is able tosolve various kinds of nonlinear functional equations. In this method the solution is considered asthe summation of an infinite series which usually converges rapidly to the exact solutions. Thismethod continuously deforms a simple problem, easy to solve, into the difficult problems understudy.For the purpose of applications illustration of the methodology of the proposed method, usinghomotopy perturbation method, we consider the following nonlinear differential equation,

A(u)− f(r) = 0, r ∈ Ω, (1.1)

B(u, ∂u/∂n) = 0, r ∈ Γ, (1.2)

where A is a general differential operator, f(r) is a known analytic function, B is a boundarycondition and Γ is the boundary of the domain Ω.

The operator Acan be generally divided into two operators, L and N , where L is a linear, whileN is a nonlinear operator. Equation (1.1) can be, therefore, written as follows:

L(u) + N(u)− f(r) = 0 (1.3)

Corresponding author: A. Neirameh

271

Using the homotopy technique, we construct a homotopy U(r, p) : Ω× [0, 1]→ R which satisfies:

H(U, p) = (1− p)[L(U)− L(u0))] + p[A(U)− f(r)] = 0, p ∈ [0, 1], r ∈ Ω, (1.4)

orH(U, p) = L(U)− L(u0) + pL(u0) + p[N(U)− f(r)] = 0 (1.5)

Wherep ∈ [0, 1], is called homotopy parameter, and u0is an initial approximation for the solutionof Eq.(1.1), which satisfies the boundary conditions. Obviously from Esq. (1.4) and (1.5) we willhave.

H(U, 0) = L(U)− L(u0) = 0, (1.6)

H(U, 1) = A(U)− f(r) = 0, (1.7)

we can assume that the solution of (1.4) or (1.5) can be expressed as a series inp, as follows:

U = U0 + pU1 + p2U2 + . . . (1.8)

Setting p = 1, results in the approximate solution of Eq. (1.1)

u = limp→1

U = U0 + U1 + U2 + . . . (1.9)

Example : fifth order KdV equation [10]

∂u

∂t+ 45u2 ∂u

∂x− 15

∂u

∂x

∂2u

∂x2− 15u

∂3u

∂x3+

∂5u

∂x5= 0. (1.10)

With initial condition,

u(x, 0) = a0 − 2c2 sech2(√c2x). (1.11)

With the exact solution

u = a0 − 2c2 sech2(√c2(x− (45a20 − 60a0c2 + 16c22)t)), c2 > 0.

Where a0, c2are arbitrary constants.To solve Eq. (1.10) by homotopy perturbation method, we construct the following homotopy

(1− p)(∂U

∂t− ∂u0

∂t) + p(

∂U

∂t+ 45U2 ∂U

∂x− 15

∂U

∂x

∂2U

∂x2− 15U

∂3U

∂x3+

∂5U

∂x5) = 0,

or

∂U

∂t− ∂u0

∂t+ p(

∂u0

∂t+ 45U2 ∂U

∂x− 15

∂U

∂x

∂2U

∂x2− 15U

∂3U

∂x3+

∂5U

∂x5) = 0, (1.12)

Suppose the solution of Eq. (1.12) has the following form

U = U0 + pU1 + p2U2 + . . . (1.13)

Substituting (1.13) into (1.12) and equating the coefficients of the terms with the identical powersof p leads to

p0 : ∂U0∂t

− ∂u0∂t

= 0,

p1 : ∂U1∂t

+ 45U20

∂U0∂x

− 15 ∂U0∂x

∂2U0∂x2 − 15U0

∂3U0∂x3 + ∂5U0

∂x5 = 0,

p2 : ∂U2∂t

+ 90U1U0∂U0∂x

+ 45U20

∂U1∂x

− 15 ∂U1∂x

∂2U0∂x2 − 15 ∂U0

∂x∂2U1∂x2 − 15U1

∂3U0∂x3 − 15U0

∂3U1∂x3 + ∂5U1

∂x5 = 0,...

pj :∂Uj

∂t+ 45

∑j−1i=0

∑j−i−1k=0 UiUk

∂Uj−k−i−1

∂x− 15

∑j−1k=0

∂Uk∂x

∂2Uj−1−k

∂x2 − 15∑j−1

k=0 Uk∂3Uj−1−k

∂x3 +∂5Uj

∂x5 = 0,...

272

We take

U0 = u0 = a0 − 2c2 sech2(√c2x) (1.14)

We have the following recurrent equations for j = 1, 2, 3 . . . .

Uj = −∫ t

0

(45

j−1∑i=0

j−i−1∑k=0

UiUk∂Uj−k−i−1

∂x− 15

j−1∑k=0

∂Uk

∂x

∂2Uj−1−k

∂x2− 15

j−1∑k=0

Uk∂3Uj−1−k

∂x3+

∂5Uj

∂x5) dt = 0

(1.15)

With the aid of the initial approximation given by Eq. (1.14) and the iteration formula (1.15) weget the other of component as follows

U1 = 720ta0√

c52 sech2(√c2x) tanh3(

√c2x)− 480ta0

√c52 sech2(

√c2x) tanh(

√c2x)

−2160t√

c72 sech4(√c2x) tanh3(

√c2x) +1200t

√c72 sech4(

√c2x) tanh(

√c2x)

−180√c32 sech2(

√c2x) tanh(

√c2x)ta20+720

√c52 sech4(

√c2x) tanh(

√c2x)ta0

−720√c72 sech6(

√c2x) tanh(

√c2x)t−1440

√c72 sech2(

√c2x) tanh5(

√c2x)t

+1920√c72 sech2(

√c2x) tanh3(

√c2x)t −544

√c72 sech2(

√c2x) tanh(

√c2x)t,

U2 = 353792t2c62 sech2(√c2x) + 162000t2a30c

32 sech2(

√c2x) tanh4(

√c2x)

+238080t2a0c52 sech2(

√c2x) − 43480800t2c62 sech6(

√c2x) tanh6(

√c2x)

−74606400t2c62 sech4(√c2x) tanh8(

√c2x) − 130394880t2c62 sech2(

√c2x) tanh6(

√c2x)

+85680t2a20c42 sech2(

√c2x) + 156470400t2c62 sech4(

√c2x) tanh6(

√c2x)

+5702400t2c62 sech8(√c2x) tanh2(

√c2x)− 9504000t2c62 sech8(

√c2x) tanh4(

√c2x)

+119750400t2c62 sech2(√c2x) tanh8(

√c2x)− 17540640t2c62 sech6(

√c2x) tanh2(

√c2x)

−129600t2c52 sech8(√c2x)a0 − 10830336t2c62 sech2(

√c2x) tanh2(

√c2x)

+61036800t2c62 sech2(√c2x) tanh4(

√c2x) + 97200t2c42 sech6(

√c2x)a20

−172800t2c42 sech4(√c2x)a20 − 544320t2c52 sech4(

√c2x)a0

+21600t2a30c32 sech2(

√c2x) + 4050t2a40c

22 sech2(

√c2x) + 432000t2c52 sech6(

√c2x)a0

+23326080t2c62 sech4(√c2x) tanh2(

√c2x)− 32400t2c32 sech4(

√c2x)a30

−39916800t2c62 sech2(√c2x) tanh10(

√c2x)− 712800t2c62 sech10(

√c2x) tanh2(

√c2x)

+55663200t2c62 sech6(√c2x) tanh4(

√c2x)− 104843520t2c62 sech4(

√c2x) tanh4(

√c2x)

−5068800t2a0c52 sech(

√c2 tanh2(

√c2x) + 19353600t2a0c

52 sech2(

√c2x) tanh4(

√c2x)

−25401600t2a0c52 sech(

√c2 tanh6(

√c2x)− 1587600t2c42 sech2(

√c2x) tanh6(

√c2x)a20

+10886400t2a0c52 sech(

√c2 tanh8(

√c2x) + 18273600t2c52 sech4(

√c2x) tanh6(

√c2x)a0

−26481600t2a0c52 sech4(

√c2x) tanh4(

√c2x) + 9720000t2a0c

52 sech4(

√c2x) tanh2(

√c2x)

−2268000t2c42 sech4(√c2x) tanh4(

√c2x)a20 + 1836000t2c42 sech4(

√c2x) tanh2(

√c2x)a20

+8553600t2c52 sech6(√c2x) tanh4(

√c2x)a0 − 5875200t2c52 sech6(

√c2x) tanh2(

√c2x)a0

−162000t2a30c32 sech2(

√c2x) tanh2(

√c2x)− 12150t2a40c

22 sech2(

√c2x) tanh2(

√c2x)

−1164240t2a20c42 sech2(

√c2x) tanh2(

√c2x) + 2646000t2a20c

42 sech2(

√c2x) tanh4(

√c2x)

+162000t2c32 sech4(√c2x)a30 tanh2(

√c2x)− 680400t2c42 sech6(

√c2x)a20 tanh2(

√c2x)

+1166400t2c52 sech8(√c2x)a0 tanh2(

√c2x)− 868800t2c62 sech4(

√c2x)

+64800t2c62 sech10(√c2x) + 796320t2c62 sech6(

√c2x)− 345600t2c62 sech8(

√c2x),

...

Approximate solution of (1.10) can be obtained by setting p = 1

u = limp→1

U = U0 + U1 + U2 + . . . .

Suppose u∗ =∑3

j=0 Uj , the results are presented in Table 1.

273

Table 1: The numerical results, whena0 = c2 = 0.01 for solutions of Eq. (1.10) for initial condition (1.11).

x t u∗(x, t) |u∗ − u|0.1 0.1 -0.00999800053325245730 1.0× 10−19

0.1 0.15 -0.00999800073320080290 1.0× 10−19

0.15 0.1 -0.00999550127471400001 1× 10−20

0.2 0.3 -0.00999200453139060139 1× 10−20

0.35 0.25 -0.00997552348861796656 4× 10−20

0.45 0.45 -0.00995956269009242584 4× 10−20

0.5 0.3 -0.00995008919529557727 3× 10−20

0.45 0.7 -0.00995956717739295321 1× 10−20

0.4 0.8 -0.00996804687387015724 4× 10−20

0.8 0.8 -0.00987256954075552987 3× 10−20

0.85 0.95 -0.00985622515843887668 8× 10−20

0.9 0.9 -0.00983890285245324707 7× 10−20

0.95 0.9 -0.00982061427034014788 2× 10−20

1 1 -0.00980136528619636983 3× 10−20

2 Conclusion

In this article, He’s homotopy perturbation method has been successfully applied to find thesolution of nonlinear fifth order KdV equation are presented in table 1 , for differential results ofx and t to show the stability of the method for this two problems. The approximate solutionsobtained by the homotopy perturbation method are compared with exact solutions. Revealingthat the obtained solutions are more accurate then variatioal method. This method is introducedto overcome the difficulty arising in calculating Adomain polynomial in Adomian method. In ourwork, we use the maple package to carry the computations.

References

[1] J. H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathe-matics and Computation 151 (2004) 287-292.

[2] J. H. He, Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons andFractals 26 (2005) 695–700.

[3] J. H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A 350 (2006)87-88.

[4] J. H. He, Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons and Fractals 26 (1.3) (2005)827-833.

[5] J. H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178(1999) 257–262.

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Role of cloud droplets on the removal of gaseous pollutantsfrom the atmosphere: A nonlinear model

Shyam Sundar1 and Ram Naresh2

1Department of Mathematics, P.S. Institute of Technology,Bhaunti, Kanpur-208020, IndiaEmail: ssmishra [email protected]

2 Department of Mathematics, H.B.Technological InstituteKanpur-208002, IndiaEmail: [email protected]

Abstract:

In this paper, a four-dimensional nonlinear mathematical model is proposed to study theeffect of density of cloud droplets on the removal of gaseous pollutants by rain in the atmo-sphere. The atmosphere, during rain, is assumed to consist of four nonlinearly interactingphases i.e. the phase of cloud droplets, the phase of raindrops, the phase of gaseous pollutantsand the absorbed phase of gaseous pollutants in the raindrops. It is further assumed thatthese phases undergo ecological type growth and nonlinear interactions. The proposed modelis analyzed by stability theory of differential equations and computer simulations. It is shownthat the cumulative concentration of pollutants decreases due to rain and its equilibrium leveldepends upon the density of cloud droplets, the rate of formation of raindrops, emission rateof pollutants, the rate of falling absorbed phase on the ground, etc. It is noted here that ifdue to unfavorable conditions, cloud droplets are not formed, gaseous pollutants would not beremoved due to non-occurrence of rain.

Key words: Gaseous pollutants; precipitation; cloud droplets; stability; computer simu-lation.

1 Introduction

It is well known that the intensity of rainfall depends upon the density of droplets in clouds, moredense the cloud, more intense is the rain fall. During rain gaseous pollutants are removed byabsorption/impaction by raindrops falling on the ground. Some investigations have been madein India regarding removal of air pollutants from the atmosphere and it has been found thatthe atmosphere becomes cleaner during and after rain [1 - 5]. In particular, Sharma et al. [5]measured the concentration of particulate matters in Kanpur city, India and found considerabledecrease in their concentrations during monsoon season. Some experimental observations regardingrain washout have also been made for industrial cities elsewhere in the world [6 - 11]. For example,Flossmann [9] presented the interactions of clouds and pollution and found that clouds take uppollution mass that is removed when cloud precipitates in the form of rain.

Several investigations have been conducted to study the phenomenon of pollutants removal byprecipitation scavenging due to rain, snow or fog using mathematical models [12 - 17]. In particular,

Corresponding author: Shyam Sundar

275

Hales [12] has proposed a theory of gas scavenging by rain. Hales et al. [13] have also proposeda model for predicting the rain washout of gaseous pollutants from the atmosphere. Kumar [14]has given an Eulerian model to describe the simultaneous process of trace gas removal from theatmosphere and absorption of these gases in raindrops by considering the precipitation scavengingof these gases present below the clouds.

It is pointed out here that the above mentioned models are linear but in real situations duringprecipitation, the densities of cloud droplets and raindrops change, which affect the interactionprocess. [18 - 21]. In this direction, Naresh et al. [20] proposed a nonlinear model for removal ofpollutants from the atmosphere of a city. They have shown that, under appropriate conditions,gaseous pollutants and particulate matters, emitted with a constant rate, can be washed out fromthe atmosphere. This model, however, does not incorporate the role of cloud droplets in themodeling process.

In view of above, in this paper, we propose a nonlinear model for the removal of gaseouspollutants from the atmosphere by precipitation incorporating the phase of cloud droplets. Theproposed model is analyzed to see the effect of cloud density on the equilibrium level of gaseouspollutants in the atmosphere. Numerical simulation of the model is also carried out to support theanalytical results.

2 Mathematical Model

To model the phenomena, let Cd(t) be the density of cloud droplets, Cr(t) be the density ofraindrops in the atmosphere, C(t) be the cumulative concentration of gaseous pollutants in theatmosphere and Ca(t) be the cumulative concentration of gaseous pollutants in the absorbed phase.It is considered that Q is the cumulative emission rate of gaseous pollutants with natural depletionrate δC. It is also assumed that the absorption of gaseous pollutants by raindrops is proportionalto the cumulative concentration of the pollutants and the density of raindrops (i.e. αCCr). Itis considered that the gaseous pollutants with concentration Ca in the absorbed phase may beremoved due to falling of raindrops with rate kCa. It is also assumed that the removal of absorbedgaseous pollutants due to falling of rain drops on the ground is proportional to the density of raindrops as well as their concentration in absorbed phase (i.e. νCaCr).

Thus, the dynamics of the removal processes is governed by the following system of nonlineardifferential equations involving bilinear interactions of various phases,

dCddt

= λ− λ0Cd + π r CrC

dCrdt

= µCd − r0Cr − r CrC

dC

dt= Q− δ C − αCCr

dCadt

= αCCr − k Ca − ν CaCr (2.1)

with Cd(0) ≥ 0, Cr(0) ≥ 0, C(0) ≥ 0, Ca(0) ≥ 0In the first equation of model (2.1), λ is the rate of formation of water droplets in the cloud

with the natural depletion rate coefficient λ0. In the second equation of model (2.1), µ is thegrowth rate coefficient of Cr due to cloud droplets, r0 is the natural depletion rate coefficient ofraindrops from the atmosphere. It is assumed that if the pollutant species are hot, raindrops mayvaporize to enhance the growth of clouds. Thus, the depletion of raindrops is taken to be in directproportion to the number density of raindrops as well as the concentration of gaseous pollutants(i.e. r CrC) and a part of it (i.e. π r CrC with 0 ≤ π ≤ 1) may re-enter into the atmosphereenhancing the growth of clouds. In the third and fourth equations of model (2.1), the constantsδ and k are natural removal rate coefficients of C and Ca respectively, α and ν are removal rate

276

coefficients of C and Carespectively due to interactions with Cr. All the constants considered hereare taken to be non-negative.

It is remarked here that if λ0 is very large for a given concentration C due to atmosphericconditions then dCd

dt may become negative. In such a case no cloud droplets formation takes place,then precipitation does not occur and pollutants would not be removed.

In the following, we analyze the nonlinear model (2.1) by using the stability theory of differentialequations under the situations Q(t) = 0 (instantaneous emission of pollutants) and Q(t) = Q(continuous emission of pollutants with a constant rate).

We state the region of attraction of the model (2.1) as follows, to describe the bounds ofdependent variables. The set

Ω =

(Cd, Cr, C, Ca) : 0 ≤ Cd + Cr ≤

λ

λm, 0 ≤ C + Ca ≤

Q

δm

(2.2)

attracts all solutions initiating in the interior of the positive octant, where λm = min λ0 − µ, r0and δm = min δ, k.

3 Stability Analysis

3.1 Case 1: When Q(t) = 0 (Instantaneous emission)

The concentration of pollutants in the atmosphere is C (0) >0. In this case the model has only

one non-negative equilibrium E0

(λλ0, µλr0λ0

, 0, 0)

in Cd − Cr − C − Ca space.

After computing variational matrix of the model (2.1) corresponding toE0, we have found thatall the eigen values are negative proving that E0is locally asymptotically stable.

Theorem 3.1. IfCd(0) > 0, then E0is globally asymptotically stable in the positive octant.

Proof. From the first two equations of model (2.1), we have

dCddt

+dCrdt≤ λ− (λ0 − µ)Cd − r0Cr (3.1)

From this we get limt→∞

sup Cd(t) + Cr(t) ≤ λλm

, where λm = min λ0 − µ, r0Again from the third and fourth equations of model (2.1), we get

dC

dt+dCadt

= −δ C − k Ca − ν CrCa

≤ −δ C − k Ca

≤ −δm(C + Ca) (3.2)

where, δm = minδ, kThus, we get

limt→∞

sup C(t) = limt→∞

sup Ca(t) = 0 (3.3)

Hence the proof.

This theorem implies that in the case of instantaneous emission, gaseous pollutants are removedcompletely from the atmosphere by rain and the time taken for removal will depend upon the ratesof cloud droplets and raindrops formation.

277

3.2 Case 2: Q(t) = Q (Constant emission)

In this case also, the model (2.1) has only one equilibrium namely E∗ (C∗d , C∗r , C∗, C∗a), whereC∗d , C∗r , C∗ and C∗a are positive solutions of the following equations,

Cd =λ+ π r Crf(Cr)

λ0(3.4)

µCd − r0Cr − r CrC = 0 (3.5)

C =Q

δ + αCr= f(Cr) (3.6)

Ca =αCrf(Cr)

k + ν Cr(3.7)

To show the existence of E∗, we write eq.(3.5) as

F (Cr) =µ

λ0λ+ π r Crf(Cr) − r0Cr − r Crf(Cr) (3.8)

From eq.(3.8) we note that,

F (0) =µλ

λ0> 0 (3.9)

Again, using the maximum value of Cr, we get

F

(λ

λm

)= −

[λ r0

λm

(1− µλm

r0λ0

)+r λ

λm

(1− µπ

λ0

)f

(λ

λm

)]Since µ < λ0, λm < r0 and 0 ≤ π ≤ 1, therefore

F

(λ

λm

)< 0 (3.10)

Hence there exist a root C∗r in 0 < Cr <λλm

. For this root to be unique we have F ′(Cr) < 0. Here,

F ′(Cr) = −[r0 + r Crf

′(Cr)

(1− µπ

λ0

)+ r f(Cr)

(1− µπ

λ0

)]< 0 (3.11)

Thus, F (Cr) = 0 has a unique positive root (sayC∗r ) without any condition. Using C∗r we cancalculate C∗d , C

∗ and C∗a from eqs.(3.4), (3.6) and (3.7) respectively.From eqs.(3.6) and (3.7), we note that Cand Ca both decrease with increase in the density of

raindrops and they may even tend to zero for its larger values. We also note here that the removalof gaseous pollutants would depend upon different removal parameters.

In the following, we check the characteristics of various phases with respect to relevant param-eters analytically.

3.2.1 Variation of Cr with µ

From eq. (3.5), we have

µ

λ0λ+ π r Crf(Cr) − r0Cr − r Crf(Cr) = 0

Differentiating with respect to ‘µ’ we get,

dCrdµ

=

(λλ0

+ π rλ0Crf(Cr)

)[r0

(1− π rµ

λ0r0

)+ r Crf ′(Cr)

(1− π µ

λ0

)+ r f(Cr)

]This implies, dCr

dµ > 0. Thus , Cr increases as µ increases.

278

3.2.2 Variation of C with µ

From eq. (3.6), we note that dCdCr

< 0. Now,

dC

dµ=

dC

dCr

dCrdµ

< 0

since dCr

dµ > 0

Therefore C decreases as µ increases. Similarly, we can also show that dCd

dλ1< 0, dC

dα < 0, dCa

dα >

0, dCdλ < 0, dCa

dλ < 0, etc. Thus, it may be concluded that the removal rate of gaseous pollutantsincreases as the density of cloud droplets increases. We also note that,

1. If the coefficient λ0 is so large that dCd

dt becomes negative, the cloud droplets formation willnot take place and rain will not occur.

2. If the coefficient α is so large that dCdt < 0, all the gaseous pollutants will be removed from

the atmosphere.

3. For large k and ν, dCa

dt < 0 and the formation of absorbed phase is very transient and it maynot exist.

To see the stability behavior of E∗, we state the following theorems.

Theorem 3.2. Let the following inequalities

π r µC∗ <λ0

6(r0 + r C∗) (3.12)

rαC∗r C∗ <

1

9(r0 + r C∗)(δ + αC∗r ) (3.13)

π αµC∗r <2

3λ0(δ + αC∗r ) (3.14)

hold, then E∗is locally stable. (See appendix A for proof)

Theorem 3.3. Let the following inequalities

π r µC∗ <λ0r0

6(3.15)

rαC∗r C∗ <

r0δ

9(3.16)

π αµC∗r <2

3λ0δ (3.17)

hold inΩ, then E∗is globally asymptotically stable with respect to all solutions initiating in theinterior of the positive octant. (See appendix B for proof)

The above theorems imply that under certain conditions, the gaseous pollutants from the atmo-sphere would be removed and the removal rate increases as the density of cloud droplets increases.Remarks:

1. If π = 0, r = 0, then the inequalities (3.12) – (3.17) are satisfied automatically. This impliesthat these parameters have destabilizing effect on the system dynamics. It shows that, inabsence of these parameters, the gaseous pollutants would be washed out completely fromthe atmosphere by precipitation.

2. If µ = 0, α = 0, then the inequalities (3.12) – (3.17) are again satisfied automatically. Itshows that the gaseous pollutants would be removed completely from the atmosphere due togravity.

279

Figure 1: Global stability in Cd − Cr plane Figure 2: Variation of Crwith time ′t′for differentvalues of λ

Figure 3: Variation of Cwith time ′t′for differentvalues of λ

Figure 4: Variation of Cawith time ′t′for differentvalues of λ

4 Numerical Simulation

To see the effect of various removal parameters on the system dynamics, we conduct numericalsimulation of the model system by considering the following set of parameter values,

λ = 5, λ0 = 0.4, π = 0.001, µ = 0.3, r0 = 0.06, r = 0.08,

Q = 20, δ = 0.15, α = 0.65, k = 0.30, ν = 0.55

The equilibrium E∗ is calculated as,

C∗d = 12.506089, C∗r = 21.931986, C∗ = 1.388330, C∗a = 1.600938

Eigen values corresponding to E∗are obtained as,

−12.362592, −0.060690, −14.516168, −0.399998

Since all the eigen values corresponding to E∗are negative, therefore E∗is locally asymptoticallystable.

The global stability behavior of E∗ in Cd − Cr plane is shown in the fig.1. In figs.2-4, thevariation of density of raindrops, cumulative concentrations of gaseous pollutants Cand its absorbedphase Cawith time ′t′ is shown for different values of growth rate of cloud droplets λ (i.e. at

280

Figure 5: Variation of Cwith time ′t′for differentvalues of µ

Figure 6: Variation of Cawith time ′t′for differentvalues of µ

λ = 4, 5, 6) respectively. From these figures, it is seen that the density of raindrops increaseswhile the cumulative concentrations of gaseous pollutants C and that in absorbed phase Ca decreaseas growth rate of cloud droplets increases. Thus, if the density of cloud droplets is higher, theremoval of gaseous pollutants as well as the pollutants in absorbed phase is quite significant dueto enhanced rainfall. In figs.5-6, the variation of cumulative concentrations of gaseous pollutantsand the pollutants in absorbed phase with time ′t′ is shown for different values of growth rateof raindrops µ (i.e. atµ = 0.1, 0.2, 0.3) at λ = 5 respectively. It is seen that the cumulativeconcentrations of gaseous pollutants and that in absorbed phase decrease with time as the growthrate of raindrops increases.

5 Conclusion

A nonlinear mathematical model is proposed to see the effect of density of cloud droplets on theremoval of gaseous pollutants from the atmosphere by precipitation. The model is analyzed usingstability theory of differential equations and numerical simulations. The model analysis showsthat the density of raindrops increases while the cumulative concentrations of gaseous pollutantsand pollutants in absorbed phase decrease as growth rate of cloud droplets increases. It is alsoshown that the magnitude of gaseous pollutants removed by rainfall depends upon the intensityof rain caused by cloud formation. It is noted that the gaseous pollutants are removed from theatmosphere, but the remaining equilibrium amount would depend upon the rate of emission ofgaseous pollutants, growth rate of cloud droplets and raindrops, the rate of falling raindrops onthe ground and other removal parameters.Acknowledgements: The financial support for this research from University Grants Commission,New Delhi, India through project F. No. 39-33/2010 (SR) is gratefully acknowledged.

References

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[2] A. G. Pillai, M.S. Naik, G. Momin, P. Rao, K. Ali, H. Rodhe and L. Granat, Studies of wet deposition anddustfall at Pune, India, Water, Air and Soil Pollution 130 (1-4) (2001) 475-480.

[3] P.S. Prakash Rao, L.T. Khemani, G.A. Momin, P.D. Safai and A.G. Pillai, Measurements of wet and drydeposition at an urban location in India, Atmos. Environ. 26B (1992) 73-78.

[4] K. Ravindra, S. Mor, J. S. Kamyotra and C. P. Kaushik, Variation in spatial pattern of criteria air pollutantsbefore and during initial rain of monsoon, Environmental Monitoring and Assessment 87 (2.2) (2003) 145 –153.

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[5] V.P. Sharma, H.C. Arora and R.K. Gupta, Atmospheric pollution studies at Kanpur- suspended particulatematter, Atmos. Environ. 17 (1983) 1307-1314.

[6] J.D. Blando and B.J. Turpin, Secondary organic aerosol formation in cloud and fog droplets: a literatureevaluation of plausibility, Atmos. Environ. 34 (2000) 1623-1632.

[7] T. D. Davies, Precipitation scavenging of SO2 in an industrial area, Atmos. Environ., 10 (1976) 879 – 890.

[8] T. D. Davies, Sulphur dioxide precipitation scavenging, Atmos. Environ. 17 (1983) 797-805.

[9] A. I. Flossmann, Clouds and Pollution, Pure and App. Chem. 70 (3.5) (1998) 1345 – 1352.

[10] G. A. Loosmore and R. T. Cederwall, Precipitation scavenging of atmospheric aerosols for emergency responseapplications: testing an updated model with new real time data, Atmos. Environ. 38(3.5) ( 2004) 93-1003.

[11] K.F. Moore, D.E. Sherman, J.E Reilly and J.L. Collett, Drop size dependent chemical composition in cloudand fog, part 1, observations, Atmos. Environ. 38(10) (2004) 1389-1402.

[12] J.M. Hales, Fundamentals of the theory of gas scavenging by rain, Atmos. Environ. 6 (1972) 635-650.

[13] . J.M. Hales, M.A Wolf and M.T. Dana, A linear model for predicting the washout of pollutant gases fromindustrial plume, AICHE Journal 19 (1973) 292-297.

[14] S. Kumar, An Eulerian model for scavenging of pollutants by rain drops, Atmos. Environ. 19 (1985) 769-778.

[15] R. Naresh, An analytical approach to study the problem of air pollutants removal in a two patch environment,Ultra Science, (Int. J. Physical Sc.) 16(2.1) M, (2004) 83-96.

[16] J.B. Shukla, R. Nallaswany, S. Verma and J.H. Seinfeld, Reversible absorption of a pollutant from an areasource in a stagnant fog layer, Atmos. Environ. 16 (1982) 1035-1037.

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Appendix A

Proof of Theorem 2Using the following positive definite function in the linearized system of (2.1),

(A1) V = 12

(k0C2d1 + k1C2

r1 + k2C21 + k3C2

a1)where Cd1, Cr1, C1, Ca1 are small perturbations from E∗, as followsCd = C∗d + Cd1, Cr = C∗r + Cr1,C = C∗ + C1, Ca = C∗a + Ca1

Differentiating (A1) with respect to′t′we get, in the linearized system corresponding to E∗

V = −k1λ0C2d1 − k2(r0 + r C∗)C2

r1 − k3(δ + αC∗r )C21 − k4(k + ν C∗r )C2

a1

+(k1π r C∗ + k2µ) Cd1Cr1 + k1π r C

∗rCd1C1 − (k2r C

∗r + k3αC

∗)Cr1C1

+k4(αC∗ − ν C∗a)Cr1Ca1 + k4αC∗rC1Ca1

Now V will be negative definite under the following conditions,(A2) (k1π r C∗ + k2µ)2 < 2

3k1k2λ0(r0 + r C∗)

(A3) k1(π r C∗r )2 < 23k3λ0(δ + αC∗r )

(A4) (k2r C∗r + k3αC∗)2 <49k2k3(r0 + r C∗)(δ + αC∗r )

(A5) k4(αC∗ − ν C∗a)2 < 23k2(r0 + r C∗)(k + ν C∗r )

(A6) k4(αC∗r )2 < 23k3(δ + αC∗r )(k + ν C∗r )

Assuming k1 = 1, and we write the equation (A2) as

(π r C∗ − k2µ)2 + 4k2π r µC∗ <

2

3k2λ0(r0 + r C∗)

Choosing k2 = π r C∗

µ, above equation reduces to

(A7) π r µC∗ < λ06

(r0 + r C∗)

Similarly, from equation (A4), choosing k3 =π r2 C∗rαµ

it reduces to

282

(A8) rαC∗r C∗ < 1

9(r0 + r C∗)(δ + αC∗r )

From equation (A3), usingk3, we get,(A9) π αµC∗r <

23λ0(δ + αC∗r )

From equations (A5) and (A6), usingk2 and k3, we get

(A10) k4 <23

(k + ν C∗r )π rµ

min

(r0+r C∗)C∗

(αC∗−ν C∗a)2,

(δ+αC∗r ) r

α3 C∗r

Under conditions (A7) – (A9), V will be negative definite showing that V is a Liapunov’s function and hence thetheorem.

Appendix B

Proof of Theorem 3. Using the following positive definite function,

(B1) U = 12

[m1(Cd − C∗d )2 +m2(Cr − C∗r )2 +m3(C − C∗)2 +m4(Ca − C∗a)2]Differentiating with respect to‘t’ we get,

U = −m1λ0(Cd − C∗d )2 −m2(r0 + rC)(Cr − C∗r )2 −m3(δ + αCr)(C − C∗)2 −m4(k + ν Cr)(Ca − C∗a)2

+(m1π r C∗ +m2µ)(Cd − C∗d )(Cr − C∗r ) +m1π r Cr(Cd − C∗d )(C − C∗) − (m2r C

∗r +m3αC

∗)(Cr − C∗r )(C − C∗)

+m4(αC∗ − ν C∗a)(Cr − C∗r )(Ca − C∗a) + (m4αCr)(C − C∗)(Ca − C∗a)

Now U will be negative definite under the following conditions,(B2) (m1π r C∗ +m2µ)2 < 2

3m1m2λ0(r0 + r C)

(B3) m1(π r Cr)2 <23m3λ0(δ + αCr)

(B4) (m2r C∗r +m3αC∗)2 <49m2m3(r0 + r C)(δ + αCr)

(B5) m4(αC∗ − ν C∗a)2 < 23m2(r0 + r C)(k + ν Cr)

(B6) m4(αCr)2 <23m3(δ + αCr)(k + ν Cr)

Maximizing LHS and minimizing RHS, choosing m1 = 1and

m4 <2

3

π r k

µmin

r0C∗

(αC∗ − ν C∗a)2,δ r C∗rα3

(λm

λ

)2

Equations (B2) – (B6) reduce to

(B7) π r µC∗ < λ0r06

(B8) rαC∗r C∗ < r0δ

9

(B9) π αµC∗r <23λ0δ

Under conditions (B7) – (B9), U will be negative definite showing that U is a Liapunov’s function and hence thetheorem.



Effect of ion and electron streaming on the formation ofion-acoustic solitons in weakly relativistic magnetizedion-beam plasmas

S. N. Barman1 and A. Talukdar2

1,2Department of Mathematics, Arya Vidyapeeth College,Guwahati, Assam, India, 781016.E-mail: [email protected]

Abstract:

Existence of both compressive and rarefactive solitons is established in weakly relativisticbut magnetized plasma with cold ions and ion-beams in presence of electron inertia. It isobserved that rarefactive (compressive) solitons exist for smaller (higher) difference of electronand ion initial streaming speeds for different ion to ion-beam mass ratio and wave speed.Interestingly, the amplitude of the rarefactive solitons is seen to increase with the increase ofthe difference of initial streaming speeds to certain limit before turning out to be compressiveafter that limit.

Key words: KdV equation; soliton; relativistic plasma; nonlinear phenomena

1 Introduction

Nonlinear waves may play important roles in rarefied space plasmas in the Earth’s auroral zone[1,2,3], the physics of solar atmosphere [4], and other astrophysical plasmas[5,6]. Plasmas are anintrinsically nonlinear medium that can support a great variety of diverse waves. It has beenwell established both theoretically and experimentally that the behavior of ion-acoustic solitarywaves in collisionless plasmas can be described by the Korteweg-de Vries (KdV) equation in thesmall amplitude and long-wavelength region. During the last three decades or so, many workershave investigated the existence of solitary waves both theoretically [7,8,9,10,11] and experimentally[12,13,14,15,16] in different physical situations of plasma compound.

In recent space observations, it has been investigated that the high-speed streaming ions aswell as the electrons play a major role in the physical mechanism of the nonlinear wave structures.When we assume that the ion and electron energies depend only on the kinetic energy, velocities ofplasma particles in the solar atmosphere and the magnetosphere have to attain relativistic speeds[17,18]. Thus, by considering such relativistic effects on ion and electron velocities, one can takethe relativistic motion of such particles into consideration in the study of nonlinear plasma waves.When the velocity of the plasma particles approaches near to that of light, the nonlinear waveswhich occur in the space, exhibit a peculiar feature due to the effect of the high speed ions [19,20].Starting from the works of Das and Paul [21], many workers like Nejoh [20], Das et al.[21], Pakiraet al.[23], Kalita et al.[24], El-Labany and Shaaban [25], Singh et al.[26], Gill et al.[,27], Lee andChoi [28] have considered relativistic effects and investigated the existence of ion-acoustic solitarywaves under various physical situations in plasmas.

Corresponding author: S. N. Barman

284

In this paper, the investigation of ion-acoustic solitary waves in weakly relativistic magnetizedplasmas with cold ions and ion-beams together with isothermal electrons is carried out. Theexternal magnetic field B0z is taken along the z-direction, while the propagation of the wave isassumed to take place in a ξ-direction in the (x, z) plane, inclined an angle θ to the direction of themagnetic field. We assume that the ion-beam has a constant drift velocity Ud along the x-directionperpendicular to the direction of the magnetic field. The layout of this paper is as follows: Insection 2, we present the basic equations of motions for a relativistic three components plasmanamely cold ions, ion-beams and electrons together with the Poisson equation. Using perturbationmethod, KdV equation is derived from this basic set of equations. In section 3, condition for theexistence of solitons and solitary wave solutions are discussed. The last section 4 is devoted to theconcluding discussion.

2 Equations of motion and derivation of KdV equation

We consider a model of weakly relativistic magnetized plasma consisting of cold ions and ion-beamstogether with the isothermal electrons. The equations of motion governing the state of this plasmain a ξ- direction (inclined at an angle θ to the direction of the magnetic field) are as follows:

For the cold ions,∂ns∂t

+∂

∂ξ(nsvsξ) = 0, (2.1)(

∂

∂t+ vsξ

∂

∂ξ

)βsvsx = − e

mi0

(sin θ

∂φ

∂ξ−B0vsy

), (2.2)(

∂

∂t+ vsξ

∂

∂ξ

)βsvsy = − e

mi0B0vsx, (2.3)(

∂

∂t+ vsξ

∂

∂ξ

)βsvsz = − e

mi0cos θ

∂φ

∂ξ, (2.4)

For the ion-beams,∂nb∂t

+ Ud sin θ∂nb∂ξ

+∂

∂ξ(nbvbξ) = 0, (2.5)[

∂

∂t+ (vbξ + Ud sin θ)

∂

∂ξ

]βbvbx = − e

mb0

(sin θ

∂φ

∂ξ−B0vby

), (2.6)[

∂


∂

∂ξ

]βbvby = − e

mb0vbxB0, (2.7)[

∂


∂

∂ξ

]βbvbz = − e

mb0cos θ

∂φ

∂ξ, (2.8)

For the electrons,∂ne∂t

+∂

∂ξ(neveξ) = 0, (2.9)(

∂

∂t+ veξ

∂

∂ξ

)βevez =

1

me0

(e cos θ

∂φ

∂ξ− Tene

cos θ∂ne∂ξ

), (2.10)

With the Poisson equation,∂2φ

∂ξ2= 4π e(ne − ns − nb), (2.11)

where,

βr =

(1−

v2rξ

c2

)−1/2

≈ 1 +v2rξ

2c2, r = s, b, e.

285

We have normalized the physical quantities appearing in the set of equations (2.1) – (2.11) as:densities by the equilibrium plasma density n0, velocities by csσυ, potential φ by Te/e, distances

by the Debye length λD =(Te/4π n0e

2)1/2

and time by the ion plasma period ω−1pi = csσυ/λD =(

4π n0e2/mi0

)−1/2. We denote the ratio cold-ion rest mass (mi0) to ion-beam rest mass (mb0)

by α = mi0/mb0, electron rest mass (me0) to ion rest mass by Q = me0/mi0, the ion-acoustic

speed by cs = (Te/mi0)1/2

, the cyclotron frequency by ωBi = eB0/mi0, the ratio of plasmafrequency to cyclotron frequency by µ = ωpi/ωBi, and the direction of the wave propagationby συ(= cos θ). For fast ion-acoustic waves the ratio of plasma frequency to wave frequency islargeµ >> 1 i.e. ωpi >> ωBi( ωpi >> ωBi suggests weak magnetic resistance to the plasma wavesand so it is termed as “fast ion-acoustic” ignoring presence of strong magnetic field). In thissituation, we choose µ = bε−3/2, where b is greater than zero and depends upon the distributionof cold ions and ion-beams.

To derive the KdV equation, we introduce the stretched coordinates as

η = ξ −Mt, τ = ε t (2.12)

where M is the phase velocity of the ion-acoustic wave in (ξ, t)space, and ε is a small dimensionlessexpansion parameter. We expand the flow variables asymptotically about the equilibrium state interms of this parameter as follows:

ns = (1−Nb) + ε ns1 + ε2ns2 + .....,nb = Nb + ε nb1 + ε2nb2 + .....,ne = 1 + ε ne1 + ε2ne2 + .....,φ = εφ1 + ε2φ2 + .....,vez = vez0 + ε vez1 + ε2vez2 + .....,veξ = veξ0 + ε veξ1 + ε2veξ2 + .....,vjξ = vjξ0 + ε vjξ1 + ε2vjξ2 + .....,vjx = vjx0 + ε vjx1 + ε2vjx2 + .....,vjy = ε vjy1 + ε2vjy2 + .....,vjz = vjz0 + ε vjz1 + ε2vjz2 + ....., j = s, b.

(2.13)

With the use of the transformation (2.12) and the expansions (2.13) in the normalized set ofequations (2.1) – (2.11) subject to the boundary conditions

vjx1 = 0, vjy1 = 0, vjz1 = 0, vjξ1 = 0, (j = s, b), vez1 = 0, and φ1 = 0 as |η| → ∞ (2.14)

We get, from the ε-order equations, the following quantities:

vsξ1 = φ1

βs1(M−vsξ0)σ2υ, vbξ1 = αφ1

βb1(M−Ud sin θ−vbξ0)σ2υ, veξ1 =

(M−veξ0)φ1

1−Qβe1(M−veξ0)2 ,

ns1 = (1−Nb)φ1

βs1(M−vsξ0)2σ2υ, nb1 = α Nbφ1

βb1(M−Ud sin θ−vbξ0)2σ2υ, ne1 = φ1

1−Qβe1(M−veξ0)2 ,

ne1 − ns1 − nb1 = 0.

where, βr1 = 1 +

3v2rξ02c2 , r = s, b, e.

Using the values of ne1, ns1, nb1 in the last equation of above, we obtain the expression for thephase velocity Mas

1

1−Qβe1(M − veξ0)2− 1−Nbβs1(M − vsξ0)2σ2

υ

− α Nbβb1(M − Ud sin θ − vbξ0)2σ2

υ

= 0. (2.15)

Again from ε2- order equations obtained from the equations (2.1) – (2.11), with the use of the firstorder quantities, we can find the following equations:

2(1 −Nb)

βs1σ2υ(M − vsξ0)3

∂φ1

∂τ− ∂ns2

∂η+

(1 −Nb)

βs1(M − vsξ0)2σ2υ

∂φ2

∂η

+2(1 −Nb)c

2βs1(M − vsξ0) + (1 −Nb)[βs1c2 − 3vsξ0(M − vsξ0)]

c2β3s1σ

4υ(M − vsξ0)4

φ1∂φ1

∂η= 0, (2.16)

286

2α Nbβb1σ2

υ(M − Ud sin θ − vbξ0)3∂φ1

∂τ− ∂nb2

∂η+

α Nbβb1(M − Ud sin θ − vbξ0)2σ2

υ

∂φ2

∂η

+2α2Nbc

2βb1(M − Ud sin θ − vbξ0) + α2Nb[βb1c2 − 3vbξ0(M − Ud sin θ − vbξ0)]

c2β3b1σ

4υ(M − Ud sin θ − vbξ0)4

φ1∂φ1

∂η= 0, (2.17)

2Qβe1(M − veξ0)

[1 −Qβe1(M − veξ0)2]2∂φ1

∂η+∂ne2∂η

− 1

1 −Qβe1(M − veξ0)2∂φ2

∂η

+3Qβe1c

2(M − veξ0)2 − 3Q veξ0(M − veξ0)3 − c2

[1 −Qβe1(M − veξ0)2]3= 0, (2.18)

∂2φ1

∂η2= ne2 − ns2 − nb2 (2.19)

Combining the equations (2.16) – (2.19) and using (2.15), we obtain the KdV equation

∂φ1

∂τ+Aφ1

∂φ1

∂η+B

∂3φ1

∂η3= 0, (2.20)

whereA = A1/A2, B = B1/B2,

A1 = β3b1(M − Ud sin θ − vbξ0)4[1−Qβe1(M − veξ0)2]32(1−Nb)c2βs1(M − vsξ0)+ (1−Nb)[βs1c

2 − 3vsξ0(M − vsξ0)]+ β3s1(M − vsξ0)4[1−Qβe1(M − veξ0)2]32Nb

× α2c2βb1(M − Ud sin θ − vbξ0) + α2Nb[βb1c2 − 3vbξ0(M − Ud sin θ − vbξ0)]

+ c2β3s1β

3b1σ

4υ(M − vsξ0)4(M − Ud sin θ − vbξ0)4[3Qβe1c

2(M − veξ0)2

− 3Q veξ0(M − veξ0)3 − c2],

A2 = 2c2β2s1β

2b1σ

2υ(M − vsξ0)(M − Ud sin θ − vbξ0)[1−Qβe1(M − veξ0)2](1−Nb)βb1

× (M − Ud sin θ − vbξ0)3[1−Qβe1(M − veξ0)2]2 + α Nbβs1(M − vsξ0)3

× [1−Qβe1(M − veξ0)2]2 + 2Qβe1βs1βb1σ2υ(M − veξ0)(M − vsξ0)3(M − Ud sin θ − vbξ0)3,

B1 = βs1βb1σ2υ(M − vsξ0)3(M − Ud sin θ − vbξ0)3[1−Qβe1(M − veξ0)2]2,

B2 = 2(1−Nb)βb1(M − Ud sin θ − vbξ0)3[1−Qβe1(M − veξ0)2]2 + α Nbβs1(M − vsξ0)3

× [1−Qβe1(M − veξ0)2]2 + 2Qβe1βs1βb1σ2υ(M − veξ0)(M − vsξ0)3(M − Ud sin θ − vbξ0)3.

3 Condition for existence of solitons and solitary wave solution

The expression given in equation (2.15) shows that, irrespective of the magnitude of the streamingvelocities vjξ0(j = s, b, e), the phase velocity M has the relativistic impact through βs1, βb1 andβe1 on the plasma particles and thus affects the same on the dynamics of the soliton propagation.The soliton solution of the KdV equation (2.20) is possible if A and B(> 0) are non-zero and finitefor which we have that

Qβe1(M − veξ0)2 6= 1, Ud sin θ + vbξ0 6= M. (3.1)

To find solitary wave solution of the KdV equation (2.20), we introduce the variableχ = η−V τ ,where V is the soliton speed in the linear χ- space. Using the boundary conditions

φ1 =∂φ1

∂χ=∂2φ1

∂χ2= 0 as |χ| → ∞, (3.2)

equation (2.20) can be integrated to give

φ1 =3V

Asech2

(V

4B

)1/2

χ. (3.3)

287

Figure 1: Amplitudes of both compressive and rar-efactive ion-acoustic solitons versus direction ofwave propagation συ for |veξ0 − vsξ0| = 12.5(1),13.0(2), 13.5(3), 15.5(4), 39.5(5) when V =0.1, α = 0.1, Nb = 0.01, vsξ0 = 30.

Figure 2: Amplitudes of both compressive and rar-efactive ion-acoustic solitons versus wave velocity Vfor |veξ0 − vsξ0| = 20.5(1), 22.5(2), 23.5(3), 26.5(4),29.5(5), 39.5(6) when συ = 0.8, α = 0.1, Nb = 0.01,vsξ0 = 30.

Figure 3: Soliton amplitudes versus initial stream-ing difference of ion and electron for fixedV =0.1, α = 0.1, Nb = 0.01, vsξ0 = 30and for differ-ent values ofσυ = 0.9(1), 0.8(2), 0.7(3), 0.6(4).

Figure 4: Soliton amplitudes versus ion to beam-ion mass ratio α for fixed V = 0.05, Nb =0.1, |veξ0 − vsξ0| = 29.75, vsξ0 = 30and for highervalues of συ=0.9(1), 0.95(2), 0.97(3).

The amplitude and the width of the solitary waves are given respectively by

φ0 =3V

A,∆ =

(4B

V

)1/2

.

4 Discussion

In this model of plasma, both compressive and rarefactive solitons are established to exist whichare predicted to propagate depending on a definite range of difference of electron and ion streamingspeeds in various ξ-directions making an angle θ to the direction of the magnetic field B0z. Com-putational work reveals the existence range of solitons as 9 < |veξ0 − vsξ0| < 42.5. It is observedthat rarefactive (compressive) solitons exist for the smaller (higher) difference of electron and ion(or ion-beam) initial streaming speeds for different values of α, V,Nb. For small α = 0.1 that isfor heavy concentration of ion-beam mass in the plasma and for small ion-beam density Nb = 0.01and wave velocityV = 0.1, the amplitude of the rarefactive solitons are found to increase with

288

Figure 5: Soliton widths versus wave velocity forfixedα = 0.1, Nb = 0.01,|veξ0 − vsξ0| = 29.75,vsξ0 = 30 and for συ = 0.6, 0.7, 0.8, 0.9.

συ but in the contracted range of existence as the difference of electron and ion (or ion-beam)initial streaming speeds increases to a critical value (figure 1) before change over to compressivesolitons. But after the critical value of this difference, between 13.5 and 15.5 (figure 1) the for-mation of soliton turns out to be compressive character attaining relatively smaller amplitudesrather in the entire range. It is seen that the compressive solitons tend to disappear in the fullyexpanded range as the difference of the initial streaming speeds increases. Both compressive andrarefactive solitons (figure 2) of high amplitudes exist near the magnetic field for small α = 0.1 andion-beam density Nb = 0.01 quite with linear growth relative to wave velocity V when vsξ0 = 30as the difference of initial streaming speeds of electrons and ions increases. The amplitude of therarefactive solitons increases as the difference of initial streaming speeds increases to certain limitbut moment it crosses that limit, the solitons turn out to be compressive being consistent withthe characters of figure 1. Unlike rarefactive solitons, the amplitude of the compressive solitonsincreasingly diminishes with the increase of |veξ0 − vsξ0| and the wave velocity V . Compressiveand rarefactive solitons (figure 3) are found to exist for fixed V = 0.1, α = 0.1 and Nb = 0.01 in alldirections συ = 0.9, 0.8, 0.7, 0.6 as the difference of initial streaming speeds increases. But at thegreater (for compressive) and smaller (for rarefactive) difference of the initial streaming speeds,the small amplitudes remain almost constant. Further, it exhibits indication of disappearance ofthe non-linearity range in every direction of propagation showing ample scope of study of modifiedKdV solitons. Besides, this range has shifted to the smaller value of |veξ0 − vsξ0| as the directionof wave propagation deviates from that of the magnetic field. Interestingly, much concentrationof ions in the plasma compound is observed to generate both compressive and rarefactive highamplitude solitons and that too near the magnetic field (figure 4). The widths of the solitonsadmitting sharp fall for small V and attaining high values gradually diminishes at the increase ofthe wave velocity V (figure 5) in all directions συ when α = 0.1, Nb = 0.01 and vsξ0 = 30. In allour numerical calculations, we consider Q = 1/1836 with reference to the lightest ion.

References

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[4] H. Alfven, The plasma universe, Phys. Today 39(1986) 22-27.

[5] H. Alfven, Double layers and circuits in astrophysics, IEEE Trans. Plasma Sci. PS-14 (1986) 779-793.

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[12] H. Ikezi, R.J. Taylor, D.R. Baker, Formation and interaction of ion-acoustic solitons, Phys. Rev. Lett. 25(1970)11-14.

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[18] F.L. Scarf , F.V. Coroniti , C.F. Kennel , R.W. Fredricks , D.A. Gurnett , E.J.Smith, ISEE-3 wave measurementsin the distant geomagnetic tail and boundary layer, Geophys. Res. Lett. 11(1984) 335-338.

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Uniformly convergent scheme for Convection Diffusionproblem

K. Sharath babu1 and N. Srinivasacharyulu2

Department of Mathematics, National Institute of Technology,warangal-507004.Email: [email protected], 2 nitw [email protected]

Abstract:

In this paper a study of uniformly convergent method proposed by Ilin Allen-South wellscheme was made. A condition was contemplated for uniform convergence in the specifieddomain. This developed scheme is uniformly convergent for any choice of the diffusion param-eter. The search provides a first- order uniformly convergent method with discrete maximumnorm. It was observed that the error increases as step size h gets smaller for mid range valuesof perturbation parameter. Then an analysis carried out by [16] was employed to check thevalidity of solution with respect to physical aspect and it was in agreement with the analyti-cal solution. The uniformly convergent method gives better results than the finite differencemethods. The computed and plotted solutions of this method are in good agreement with theexact solution.

Key words: Boundary layer; Peclet number; Uniformly convergence; Perturbation pa-rameter.

1 Introduction

Consider the elliptic operator whose second order derivative is multiplied by a parameter ε that isclose to zero. These derivatives model diffusion while first-order derivatives are associated with theconvective or transport process. In classical problems ε is not close to zero. This kind of problemthat was studied in the paper [17]. To summarize when a standard numerical method is applied toa convection-diffusion problem, if there is too little diffusion then the computed solution is oftenoscillatory, while if there is superfluous diffusion term, the computed layers are smeared. There isa lot of work in literature dealing with the numerical solution of singularly perturbed problems,showing the interest in this nature of problems in Kellog et al [10], Kadalbajoo et al [9], Bender[4], Robert E.O’s Malley, Jr [8], Mortan [13] and Miller et al [12]. We can see that the solutionof this problem has a convective nature on most of the domain of the problem, and the diffusivepart of the differential operator is influential only in the certain narrow sub-domain. In this regionthe gradient of the solution is large. This nature is described by stating that the solution has aboundary layer. The interesting fact that elliptic nature of the differential operator is disguised onmost of the domain, it means that numerical methods designed for elliptic problems will not worksatisfactorily. In general they usually exhibit a certain degree of instability.

Corresponding author: K. Sharath babu

291

2 Motivation and History

The numerical solution of convection-diffusion problems dates back to the 1950s, but only in the1970s it did acquire a research momentum that has continued to this day. In the literature thisfield is still very active and as we shall see more effort can be put in. Perhaps the most commonsource of convection-diffusion problem is the Navier Stokes equation having nonlinear terms withlarge Reynolds number. Morton [13] pointed out that this is by no means the only place wherethey arise. He listed ten examples involving convection diffusion equations that include the drift-diffusion equations of semiconductor device modeling and the BlackSholes equation from financialmodeling. He also observed that accurate modeling of the interaction between convective anddiffusive processes is the most ubiquitous and challenging task in the numerical approximation ofpartial differential equations.

In this paper, the diffusion coefficient ε is a small positive parameter and coefficient of convectiona(x) is continuously differentiable function.

Consider the convection diffusion problem

Lu(x) = −εu′′(x) + a(x)u′(x) + b(x)u(x) for0 < x < 1

with u(0) = u(1) = 0(2.1)

Where 0 < ε 1, a(x) > α > 0 and b(x) ≥ 0 on [0, 1] , Here assume that

a(x) ≤ 1

The above problem is solved by the method proposed by the Il’in Allen uniformly convergentmethod. The convergence criterion is realized through computation, based on explanation givenby Roos et al [16], for lower values of the diffusion coefficient. The reciprocal of the diffusioncoefficient is called the Piclet number. For a finite Piclet number the solution patterns matcheswith the exact solution.

3 Construction of a Uniformly Convergent Method

We describe a way of construction of uniformly convergent difference scheme. We start with thestandard derivation of an exact scheme for the convection-diffusion problem (2.1). Introduce theformal adjoint operator L∗ of L and for the sake of convenience select b = 0 in (2.1)

Let gi be local Greens function of L∗ with respective to the argument xi; i.e.,

L∗gi = − ε g′′

i −a g′

i = 0 in (xi−1, xi)⋃

(xi , xi+1 ) (3.1)

Let us impose boundary conditions

gi(xi−1) = gi(xi+1) = 0 (3.2)

And impose additional conditions

ε (g′

i (x−i ) − g′

i ( x+i ) ) = 1

Equation (2.1) is multiplied by gi , integrated with respective to x between the limits xi−1 andxi+1 to get ∫ xi+1

xi−1

(Lu) gi dx =

∫ xi+1

xi−1

f gidx∫ xi+1

xi−1

(−εu′′(x) + a u

′(x)) gi dx =

∫xi+1

xi−1

fgidx (3.3)

292

Now L.H.S of (3.3) :

=

∫ xi

xi−1

(−εu′′(x) + a u

′(x))gidx+

∫ xi+1

xi

(−ε u′′(x) + a u

′(x)gidx

= (−εu′+ au) gi(x)

xi|

xi−1

+(−εu′

+ au) gi(x) |xi+1

xi

−∫ xi

xi−1

(−ε u′

+a u) g′

i dx−∫ xi+1

xi

(−ε u′+a u) g

′

i dx

= [−ε u′(x−

i) + a u(xi)] gi(xi)− [−ε u

′(xi−1) + a u(xi−1)] gi(xi−1)

+ [−ε u′(xi+1) + a u (xi+1)] gi( xi+1)− (−ε u

′(x+i ) + a u(xi)) gi(xi)]

−∫ xi

xi−1

(a u)g′

idx −∫ xi+1

xi

(a u) g′

i dx +

∫ xi

xi−1

(ε u′) g′

i dx +

∫ xi+1

xi

(ε u′) g′

i dx

= −ε u′(x−i ) gi(xi) + ε u

′(x+i ) gi(xi) + [εu(x) g

′

i(x)]xixi−1+ [εu(x) g

′

i(x)]xi+1xi

+

∫ xi

xi−1

(−εg′′

i − ag′

i)udx +

∫ xi+1

xi

(−εg′′

i − a g′

i)udx

Since u′

is continuous on (xi−1,xi+1), we have

= [εu(xi) g′

i(x−i ) − ε u(xi−1) g

′

i(x+i−1)] + [εu(xi+1) g

′

i(x−i+1)− εu (xi) g

′

i(x+i )]

= −ε g′

i(xi−1) ui−1 + ui + ε g′

i(xi+1) ui+1 = f

∫ xi+1

xi−1

gi dx(3.4)

The difference scheme of equation (3.1) is exact. We can able to evaluate each g′

i exactlyThe solution of the equation (3.1) is given by

gi(x−) = c1 + c2 (

−εa

) e−axε on(xi−1, xi+1) (3.5)

gi(x+) = c

′

1 + c′

2 (−εa

) e−axε on(xi−1, xi+1) (3.6)

Here there are 4 unknowns c1 , c2,c′

1 , c′

2 requiring 4 equations

gi (xi−1 ) = 0 (3.7)

gi (xi+1 ) = 0 (3.8)

ε ( g′

i ( x−i ) − g′

i ( x+i ) ) = 1 (3.9)

and, from continuity of gi at x=xi

gi (x−i ) = gi (x+i ). (3.10)

On imposing boundary conditions (3.7) and (3.8) on (3.5), (3.6) it can be seen

gi (xi−1 ) = c1 + c2 (−εa

) e−axi−1

ε = 0 (3.11)

gi (xi+1 ) = c′

1 + c′

2 (−εa

) e−axi+1

ε = 0 (3.12)

293

On differentiation of equations (3.5), (3.6)

g′

i(x−i ) = c2(− ε

a)(−aε

) e−axiε , g

′

i(x+i ) = c

′

2(− εa

)(−aε

) e−axiε

Then the equation (3.9) can be written in the following form

ε(c2 e−axiε −c

′

2 e−axiε ) = 1⇒ c2 − c

′

2 =1

εeaxiε (3.13)

Using the fact gi (x−i ) = gi (x+i ) at x= xi in (3.11) ,(3.12) it follows

⇒ c1 + c2(−εa ) e−axiε − [c

′

1 + c′

2(−εa ) e−axiε ] = 0 (3.14)

On assumption that αi = a xiε , ρi = ah

ε , above equations may be rewritten as

eaxi+1ε = e

a(xi+ h)

ε = eαi +ρi , eaxi−1ε = eαi −ρi

Hence on transformation of the equations (3.11) to (3.14) in to the equations (3.15) to (3.18)

c1 + c2 (−εa

) e−αi +ρi = 0 (3.15)

c′

1 + c′

2(−εa

) e− (αi +ρi ) = 0 (3.16)

c2 − c′

2 =1

εeαi (3.17)

(c1 − c′

1) + (c2 − c′

2)(− εa

) e−αi = 0 (3.18)

On insertion of (3.17) into the equation (3.18)

(c 1 − c′

1) +1

εeαi(−εa

) e−αi = 0

(c 1 − c′

1) =1

a(3.19)

Subtracting the equation (3.16) from the equation (3.15) , then by using equations (3.17) & (3.19)it may be obtained

(c 1 − c′

1) + (c2 e−αi +ρi − c

′

2 e−αi−ρi)(

−εa

) = 0

1

a+ (c2 e

−αi + ρi −(c2 −1

εeαi)(e−(αi +ρi)(

−εa

) = 0

1

a+ (c2 e

−αi + ρi − c2 e−(αi +ρi ) +1

εeαi e−αi −ρi )(

−εa

) = 0 (3.20)

From (3.20) it follows

c2 =eαi

ε

(1− e−ρi)

(eρi − e−ρi )(3.21)

To find c′

2 the value of c2 is substituted in (3.17) , to get

c′

2 =eαi

ε

(1− eρi)

(eρi − e−ρi )(3.22)

294

Again employing the value of c2 in (3.15) the value of c1 can be obtained as

c1 =1

a

eρi − 1

(eρi − e−ρi )(3.23)

Next the value of c1 is used in (3.19) to obtain c′

1

c′

1 =1

a

e−ρi − 1

(eρi − e−ρi)(3.24)

Now on imposition of equations (3.21)- (3.24) , on (3.5) , (3.6) they may be rewritten as

gi( x− ) =

1

a

eρi − 1

(eρi − e−ρi)+eαi

ε

(1− e−ρi)(eρi − e−ρi)

(−εa

) e−axε (3.25)

gi( x+ ) =

1

a

e−ρi −1

(eρi − e−ρi)+eαi

ε

(1− eρi )

(eρi − e−ρi)(−εa

) e−axε (3.26)

The derivatives of equations (3.25) , (3.26) are

g′

i(x−) =

1

εe−axε e

axiε

(1− e−ρi )

(eρi − e−ρi)(3.27)

g′

i(x+) =

1

εe−axε e

axiε

(1− eρi )

(eρi − e−ρi)(3.28)

Now from (3.27) , (3.28) and (3.9) it follows.

g′

i (x−i−1 ) =1

εeahε

(1− e−ρi )

(eρi − e−ρi)i.e.

g′

i (x−i−1 ) =1

ε

(eρi − 1)

(eρi − e−ρi)(3.29)

g′

i (x+i+1) =1

ε

(e−ρi − 1)

(eρi − e−ρi)(3.30)

Now by inserting values of gi+ and gi

−− from (3.29) , (3.30) in (3.2) & (3.3) it may be obtained

f

∫ xi+1

xi−1

gi dx = f [

∫ xi

xi−1

g−i dx +

∫ xi+1

xi

g+i dx ] where ρi =

ah

ε, αi =

axiε

=

∫ xixi−1

[ 1a

eρi −1(eρi −e−ρi )

+ eαi

ε( 1− e−ρi )(eρi −e−ρi ) (−εa ) e

−axε ] dx +

∫ xi+1

xi

[1

a

e−ρi − 1

(eρi − e−ρi )+eαi

ε

( 1− eρi)

(eρi − e−ρi)(−εa

) e−axε ] dx[

h

a

(eρi − 1 )

(eρi − e−ρi)

]+ [

ε

a2eαi e

−axiε

(1− e−ρi )

(eρi − e−ρi)(1 − e

ahε ) ] +[

h

a

(e−ρi − 1 )

(eρi − e−ρi )

]+ [

ε

a2eαi e

−axiε

(1− eρi )

(eρi − e−ρi)( e− ahε − 1) ]

=h

a

(eρi + e−ρi − 2)

(eρi − e−ρi)+ [

ε

a2eαi e

−axi

ε ((1− e−ρi) (1− eρi) + (1− eρi)(e−ρi − 1 )

(eρi − e−ρi )) ]

=h

a

(eρi2 − e

−ρi2 )2

(eρi2 − e

−ρi2 ) (e

ρi2 + e

−ρi2 )

=h

a

(eρi − 1 )

(eρi + 1)

295

Finally, it can be represented as follows

f

∫ xi+1

xi−1

gi dx = fh

a

( eρi − 1 )

(eρi + 1)This gives the final scheme as

− (eρi − 1 )

(eρi − e−ρi)ui−1 + ui −

1− e−ρi(eρi − e−ρi)

ui+1 = fh

a

( eρi − 1 )

(eρi + 1)(3.31)

here ρi =ah

ε.

The equation (3.31) is the Il’in-Allen scheme.This method is tested for a linear problem by applying various perturbation parameter values

with in the defined range. I t is observed from the numerical results that Il’in-Allen scheme isconverging uniformly in the entire domain. In the boundary layer region , it is appreciable thingthat the scheme is uniformly converging one. For testing the algorithm outlined above the two-point boundary value problem

−ε u′′

(x) + u′(x) = 2xwithu(0) = u(1) = 0 (3.32)

Is considered with ‖a(x)‖ ≤ 1The analytical solution of (3.32) is

u(x) =(1 + 2ε)

(e1ε − 1)

− (1 + 2ε )

(e1ε − 1)

exε + x2 + 2 ε x , 0 < ε << 1 (3.33)

The computational method is executed with various choices of the diffusion co-efficient byapplying forward difference method, upwind method, central difference method and the Il’in-Allenscheme. The results obtained are presented in the table.

4 Error Analysis:

The present scheme is first-order uniformly convergent in the discrete maximum norm, i.e.,

Maxi|u(xi)− ui | ≤ Ch

The region of solution u is divided into two parts, (2.1) smooth region with bounded derivatives2) boundary layer region with chaotic behavior where in u = v + z , where v is a boundary layerfunction and the bound on the smooth function

∣∣zj ∣∣ has a factor ε1−j

The calculation of |z (xi) − zi | is now considered. The corresponding consistency error |τi| isestimated with the help of Taylor series, proposed by H.G. Roos et al [16] which give the inequality

|τ i | ≤ C

∫ xi+1

xi−1

(ε∣∣z3 (t)

∣∣ + a∣∣∣z′′ (t)

∣∣∣ ) dt≤ Ch + C ε−1

∫ xi+1

xi−1

exp(−a01− tε

)dt

≤ Ch + C sinh(a0 h

ε) exp ( − ao

1− xiε

).

An application of the discrete comparison principle indicates the increase of power of εi.e., |z (xi) − zi | ≤ Ch + C sinh(a0 hε ) exp ( − ao 1−xi

ε ) for i= 1,2 ,3,. . . . . .nfor ε ≤ h that can be easily obtained

| z(xi ) − zi | ≤ Ch.

In the second case h ≤ ε , using the inequality 1− e−t ≤ ct for t > 0 the desired desired estimatecan be put as | z(xi) − zi | ≤ Ch

296

Table 1: Case1 : ε = 0.05

x Forwardscheme Backwardscheme CentralScheme Allen-Il’in scheme Exact solution0 0 0 0 0 00.01 0.001000 0.001200 0.001100 0.001103 0.00109990.02 0.002200 0.002600 0.002400 0.002407 0.00239990.03 0.003600 0.004200 0.003900 0.005613 0.00389990.04 0.005200 0.006000 0.005600 0.005613 0.0055990.05 0.007000 0.008000 0.007500 0.007517 0.00749990.06 0.009000 0.010200 0.009600 0.009620 0.00959990.07 0.011200 0.012600 0.011900 0.011923 0.01189990.08 0.013600 0.015200 0.014400 0.014427 0.01439990.09 0.016200 0.018000 0.017100 0.017130 0.01709990.1 0.019000 0.02100 0.020000 0.020033 0.0199990.2 0.058000 0.062000 0.060000 0.060067 0.065099850.3 0.123999 0.122998 0.119999 0.120100 0.1270980.4 0.204997 0.203985 0.19995 0.200129 0.2090910.5 0.305981 0.304899 0.299960 0.300127 0.29995000.6 0.426848 0.425363 0.419700 0.419895 0.419630990.7 0.566617 0.563036 0.557771 0.557961 0.55727330.8 0.716180 0.703420 0.703422 0.703453 0.69985270.9 0.790694 0.756763 0.776690 0.756044 0.751131190.91 0.780071 0.745514 0.768388 0.767636 0.7372710.92 0.762193 0.728374 0.754197 0.753337 0.74631380.93 0.735194 0.704127 0.732763 0.731799 0.71664330.94 0.696747 0.671311 0.702433 0.701373 0.7062860.95 0.643937 0.628171 0.661185 0.660049 0.68661330.96 0.573125 0.572603 0.606549 0.605369 0.6462860.97 0.479760 0.502081 0.535505 0.534331 0.5928320.98 0.358154 0.413575 0.444363 0.44321 0.43420720.99 0.20119 0.167335 0.182736 0.182179 0.178496171 0 0 0 0 0

Similarly | v(xi ) − vi | ≤ C h2

h+ε ≤ Ch as proposed by Kellog et al [10].This shows that Il’in-Allen scheme is uniformly convergent of first order.In the above scheme the absolute value of a(x) the convection coefficient is less than or equal

to unity, the scheme converges faster to the exact solution.

5 Result Analysis

We have solved the problem by using forward difference scheme, upwind scheme, central differencescheme and Il’in- Allen scheme by selecting the step width h = 0.01 and varying the perturbationparameter or diffusion coefficient . We have selected ε = 0.05, 0.001, 0.0001, 0.00001.

1. for ε =0.05 all the schemes behaves similarly in the smooth region as well as in the boundarylayer region.

2. for ε = 0.001 forward scheme is not matching with the exact solution , upwind schemeconverging to exact solution well and the central difference scheme converges in the smoothregion and oscillates in the boundary layer. where as Il’in scheme converges uniformly in theentire region.

3. for ε = 0.0001, 0.00001 forward scheme diverges , central scheme oscillates . Upwind schemehas given good numeric results in the specified domain. But at the boundary i.e near tothe point x=1 the upwind scheme is not matching with the exact solution. The solution ofthe upwind scheme is not uniformly convergent in the discrete maximum norm due to itsbehavior in the layer, where as the proposed scheme is uniformly convergent of first ordereven for lower values of ε through out the domain.

297

Table 2: Case2 : ε = 0.001 = 10−3

x Forwardscheme Backwardscheme CentralScheme Allen-Il’in scheme Exact solution0 0 0 0 0 00.01 -0.12447 0.000220 0.000120 0.00020 0.000120.02 -0.99928 0.000640 0.000440 0.000600 0.000440.03 -1.01274 0.001260 0.000960 0.001200 0.000960.04 -1.01058 0.002080 0.001680 0.002 0.001680.05 -1.00993 0.003100 0.002600 0.0030 0.0026000.06 -1.00080 0.004320 0.003720 0.04200 0.00371990.07 -0.00858 0.005740 0.005040 0.005600 0.0050400.08 -0.99616 0.007360 0.006560 0.007200 0.0065600.09 -0.99354 0.009180 0.008280 0.009000 0.008280.1 -0.99072 0.011200 0.010200 0.01100 0.010200.2 -0.97362 0.042400 0.040400 0.042000 0.040400.3 -0.92442 0.093600 0.090600 0.093000 0.09060.4 -0.85522 0.164800 0.160800 0.164000 0.1608000.5 -0.76602 0.256000 0.251000 0.255000 0.2510000.6 -0.65682 0.367200 0.361200 0.366001 0.36119990.7 -0.52762 0.498400 0.491404 0.497001 0.491400.8 -0.37842 0.649600 0.641805 0.648001 0.6416000.9 -0.20922 0.820800 0.823617 0.819001 0.811800.91 -0.19120 0.839020 0.812195 0.837201 0.8299200.92 -0.17298 0.857440 0.874828 0.855601 0.8482400.93 -0.15456 0.876060 0.826879 0.874201 0.8667600.94 -0.13594 0.894880 0.945302 0.893001 0.8854800.95 -0.11712 0.913899 0.814667 0.912001 0.9044000.96 -0.00981 0.933114 1.058120 0.931201 0.923520.97 -0.07888 0.952469 0.740940 0.950601 0.94283990.98 -0.05946 0.971384 1.265210 0.970201 0.96235990.99 -0.02002 0.91816 1.683413 0.990001 0.98203451 0 0 0 0 0

Table 3: Case3 : ε = 0.0001 = 10−4

x Forwardscheme Backwardscheme CentralScheme Allen-Il’in scheme Exact solution0 0 0 0 0 00.01 -1.020404 0.000202 -0.03588 0.000200 0.0001020.02 -1.009895 0.000604 0.00187 0.000600 0.0004040.03 -1.009597 0.001206 -0.03661 0.001200 0.0009060.04 -1.008994 0.002008 0.00466 0.002000 0.0016080.05 -1.008192 0.003010 -0.03666 0.003000 0.00251000.06 -1.00719 0.004212 0.00839 0.004200 0.00361990.07 -1.005988 0.005614 -0.03605 0.007200 0.00491400.08 -1.004586 0.007216 -0.01306 0.007200 0.0064160.09 -1.002984 0.009018 -0.03479 0.009000 0.008180.1 -1.001182 0.011020 0.01869 0.011000 0.010020.2 -0.972162 0.042040 0.06165 0.042000 0.0400400.3 -0.923142 0.093060 0.13098 0.093000 0.090060.4 -0.854122 0.164080 0.22981 0.164000 0.160080.5 -0.765102 0.255100 0.36280 0.255000 0.2501000.6 -0.656082 0.366120 0.53694 0.366000 0.3601200.7 -0.527062 0.497140 0.76261 0.497000 0.4901400.8 -0.378042 0.648100 1.05533 0.648000 0.6401600.9 -0.209022 0.819180 1.43826 0.819000 0.810180.91 -0.191020 0.837382 0.13857 0.837200 0.8282820.92 -0.172818 0.855784 1.52845 0.855600 0.8465840.93 -0.154416 0.874386 0.11939 0.874200 0.8650860.94 -0.135814 0.893188 1.62392 0.893000 0.8837880.95 -0.117012 0.912190 0.09635 0.912000 0.9026900.96 -0.098010 0.931216 1.72504 0.931200 0.9217920.97 -0.078808 0.950616 0.06906 0.950600 0.9410940.98 -0.059406 0.970216 1.83223 0.970200 0.96059590.99 -0.020002 1.008987 0.03709 0.990000 0.9802981 0 0 0 0 0

298

Table 4: Case4 : ε = 0.00001 = 10−5

x Forwardscheme Backwardscheme CentralScheme Allen-Il’in scheme Exact solution0 0 0 0 0 00.01 -1.011037 0.000200 -0.818348 0.00200 0.0001000.02 -1.009825 0.000600 0.003681 0.000600 0.01210220.03 -1.009426 0.001201 -0.820841 0.001200 0.01440240.04 -1.008825 0.002001 0.008188 0.002000 0.01690260.05 -1.008025 0.003001 -0.822561 0.003000 0.01960280.06 -1.007025 0.004201 0.013522 0.004200 0.0225030.07 -1.005825 0.005601 -0.082350 0.005600 0.02560320.08 -1.004424 0.007202 0.019682 0.007200 0.02890340.09 -1.002824 0.009002 -0.823680 0.009000 0.03240360.1 -1.001024 0.011002 0.026669 0.01100 0.03610280.2 -0.972021 0.042004 0.074019 0.042000 0.08410580.3 -0.923018 0.093006 0.142077 0.193000 0.15210780.4 -0.854015 0.164008 0.230871 0.264000 0.24010970.5 -0.765012 0.255010 0.340432 0.355000 0.34811170.6 -0.656009 0.366011 0.470792 0.466000 0.47611380.7 -0.527007 0.497013 0.621983 0.697000 0.62411580.8 -0.378005 0.648014 0.794038 0.74800 0.79211780.9 -0.209002 0.819016 0.986993 0.819000 0.810010.91 -0.191002 0.837216 -0.168016 0.837200 0.8270040.92 -0.172802 0.855616 1.028095 0.855600 0.8482820.93 -0.154402 0.874216 -0.135937 0.874200 0.86506800.94 -0.135801 0.89016 1.070035 0.893000 0.8737880.95 -0.117001 0.912016 -0.103095 0.912000 0.9006910.96 -0.098001 0.931216 1.112814 0.931200 0.9200060.97 -0.078801 0.950616 -0.069491 0.950600 0.9400020.98 -0.059401 0.970216 1.156430 0.970200 0.96002310.99 -0.039800 1.008987 -0.035126 0.99000 0.9800981 0 0 0 0 0

4. For finite value of the Peclet number Il’in-Allen scheme behaves well with the exact solutionin the region [0,1] .

5. The standard finite difference scheme of upwind and central scheme on equally spaced meshdoes not converge uniformly. Because, the point wise error is not necessarily reduced bysuccessive uniform improvement of the mesh in contrast to solving unperturbed problems.The standard central difference scheme is of order O(h2) .It is numerically unstable in theboundary layer region and gives oscillatory solutions unless the mesh width is small compar-atively with the diffusion coefficient but it is practically not possible as diffusion coefficientis very small.

6. For any value of x in [0,1] , a(x)=1 Il’in- Allen scheme converges uniformly. This has beenthoroughly verified through computation.

References

[1] V.B. Andreev and N.V. Kopteva, Investigation of difference Schemes with an approximation of the first deriva-tive by a central difference relation, Zh. Vychisl. Mat.i Mat. Fiz. 36 (1996), 101–117.

[2] Arthur E.P. Veldman, Ka-Wing Lam, Symmetry-preserving upwind discretization of convection on non-uniformgrids. Applied Numerical Mathematics 58 (2008).

[3] A. Brandt and I. Yavneh, Inadequacy of first-order upwind difference schemes for some recirculating flow, J.Comput. Phys. 93 (1991), 128-143.

[4] C.M. Bender , S.A.Orszag, Advanced Mathematical Methods for Scientists and Engineers , McGraw-Hill , NewYork, (1979).

[5] J. C. Butcher, Numerical Methods for Ordinary Differential Equations , Second edition ,John wiley & Sons,Ltd.

[6] D. Gilbarg, N.S.Trudinger, Elliptic partial differential equation of second order, springer, Berlin, (1983).

299

Figure 1Figure 2

Figure 3 Figure 4

[7] A. M. Il’in, A difference scheme for a differential equation with a Small Parameter multiplying the highestderivative, Mat. Zametki, 6 (1969),237–248.

[8] E.O. Robert, Introduction to singular Perturbationproblems, Malley,Jr, Academic press.

[9] M. K. Kadalbajoo, Y.N. Reddy, Asymptotic and numerical analysis of singular Perturbation problems: asurvey, Appl. Math.Comp. 30 (1989), 223-259.

[10] R.B. Kellog , A. Tsan, Analysis of some difference approximations for a singularly Perturbed problem withoutturning points. Math. Comp., 32 (1978), 1025–1039.

[11] Martin Stynes, Steady-state convection-diffusion problems, Acta Numerica (2005), 445–508.

[12] J. Miller, E. O’Riordan, G. Shishkin, Fitted Numerical Methods for Singularly Perturbed problems, WorldScientific, Singapore, (1996).

[13] K. W. Morton (1996), Numerical solution of Convection-Diffusion problems, Applied Mathematics and Math-ematical Computation, Vol. 12, Chapman & Hall, London , 1995.

[14] Mikhail Shashkov, Conservative finite difference methods on General grids, CRS Press(Tokyo), (2005).

[15] Dennis G. Roddeman, Some aspects of artificial diffusion in flow analysis, TNO Building and ConstructionResearch , Netherlands.

[16] H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Con-vection –Diffusion and Flow Problems Springer,Berlin, (1996).

[17] N. Srinivasacharyulu, K. Sharath babu, Computational method to solve Steady-state convection-diffusion prob-lem, Int. J. of Math, Computer Sciences and Information Technology, 1 (2008), 245–254.

[18] M. Stynes and L. Tobiska, A finite difference analysis of a streamline Diffusion method on a Shishkin meshes,Numer. Algorithms, 18 (1998), 337–360.



Thermal radiation effects on hydro-magnetic flow due to anexponentially stretching sheet

P. Bala Anki Reddy1 and N. Bhaskar Reddy2

1Department of Mathematics, NBKRIST,Vidyanagar, Nellore, A.P., India-524413.

2Department of Mathematics, S.V. University,Tirupati-517502, A.P. (INDIA).Email: [email protected], [email protected]

Abstract:

A steady laminar two-dimensional boundary layer flow of a viscous incompressible ra-diating fluid over an exponentially stretching sheet, in the presence of transverse magneticfield is studied. The non-linear partial differential equations describing the problem underconsideration are transformed into a system of ordinary differential equations using similar-ity transformations. The resultant system is solved by applying Runge-Kutta fourth ordermethod along with shooting technique. The flow phenomenon has been characterized by thethermo physical parameters such as magnetic parameter (M), radiation parameter (R) andEckert number (E). The effects of these parameters on the fluid velocity, temperature, wallskin friction coefficient and the heat transfer coefficient have been computed and the resultsare presented graphically and discussed quantitatively.

Key words: Thermal radiation; MHD; Boundary layer flow; exponentially stretchingsheet.

1 Introduction

The study of viscous incompressible flow over a stretching surface has become increasing importantin the recent years due to its numerous industrial applications such as the aerodynamic extrusionof plastic sheets, the boundary layer along a liquid film, condensation process of metallic plate ina cooling bath and glass, and also in polymer industries. Since the pioneering work of Sakiadis[1], various aspects of the stretching flow problem have been investigated by many authors likeCortell [2], Xu and Liao [3], Hayat et al [4] and Hayat and Sajid [5]. On the other hand, Guptaand Gupta [6] stressed that realistically, stretching surface is not necessarily continuous. Due tothe fact that the rate of cooling influences the quality of the product with desired characteristics,Ali [7] has investigated the thermal boundary layer flow by considering the nonlinear stretchingsurface. Further, a new dimension has been added to this investigation by Elbashbeshy [8] whoexamined the flow and heat transfer characteristics by considering an exponentially stretchingcontinuous surface. He considered an exponential similarity variable and exponential stretchingvelocity distribution on the coordinate considered in the direction of stretching.

Corresponding author: P. Bala Anki Reddy

301

For some industrial applications such as glass production and furnace design, electrical powergeneration, Astrophysical flows, solar power technology, which operates at high temperatures,radiation effects, can be significant.

An extensive literature that deals with flows in the presence of radiation is now available. Cortell[9] has solved a problem on the effect of radiation on Blasius flow by using fourth –order Runge-Kutta approach. Later, Sajid and Hayat [10] considered the influence of thermal radiation on theboundary layer flow due to an exponentially stretching sheet by solving the problem analyticallyvia homotopy analysis method (HAM). Recently, El-Aziz [11] studied effects on the flow and heattransfer an unsteady stretching sheet. Bidin and Nazar [12] studied the boundary layer flow overan exponential stretching sheet with thermal radiation, using Keller-box method.

There has been a renewed interest is studying magnetohydrodynamic flows and heat transfer dueto the effect of magnetic fields on the boundary layer flow control and on the performance of manysystems involving electrically conductive flows. In addition, this type of flow finds applicationsin many engineering problems such as MHD generators, Plasma studies, Nuclear reactors, andGeothermal energy extractions.

Raptis et al. [13] studied the effect of thermal radiation on the magnetohydrodynamic flow ofa viscous fluid past semi-infinite stationary plate and Hayat et al. [14] extended the analysis for asecond grade fluid. Later Aliakbar et al. [15] analyzed the influence of thermal radiation on MHDflow of Maxwellian fluids above stretching sheets.

In this paper an attempt is made to investigate the effects of thermal radiation on the studylaminar two dimensional boundary layer flow of a viscous incompressible electrically conductiveand radiating fluid over an exponentially stretching sheet. The governing boundary layer equationsare solved using Runge–Kutta fourth order along with shooting technique.

2 Mathematical Formulation

A two dimensional boundary layer flow of a viscous incompressible electrically conductive and ra-diative fluid bounded by a stretching surface is considered. The x-axis is taken along the stretchingsurface in the direction of the motion and y-axis perpendicular to it. The fluid is assumed to begray, absorbing-emitting but non scattering. A uniform magnetic fluid is applied in the direc-tion perpendicular to be stretching surface. The transverse applied magnetic field and magneticReynolds number are assumed to be very small, so that the induced magnetic field is negligible.Then under the above assumptions, in the absence of an input electric field, the governing boundarylayer equations are:

∂u

∂x+∂v

∂y= 0 (2.1)

u∂u

∂x+ v

∂u

∂y= υ

∂2u

∂y2− σB2

0

ρu (2.2)

ρcp

(u∂T

∂x+ v

∂T

∂y

)= k

∂2T

∂y2− ∂qr∂y

+ µ

(∂u

∂y

)2

(2.3)

where u and v are the velocities in the x-and y-directions respectively, υ-the kinematic viscosity,σ-the electrical conductivity,B0-the magnetic induction, ρ-the fluid density,cp-the specific heat atconstant pressure, T- the temperature,k- the thermal conductivity, qr- the radiative heat flux andµ- the dynamic viscosity.

The second and third terms on the right hand side of Equation (2.3) represent the radiativeheat flux and the viscous dissipative heat.

The boundary conditions for the velocity and temperature fields are:

u (0) = U0exL , v (0) = 0, T (0) = T∞ + T0e

2xL ,

u→ 0, T → 0 as y →∞ (4)

302

in which U0- the reference velocity, T0 and T∞- the temperatures at far away from the plateand L- the constant. By employed Rosseland approximation (Sajid and Hayat 2008), the radiativeheat flux is given by

qr = −4σ∗

3k∗∂T 4

∂y(2.4)

where σ∗ is the Stefan- Boltzmann constant and k∗- the mean absorption coefficient.We should be noted that the by using the Rosseland approximation, the present analysis is

limited to optically thick fluids. If the temperature differences with in the flow field are sufficientlysmall, then Equation (2.4) can be linearized by expanding T 4 into the Taylor series aboutT∞,which after neglecting higher order forms takes the form

T 4 ∼= 4T 3∞T − 3T 4

∞ (2.5)

Invoking Equations (2.3),(2.4) and (2.5), it can be written as

ρcp

(u∂T

∂x+ v

∂T

∂y

)=

(k +

16σ∗T 3∞

3k∗

)∂2T

∂y2+ µ

(∂u

∂y

)2

(2.6)

Introducing the following non-dimensional quantities

u = U0exL f ′(η),

v = −√υU0

2Le

x2L

f (η) + ηf ′(η)

,

T = T∞ + T0e2xL ,

η =

√U0

2υLe

x2L y, M =

σB202L

ρU0exL, Pr =

µcpk, R =

4σ∗T 3∞

k∗k, E =

U20

T0cp, (2.7)

Equation (2.1) is automatically satisfied, and Equations (2.2) and (2.6) reduce to

f ′′′ − 2(f ′)2 + ff ′′ −Mf ′ = 0 (2.8)(1 +

4

3R

)θ′′ Pr

(fθ′ − 4f ′θ + E(f ′′)2

)= 0 (2.9)

WhereM ,R,Prand E are the magnetic parameter, radiation parameter, Prandtl number and Eckertnumber, respectively and primes denote the differentiation with respect to η.

The corresponding boundary conditions are

f(0) = 0, f ′(0) = 1, θ(0) = 1,

f ′ → 0, θ → 0 as η →∞ (11)For the type of boundary layer flow, the skin-friction coefficient and heat transfer coefficient

are important physical parameters.Knowing the velocity field, the skin-friction at the stretching surface can be obtained, which in

non-dimensional form is given by

cf = −2 (Re)− 1

2 f′′

(0) (2.10)

Knowing the temperature field, the rate of heat transfer coefficient at the stretching surface canbe obtained, which in non-dimensional form, in terms of the Nusselt number, is given by

Nu = − (Re)12 θ′ (0) (2.11)

Where Re = U0Lυ is the Reynolds number.

303

Figure 1: Profiles for f ’(η), f(η) and θ(η) Figure 2: Effect of M on the Veloci

Figure 3: Effect of M on the Temperature Figure 4: Effect of Pr on the Temperature

3 Solution of the problem

The governing boundary layer Equations (2.8) and (2.9) subjects to boundary conditions (11) aresolved numerically by using Runge-Kutta fourth order technique along with shooting method. Firstof all higher order non-linear differential Equations (2.8) and (2.9) are converted into simultaneouslinear differential equations of first order and they are further transformed into initial value problemby applying the shooting technique (Jain et al. [16]). The resultant initial value problem is solvedby employing Runge-Kutta fourth order technique. The step size ∆η = 0.05 is used o obtainthe numerical solution with five decimal place accuracy as the criterion of convergence. From theprocess of numerical computation, the skin-friction coefficient and the Nusselt number which arerespectively proportional to f ′′(0) and −θ′(0), are also sorted out and their numerical values arepresented in a tabular form.

4 Results and Discussion

In order to get a physical insight of the problem, a representative set of numerical results weshown graphically in Figs.1-12 to illustrate the influence of physical parameters viz., the magneticparameter M , Prandtl number Pr, Radiation parameter R and Eckert number E on the velocityf ′ (η), f (η) and temperatureθ (η).

Fig.1 depicts the profiles of velocity, f (η) and θ (η) profile for M=1, Pr = 1, R = 1 andE = 0.2. It is observed that the profiles of the velocity f ′ (η) and f (η) are inversely proportional

304

Figure 5: Effect of R on the Temperature Figure 6: Effect of E on the Temperature

Figure 7: Effects of R and E on the Temperature Figure 8: Effects of Pr and E on the Temperature

to each other. The velocity profile is unique for all values of M ,Pr, Rand E due to the decoupledEquations (2.8) and (2.9).

Figs.(2.2) and (2.3) illustrate the velocity and temperature profiles for different values of themagnetic parameterM . It is observed that the velocity decreases as the magnetic parameterincreases (Fig.2). This is because that the application of transverse magnetic field will result aresistive type force (Lorentz force) similar to drag force which tends to resist the fluid flow and thusreducing its velocity. Also, the boundary layer thickness decreases with an increase in the magneticparameter. From Fig.(2.3), it is noticed that an increase in the magnetic parameter results in anincrease in the temperature.

The effect of the Prandtl number Pr on the temperature field is shown in Fig.4. The Prandtlnumber defines the ratio of momentum diffusivity to thermal diffusivity. It is noticed that asPr increases, the temperature decreases. This is because, physically, if Pr increases, the thermaldiffusivity decreases and these phenomena lead to the decreasing of energy ability that reduces tothermal boundary layer.

The influence of the thermal radiation parameter R on the temperature is shown in Fig.5. Theradiation parameter R defines the relative contribution of conduction heat transfer to thermal ra-diation transfer. It is observed that as R increases, the temperature profiles and thermal boundarylayer thickness also increase.

For different values of the viscous dissipation parameter i.e., the Eckert number E on thetemperature is shown in Fig.6. The Eckert number E express the relationship between the kineticenergy in the flow and the enthalpy. It embodies the conservation of kinetic energy into internal

305

Figure 9: Effects of M and Pr on the Temperature Figure 10: Effects of M and R on the Temperature

Table 1: Nu for various values of M, R, E and Pr

M R E Pr Nu0 0.5 0.5 1.0 1.23810.5 0.5 0.5 1.0 1.19041.0 0.5 0.5 1.0 1.14850.5 1.0 0.5 1.0 0,96050.5 0.5 0.9 1.0 1.12650.5 0.5 0.5 2.0 1.81550.5 0.5 0.5 3.0 2.2924

energy by work done against the viscous fluid stress. The positive Eckert number implies coolingof the sheet i.e., loss of heat from the sheet to the fluid. It is found that the temperature profilesand thermal boundary layer thickness increase slightly with an increase in E.

For further observations, comparison is made between the various physical parameters involvedin the problem and shown in Figs.7-10. The effects of R and E with fixed M = 1 and Pr = 1 areshown in Fig.7. It is seen that as Eor R increases, the temperature profiles also increase and theeffects of R are more pronounced than the effects of E. The effects of Eand Pr, with fixed M =1 and R = 1 are illustrated in Fig. 8. It is observed that their effects are opposite in nature, inwhich the increase in Eand the decrease in Pr lead to the increase in the temperature profiles.The effects of M and Pr, with fixed E = 0.5 and R = 1 are shown in Fig. 9. It is found thattheir effects are opposite in nature, in which the increase in M and the decrease in Pr lead to theincrease in the temperature profiles. The effects of R and M with fixed E = 0.5 and Pr = 1 arepresented in Fig.10. It is noticed that as M and R increase, the temperature profiles also increaseand the effects of R are more pronounced than the effects of M.

From equations (2.8) and (2.10), it is clear that the variations in the Prandtl number Pr, ra-diation parameter R and Eckert number E do not effect the wall skin-friction coefficient due tothe decoupled equations. However, since the magnetic parameter M is coupled with the momen-tum equation, it has significant effect on the wall skin-friction coefficient. The wall skin-frictioncoefficient has unique values 1.28213 and 1.62918, for non- magnetic (M=0) and magnetic (M=1)cases respectively. It is interesting to note that the value of the wall skin-friction coefficient in non-magnetic case is in good agreement with that of Bidin and Nazar [12], whose solved the problemusing the Keller-box method.

The effects of various governing parameters on the Nusselt number Nu are shown in Table1. Itis observed that the Nusselt number increases as Pr increases, where as it decreases as M or R or

306

E increases.

References

[1] Sakiadis, B.C.(1961), Boundary-layer Behavior on Continuous Solid Surfaces: I Boundary Layer Equations forTwo Dimensional and Axisymmetric Flow, AIChE J 7, pp. 26-28.

[2] Cortell, R.(2006), Effects of viscous dissipation and work done by deformation on the MHD flow and heattransfer of a viscoelastic fluid over a stretching sheet. Physics LettersA357, pp.298-305.

[3] Xu H. and Liao S.J.(2005), Series solutions of unsteady magnetohydrodynamics flows of non- Newtonian fluidscaused by an impulsively stretching plate. Journal of Non-Newtonian Fluid Mechanics 159, p.p.46-55.

[4] Hayat T., Abbas Z. and Sajid M.(2006), Series solution for the upper-convected Maxwell fluid over a porousstretching plate, Physics Letters A 358, p.p.396-403.

[5] Hayat T. and Sajid M.(2007), Analytic solution for axisymmetric flow and heat transfer of a second grade fluidpast a stretching sheet, International Journal of Heat and Mass Transfer 50, p.p. 75-84.

[6] Gupta P.S. and Gupta A.S.(1997), Heat and mass transfer on a stretching sheet with suction or blowing,Canadian Journal of Chemical Engineering 55, p.p.744-746.

[7] Ali M.E.(1995), On thermal boundary layer on a power law stretched surface with suction or injection. Inter-national Journal of Heat and Fluid Flow16, pp.280-290.

[8] Elbashbeshy, E.M.A.(2001), Heat transfer over an exponentially stretching continuous surface with suction,Archive of Mechanics 53, p.p.643- 651.

[9] Cortell R.(2008), Radiation effects in the Blasius flow, Applied Mathematics and Computation 198, p.p.33-338.

[10] Sajid M. and Hayat T.(2008), Influence of thermal radiation on the boundary layer flow due to an exponentiallystretching sheet, International Communications in Heat and Mass Transfer 35, p.p.347-356.

[11] El-Aziz M.A.(2009), Radiation effect on the flow and heat transfer over an unsteady stretching sheet, Interna-tional Communications in Heat and Mass Transfer 36, p.p.521-524.

[12] Bidin B. and Nazar R.,(2009) Numerical solution of the boundary layer flow over an exponentially stretchingsheet with thermal radiation, European journal of scientific research, vol.33, No.4, pp.710-717.

[13] Raptis A., Perdikis C. and Takhar H.S.(2004), Effect of Thermal Radiation on MHD Flow, Int. J. Heat MassTransfer,153, p.p. 645- 649.

[14] Hayat T., Abbas Z., Sajid M. and Asghar S.(2007), The Influence of Thermal Radiation on MHD Flow of aSecond Grade Fluid, Int. J. Heat Mass Transfer,50, p.p. 931-941.

[15] Aliakbar V., Alizadeh-Pahlawan A. and Sadeghy K.(2008), The Influence of Thermal Radiation on MHD Flowof Maxwellian Fluids above Stretching Sheets, Commun. Nonlinear Science Numerical Simulation (in press).

[16] Jain M.K., Iyengar S.R.K. and Jain R.K. (1985), Numerical Methods for Scientific and Engineering Computa-tion, Wiley Eastern Ltd., New Delhi, India.

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