Exact analytic, regular perturbation and numerical solutions of symmetry reductions of a (2+1) -dimensional KdV–Burgers equation

Nonlinear Analysis: Real World Applications 14 (2013) 1265–1275

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Exact analytic, regular perturbation and numerical solutions of symmetry reductions of a (2 + 1)-dimensional KdV–Burgers equation B. Mayil Vaganan ∗ , T. Shanmuga Priya Department of Applied Mathematics and Statistics, Madurai Kamaraj University, Madurai-625021, India

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Article history: Received 23 December 2011 Accepted 13 September 2012 Keywords: A (2 + 1)-dimensional generalized KdV–Burgers equation Allowed transformations Symmetry analysis Exact, regular perturbation and numerical solutions

abstract The (2 + 1)-dimensional generalized KdV–Burgers equation, (ut + un ux + λ(x, y, t )uxx + α(x, y, t )uxxx )x + S (x, y, t )uyy = 0, is changed to its canonical form via allowed transformations and then the canonical equation is subjected to Lie’s symmetry analysis. Exact and regular perturbation solutions are obtained for the reduced partial differential equations. Regular perturbation and numerical solutions are reported for the reduced second order nonlinear ordinary differential equations. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction The Burgers equation [1] ut + uux =

δ 2

uxx ,

(1.1)

is the simplest second order nonlinear partial differential equation which balances the effect of nonlinear convection and linear diffusion. Hopf [2] and Cole [3] have shown that (1.1) may be linearized to the heat conduction equation

φt =

δ 2

φxx ,

(1.2)

via the Cole–Hopf transformation u = −δ

φx . φ

(1.3)

Another simple model equation is the KdV equation [4]

vt + 6vvx + δvxxx = 0

(1.4)

which describes long waves in shallow water. Its modified version is ut − 6u2 ux + uxxx = 0,

∗

Corresponding author. Tel.: +91 9842082214. E-mail address: [email protected] (B. Mayil Vaganan).

1468-1218/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2012.09.005

(1.5)

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and again there is the Miura transformation [5]

v = u2 + ux ,

(1.6)

between the KdV equation (1.4) and its modified version (1.5). In 2002, Liu and Yang [6] studied the bifurcation properties of the generalized KdV equation (GKdVE) ut + aun ux + uxxx = 0,

a ∈ R, n ∈ Z +

(1.7)

Recently Senthilkumaran et al. [7] reported invariant solutions of another GKdVE in the form ut + un ux + α(t )u + β(t )uxxx = 0,

(1.8)

using Lie’s group of infinitesimal transformations [8,9]. In 1969, Zabolotskaya and Khokhlove derived the ZK equation [10]

(ut + uux − β uxx )x + γ uyy = 0.

(1.9)

Yet another model equation is derived by Kadomtsev and Petviashvili [11] ut + uux + ϵ 2 uxx



 x

+ λuyy = 0,

λ = ±1.

(1.10)

The KP equation (1.10) is a generalization to two spatial dimensions, x and y, of the KdV equation. But David et al. [12,13] generalized the KP equation to describe water waves in straits or rivers. Güngör and Winternitz [14] transformed the generalized KP equation (GKPE)

(ut + p(t )uux + q(t )uxxx )x + σ (y, t )uyy + a(y, t )uy + b(y, t )uxy + c (y, t )uxx + e(y, t )ux + f (y, t )u + h(y, t ) = 0,

(1.11)

to its canonical form and established conditions on the coefficient functions under which (1.11) has an infinite-dimensional symmetry group having a Kac–Moody–Virasoro structure. In [15], they carried out the symmetry analysis of the VCKP equation in the form

(ut + f (x, y, t )uux + g (x, y, t )uxxx )x + h(x, y, t )uyy = 0,

(1.12)

and further classified it into equivalence classes under the local fiber preserving point transformations u = U (x, y, t , u˜ (˜x, y˜ , t˜)),

x = X (˜x, y˜ , t˜),

y = Y (˜x, y˜ , t˜),

t = T (˜x, y˜ , t˜),

(1.13)

with nonvanishing Jacobian determinant

∂U ̸= 0, ∂ u˜

∂(X , Y , T ) ̸= 0. ∂(˜x, y˜ , t˜)

(1.14)

In this paper, we combine the ZK and the GKP equation (1.12) to write the (2 + 1)-dimensional KdV–Burgers equation in the form

(ut + un ux + λ(x, y, t )uxx + α(x, y, t )uxxx )x + S (x, y, t )uyy = 0,

n ∈ Z +.

(1.15)

Allowed transformations (1.13)–(1.14) are applied to (1.15) to change it to its canonical form

(ut + un ux + λuxx + α uxxx )x + uyy = 0,

n ∈ Z +,

(1.16)

and we use Lie symmetries to reduce sequentially (1.16) to partial differential equations (PDEs) with two independent variables and ordinary differential equations (ODEs) in order to facilitate the determination of

• exact analytic solutions of PDEs; • regular perturbation solutions of both NLODEs and PDEs; • numerical solutions of initial and boundary value problems of ODEs.

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2. Allowed transformations We shall determine the allowed transformation in its global form (1.13) to (1.15) in order to convert it to the canonical form (1.16). Now we substitute (1.13) into (1.15) and require that the transformed equation is form invariant. We further require that the coefficient functions α(x, y, t ), λ(x, y, t ), S (x, y, t ) are replaced by constants, in particular, S → 1. Equating the coefficient of u˜ x˜ x˜ x˜ u˜ x˜ to zero yields Uu˜ u˜ = 0. We therefore assume that U = A(x, y, t )˜u + B(x, y, t ).

(2.1)

We again set the coefficients of u˜ t˜t˜t˜ , u˜ y˜ y˜ y˜ , u˜ y˜ t˜ , u˜ x˜ x˜ x˜ , u˜ u˜ x˜ , u˜ x˜ x˜ to zero to obtain t˜x = y˜ x = t˜y = x˜ xx = Ax = B = 0.

(2.2)

In view of (2.2), we rewrite the transformations (1.13), (2.1) as u = A(y, t )˜u,

x˜ = ξ (y, t )x + φ(y, t ),

y˜ = η(y, t ),

t˜ = θ (t ).

(2.3)

On inserting (2.3), Eq. (1.15) becomes Aξ t˜t u˜ x˜ ,t˜ + nAn+1 ξ 2 u˜ n−1 u˜ 2x˜ + An+1 ξ 2 u˜ n u˜ x˜ ,˜x + Aλξ 3 u˜ x˜ ,˜x,˜x

  +Aαξ 4 u˜ x˜ ,˜x,˜x,˜x + AS y˜ 2y u˜ y˜ ,˜y + ξ At + Aξt + 2SxAy ξy + 2SAy φy + AS φy,y + ASxξy,y u˜ x˜     + Asy˜ y,y + 2SAy y˜ y u˜ y˜ + Aξ y˜ t + 2ASxy˜ y ξy + 2AS y˜ y φy u˜ x˜ ,˜y   + Axξ ξt + ASx2 ξy2 + Aξ φt + 2ASxξy φy + AS φy2 u˜ x˜ ,˜x + SAyy u˜ = 0.

(2.4)

Since we demand that Eq. (2.4) for u˜ is in agreement with Eq. (1.15) for u in ‘‘form’’, we must set the coefficients of u˜ x˜ , u˜ y˜ , u˜ x˜ ,˜y , u˜ x˜ ,˜x , u˜ to zero. We thus obtain

ξ At + Aξt + 2SxAy ξy + 2SAy φy + AS φyy + ASxξyy = 0,

(2.5)

Ay˜ yy + 2Ay y˜ y = 0,

(2.6)

ξ y˜ t + 2Sxy˜ y ξy + 2S y˜ y φy = 0, xξ ξ t +

(2.7)

ξ + ξ φt + 2Sxξy φy + S φ = 0,

Sx2 y2

2 y

(2.8)

Ayy = 0.

(2.9)

In view of (2.5)–(2.9), Eq. (2.4) reduces to u˜ x˜ t˜ +

An ξ 

θ′

nu˜ n−1 u˜ 2x˜ + u˜ n u˜ x˜ x˜ +



S y˜ 2y αξ 3 λξ 2 ˜ ˜ u + u + u˜ y˜ y˜ = 0, ˜ ˜ ˜ ˜ ˜ ˜ ˜ x x x x x x x θ′ θ′ ξθ′

(2.10)

where we have divided by Aξ t˜t . It therefore follows that A ̸= 0, ξ ̸= 0, t˜t ̸= 0. The canonical form (1.16) follows from (2.10) provided that An ξ

θ′

= 1,

λξ 2 ˜ = λ, θ′

αξ 3 = α, ˜ θ′

S y˜ 2y

ξθ′

= 1,

(2.11)

where we have dropped the tildes. Thus Eq. (1.15) is transformed to its canonical form (1.16) through the allowed transformation (2.3) under the conditions (2.5)–(2.9) and (2.11). 3. Symmetry analysis of (1.16) If (1.16) is invariant under a one-parameter Lie group of point transformations [8,9] x∗ = x + ϵξ1 (x, y, t , u) + O(ϵ 2 ),

(3.1)

y = y + ϵξ2 (x, y, t , u) + O(ϵ ),

(3.2)

t = t + ϵξ3 (x, y, t , u) + O(ϵ ),

(3.3)

u = u + ϵφ1 (x, y, t , u) + O(ϵ )

(3.4)

∗

2

∗

2

∗

2

with infinitesimal generator X = ξ1 (x, y, t , u)

∂ ∂ ∂ ∂ + ξ2 (x, y, t , u) + ξ3 (x, y, t , u) + φ1 (x, y, t , u) , ∂x ∂y ∂t ∂u

(3.5)

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then the invariant condition reads as y

yy

2uuy φ1 + φ1t + u2 φ1 − αφ1xx − αφ1 = 0.

(3.6)

In (3.6), we introduced the following quantities:

φ1x = φ1y = φ1t = φ1xx =

Dφ1

− ux

Dx Dφ1

− ux

Dy Dφ1

− ux

Dt Dφ1x Dx

Dξ1 Dx Dξ1 Dy Dξ1 Dt

− uxx

y

φ1yy = D Dx D Dy D Dt

Dφ1 Dy

− uxy

− uy − uy − uy

Dξ1

Dξ2

Dξ1

Dξ2

− ut

Dt

− uyy

Dy

− ut

Dy

− uxy

Dx

− ut

Dx Dξ2

Dξ2 Dx Dξ2 Dy

Dξ3 Dx Dξ3 Dy Dξ3 Dt

− uxt − uyt

,

(3.7)

,

(3.8)

,

(3.9)

Dξ3 Dx Dξ3 Dy

,

(3.10)

,

(3.11)

=

∂ ∂ ∂ ∂ ∂ + ux + uxx + +uxy + uxt , ∂x ∂u ∂ ux ∂ uy ∂ ut

(3.12)

=

∂ ∂ ∂ ∂ ∂ + uy + uxy + +uyy + uyt , ∂y ∂u ∂ ux ∂ uy ∂ ut

(3.13)

=

∂ ∂ ∂ ∂ ∂ + ut + uxt + +uyt + utt . ∂t ∂u ∂ ux ∂ uy ∂ ut

(3.14)

The determining equations are obtained from (3.5) and solved for the infinitesimals ξ = ξ1 , η = ξ2 , τ = ξ3 , φ = φ1 :

ξ = −k4 −

k3 y 2

,

η = k2 + k3 t ,

τ = k1 and φ = 0.

(3.15)

Now, we write down the four symmetry generators corresponding to each of the constants ki , i = 1, 2, 3, 4 involved in the infinitesimals, viz., V1 = ∂t ,

V2 = ∂ y ,

V3 = −

y 2

∂x + t ∂y ,

V4 = ∂ x .

(3.16)

The one-parameter groups gi (ϵ) generated by the Vi , where i = 1, 2, 3, 4 are g1 (ϵ) : (x, y, t , u) → (x, y, t + ϵ, u), g2 (ϵ) : (x, y, t , u) → (x, y + ϵ, t , u), g3 (ϵ) : (x, y, t , u) →



x−ϵ

y 2β

 , y + ϵt , t , u ,

g4 (ϵ) : (x, y, t , u) → (x + ϵ, y, t , u), where exp(ϵ Vi )(x, y, t , u) = (˜x, y˜ , t˜, u˜ ). The transformations g1 , g2 and g4 represent the time and space invariance of the equation. Let i = 1, 2, 3, 4 and ϵ ∈ R. If u = U (x, y, t ) is a solution of (1.16) so are the functions ui (x, y, t ) = U (x, y, t ) where u1 = U (x, y, t − ϵ), u2 = U (x, y − ϵ, t ), 3

u =U



x+ϵ

y 2β



, y − ϵt , t ,

u4 = U (x − ϵ, y, t ). The commutation relations of the Lie algebra G, determined by V1 , V2 , V3 , V4 are shown in the following table. These vector fields form a Lie algebra L by:

[V1 , V3 ] = V2 ,

[V3 , V1 ] = −V2 ,

[ V2 , V3 ] =

−V 4 2

,

[V3 , V2 ] =

V4 2

.

For this four-dimensional Lie algebra the commutator table for Vi is a (4 ⊗ 4) table whose (i, j)th entry expresses the Lie Bracket [Vi , Vj ] given by the above Lie algebra L. The table is skew symmetric and the diagonal elements all vanish. The

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coefficient Ci,j,k is the coefficient of Vi of the (i, j)th entry of the commutator table and the related structure constants can be easily calculated from the table as follows: C1,3,2 = 1, C3,1,2 = −1, C2,3,4 = −21 , C3,2,4 = 12 . The Lie algebra L is solvable. The radical of G is, R = ⟨V4 ⟩ ⊕ ⟨V1 , V2 , V3 ⟩. These symmetry generators form a closed Lie algebra as is seen from the following commutator table:

[Vi , Vj ]

V1

V2

V3

V4

V1

0

0

V2

0

V2

0

0

−V4 /2

0

V3

−V 2

V4 /2

0

0

V4

0

0

0

0

In the next section we derive the reductions of (1.16) to PDEs with two independent variables and ODEs. There are four one-dimensional Lie subalgebras Ls,1 = {V1 }, Ls,2 = {V2 }, Ls,3 = {V3 }, Ls,4 = {V4 }. And corresponding to each onedimensional subalgebra we may reduce (1.16) to a PDE with two independent variables. Further reductions to ODEs are associated with two-dimensional subalgebras. It is evident from the commutator table that there are no two-dimensional solvable non-Abelian subalgebras. And there are four two-dimensional Abelian subalgebras, namely, LA,1 = {V1 , V2 } , LA,2 = {V1 , V4 } , LA,3 = {V2 , V4 } , LA,4 = {V3 , V4 }. 4. Reductions of (1.16) by one-dimensional subalgebras We reduce the (1.16) PDEs with two independent variables for the four generators, Vn , n = 1, 2, 3, 4. Case: 1 The subalgebra Ls,1 = {V1 }. The characteristic equation associated with the generator V1 is dx 0

=

dy 0

dt

=

1

=

du 0

.

(4.1)

Integration of (4.1) yields the new set of similarity variables r (x, y, t ), s(x, y, t ), W (r , s), viz., s(x, y, t ) = x,

r (x, y, t ) = y,

u(x, y, t ) = W (r , s).

(4.2)

Using (4.2) in (1.16), the latter changes to the PDE



1 n+1

W n+1 + λWs + α Wss



+ Wrr = 0.

(4.3)

ss

In Section 5, we shall determine both exact and approximate solutions of the PDE (4.3) with n = 1 from an approximate solution of the ODE to be derived as a similarity reduction of (4.3). Case: 2 The subalgebra Ls,2 = {V2 }. The characteristic equation and its solutions are dx 0

=

dy 1

s = x,

dt

=

0

=

du 0

r = t,

,

(4.4)

u = W (r , s).

(4.5)

In view of (4.5), (1.16) transforms into a generalized KdV–Burgers equation Wr + W n Ws + λ Wss + α Wsss = 0.

(4.6)

Case: 3 The subalgebra Ls,3 = {V3 }. The characteristic equation associated with V3 is dx

−y/2

=

dy t

=

dt 0

=

du 0

.

(4.7)

Integrating (4.7) we get s = tx +

y2 4

,

t = r,

u = W (r , s).

(4.8)

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Eqs. (1.16) and (4.8) together lead to a PDE r Wr + s Ws + r W n Ws + λ r 2 Wss + α r 3 Wsss +

W 2

= 0.

(4.9)

Eq. (4.9) can be transformed through W (r , s) = G(r , z ),

z (s, r ) = s/r ,

(4.10)

to rGr + Gn Gz + λGzz + α Gzzz +

1 2

G = 0.

(4.11)

We further transform (4.12) via G(r , z ) = J (θ , z ),

θ (r ) = log r ,

(4.12)

to Jθ + J n Jz + λJzz + α Jzzz +

1 2

J = 0.

(4.13)

We further transform (4.13) via K (θ , z ) = eθ/2 J ,

(4.14)

Kθ + e−nθ/2 K n Kz + λKzz + α Kzzz = 0,

(4.15)

to

a generalized KdV–Burgers equation with a variable coefficient. Thus the transformation which changes the PDE (4.9) into the generalized KdV–Burgers equation (4.13) is obtained from (4.10) and (4.12): W (r , s) = e−θ/2 J (θ , z ),

θ (r ) = log r , z (s, r ) = s/r .

(4.16)

Case: 4 The subalgebra Ls,4 = {V4 }. The characteristic equation associated with V4 is dx 1

=

dy 0

=

dt 0

=

du 0

.

(4.17)

Integration of (4.17) gives rise to s = y,

r = t,

u = W (r , s).

(4.18)

Eqs. (4.18) and (1.16) together leads to a linear PDE Wss = 0.

(4.19)

5. Reductions of (1.16) by two-dimensional subalgebras It is evident from the commutator table that there are no two-dimensional solvable non-Abelian subalgebras, and we list below the reductions of (1.16) under two-dimensional Abelian subalgebras. Case: 1 The subalgebra LA,1 = {V1 , V2 } . From the commutator table, we find that the generators V1 and V2 commute, that is, [V1 , V2 ] = 0. We can initiate the reduction procedure by taking V1 or V2 . If we begin with V1 , then (1.16) is reduced to the PDE (4.3) (cf. Section 3, case: 1). We now write below V2∗ which is V2 , but, expressed in terms of the similarity variables given in (4.2): V2∗ = ∂r .

(5.1)

The associated characteristic equation and the transformation are dr 1

=

ds 0

=

dW 0

,

and ζ = s,

W = B(ζ ).

(5.2)

Consequently, (4.3) is replaced by an ODE Bn B′ + λB′′ + α B′′′



′

= 0.

(5.3)

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We integrate (5.3) with respect to ζ twice to yield the second order equation,

α

d2 B dζ 2

+ϵ

dB dζ

+

1

(B(ζ ))n+1 = c1 ζ + c2 .

n+1

(5.4)

We shall now obtain an approximate analytical solution of (5.4) under the assumption that and λ = ϵ ≪ 1.

c1 = c2 = 0

(5.5)

Eq. (5.5) takes the form

α

d2 B dζ 2

+ϵ

dB dζ

+

1

(B(ζ ))n+1 = 0.

n+1

(5.6)

In the following subsections we shall determine both approximate analytic and numerical solutions of the nonlinear ODE (5.6). We remark that the approximate analytic solution to be obtained here turns out to be valid only for odd n. And this necessitated us obtaining numerical solutions of (5.6) when n is even.

5.1. Approximate analytic solution of (5.6) According to the regular perturbation method we seek a solution of (5.6) in the form B(ζ ) = B0 (ζ ) + ϵ B1 (ζ ) + ⃝(ϵ 2 ).

(5.7)

Inserting (5.7) into (5.6) and equating the coefficients of powers of ϵ to 0, we have

α B′′0 +

1 n+1

B0n+1 = 0

(5.8)

α B′′1 + Bn0 B1 + B′0 = 0

(5.9)

A power solution of (5.8) is B0 (ζ ) =

 −

2α(n + 1)(n + 2)

1/n

n2

ζ −2/n ,

n is odd.

(5.10)

In view of (5.10), Eq. (5.9) becomes

α B′′1 −

2 2α(n + 1)(n + 2) −2 ζ B1 = 2 n n

 1/n 2α(n + 1)(n + 2) − ζ −2/n−1 . 2 n

(5.11)

A power solution of (5.11) is easily found to be B1 (ζ ) = −



1

(n + 4)α

−

2α(n + 1)(n + 2)

1/n

n2

ζ 1−2/n .

(5.12)

Substituting from (5.10) and (5.12) into (5.7) we arrive at the following perturbation solution B(ζ ) =

 −

2α(n + 1)(n + 2) n2

1/n

 ζ −2/n 1 − ϵ

1

(n + 4)α

 ζ ,

n is odd,

(5.13)

of the ODE

α

d2 B dζ 2

+ϵ

dB dζ

+

1 n+1

(B(ζ ))n+1 = 0,

ϵ ≪ 1.

(5.14)

We recall that the regular perturbation solution (5.13) of the nonlinear second order ODE (5.14) is valid only when n is an odd integer.

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5.2. Numerical Solution of an IVP for (5.6) In 5.1, we have obtained the approximate analytic solution of the ODE (5.14) using the regular perturbation method. As this solution turned out to be valid only for odd n, we present numerical solutions of both initial and boundary value problems for the ODE (5.14) when n = 2 in 5.2 and 5.3. We choose n = 2, α = 2, ϵ = 0.1. We pose the following initial value problem (IVP) for the ODE (5.6): 2

d2 B dζ

2

+ 0.1

dB dζ

1

+ (B(ζ ))3 = 0,

(5.15)

3

B(1) = 3.4,

(5.16)

B (1) = −3.5.

(5.17)

′

The numerical solution to the IVP (5.15)–(5.17) is depicted below:

These figures reveal that the solution to the IVP (5.15)–(5.17) approaches zero as ζ → 250. 5.3. Numerical solution of a BVP for (5.6) We again choose n = 2, α = 2, ϵ = 0.1. And ζ ∈ [1, 250]. We now pose the following boundary value problem (BVP) for the ODE (5.6): 2

d2 B dζ

2

+ 0.1

dB dζ

1

+ (B(ζ ))3 = 0, 3

1 ≤ ζ ≤ 250,

(5.18)

B(1) = 3.4,

(5.19)

B(250) = −0.00223123

(5.20)

The numerical solution to the BVP (5.18)–(5.20) is similar to the ones given in Section 5.2.

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5.4. Approximate and exact analytic solutions of the PDE (4.3) 5.4.1. Approximate analytic solution of the PDE (4.3) The PDE (4.3) through the similarity transformation (5.2) is reduced to the ODE (5.6). And we have found an approximate analytic solution (5.13) of the ODE (5.6) using the regular perturbation method when

λ = ϵ ≪ 1.

(5.21)

In view of (5.21), the PDE (4.3) takes the form



1 n+1

W n+1 + ϵ Ws + α Wss



+ Wrr = 0,

ϵ ≪ 1.

(5.22)

ss

Since W (s, r ) = B(ζ ), s = ζ , the regular perturbation solution (5.13) of the ODE (5.6) is also an approximate solution of the PDE (5.22), with n = 1, s = ζ , up to the order of ϵ . Now we shall proceed to obtain the corresponding regular perturbation solution of the PDE (5.22) with n = 1, viz.,



1 2

W 2 + ϵ Ws + α Wss



+ Wrr = 0.

(5.23)

ss

When n = 1, (5.13) gives



B(ζ ) = −12α

1

ζ2

ϵ 5αζ

−



.

(5.24)

Eq. (5.24) is a regular perturbation solution of the ODE (5.14) with n = 1, viz.,

α

d2 B dζ

2

+ϵ

dB

1

+ (B(ζ ))2 = 0,

dζ

ϵ ≪ 1.

2

(5.25)

We add K (r ) to the expression given in (5.24) for B(ζ ) to write W (s, r ) as W (s, r ) = −12α



1

ϵ 5α s

−

s2



+ K (r ).

(5.26)

Substitution of (5.26) in (5.23) results in an ODE for K (r ) 72α K (r )

24ϵ K (r ) + = 0, s4 5s3 where we have omitted the term involving ϵ 2 . We seek a regular perturbation solution of Eq. (5.27) in the form K ′′ (r ) −

(5.27)

K (r ) = K0 (r ) + ϵ K1 (r ) + O(ϵ 2 ).

(5.28)

Substituting (5.28) into (5.27) and equating the coefficients of ϵ , i = 0, 1, we have the system of ODEs i

72α A(r )

= 0,

(5.29)

72α B(r ) − = 0. 5s3 s4 The general solution of the system (5.29) and (5.30) is

(5.30)

A′′ (r ) −

s4 24A(r )

B′′ (r ) +

√ √

6 2r s2

A(r ) = e

√ √

α

c1 + e

√ √

−

B(r ) =

e

6 2r s2

α

6 2r s2

√ √

 e

60sα

−

12 2r s2

α

α

c2 ,

(5.31)

√

2

s c1 − 12 2e

√ √ α

12 2r s2

√ √ √ √  6 2r α 6 2r α √ √ √ − 2 2 s s2 r α c1 + s c2 + 12 2r α c2 + e c3 + e c4 ,

(5.32)

where ci , i = 1, 2, 3, 4 are arbitrary constants. Therefore the regular perturbation solution of the ODE (5.27) is written by inserting from (5.31) and (5.32) for K0 (r ) and K1 (r ): K (r ) = e

√ √ α

6 2r s2

c1 + e √ √

 +ϵ

√ √

e

−

6 2r s2

−

α

6 2r s2

√ √ α

+e

6 2r s2

c2 √ √



60sα

α

e

12 2r s2

√ √

c3 + e

−

6 2r s2

α

α

√

s2 c1 − 12 2e

√ √

12 2r s2

α

r

√ √ √ α c1 + s2 c2 + 12 2r α c2



 c4 .

(5.33)

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B. Mayil Vaganan, T. Shanmuga Priya / Nonlinear Analysis: Real World Applications 14 (2013) 1265–1275

Thus the regular perturbation solution of the PDE (5.23) corresponding to the regular perturbation solution (5.24) of the ODE (5.14) with n = 1 is obtained if we use (5.33) in (5.26): √ √ √ √  6 2r α 6 2r α ϵ − s2 s2 c + e c2 + e 1 s2 5α s  − 6√2r √α  √ √ √  12 2r α √ 12 2r √α √ √ √ s2 e 2 2 2 2 s s +ϵ s c1 − 12 2e r α c1 + s c2 + 12 2r α c2 e 60sα  √ √ √ √

W (s, r ) = −12α

+e



1

−

α

6 2r s2

c3 + e

−

α

6 2r s2

c4 .

(5.34)

5.4.2. Exact analytic solution of the PDE (4.3) We now present an exact solution of the PDE (4.3) with n = 1, viz.,



1 2

W 2 + λWs + α Wss



+ Wrr = 0.

(5.35)

ss

The idea used here is novel as we shall use the approximate analytic solution (5.24) of the ODE (5.25) to construct an exact solution of the PDE (5.35). We remark here that the ODE (5.25) is a similarity reduction ofthe PDE (5.35).  Now we add K (r ) to (5.24), with ϵ → λ, and insert the resulting expression −12α

1 s2

−

λ

5α s

+ K (r ) into the PDE (5.35)

to obtain K ′′ (r ) −

72α K (r ) s4

+

24λK (r ) 5s3

+

72λ2 25s4

= 0.

(5.36)

Eq. (5.36) is linear and hence its general solution is

K (r ) = −

  2

3λ2

+ c1 exp 

5(−15α + sλ)

6 r 5

√

15α − sλ

s2







2

 + c2 exp −

6 r 5

√

15α − sλ

s2

 .

(5.37)

Thus the required exact solution of the PDE (5.35) is W (s, r ) = −12α



1 s2

−

 + c2 exp −

λ



5α s

2



6 r 5

− √

3λ

5(−15α + sλ)

15α − sλ

s2

2

  2

+ c1 exp 

6 r 5

√

15α − sλ



s2



 .

(5.38)

Case: 2 The subalgebra LA,2 = {V1 , V4 }. It follows from the commutator table that [V1 , V4 ] = 0. We shall begin with V1 to transform (1.16) to (4.3). Then V4 changes to V4∗ = ∂s . Integration of the characteristic equations associated with V4∗ gives W = B(ζ ), ζ = r which reduces (4.3) to B′′ = 0. Case: 3 The subalgebra LA , 3 = {V2 , V4 } . Since [V2 , V4 ] = 0, we begin with V2 and arrive at the PDE (4.6). We express V4 in terms of the similarity variables defined in (4.5) as V4∗ = ∂s . The characteristic equation for V4∗ is ds 1

=

dr 0

=

dW 0

.

(5.39)

Integrating (5.5) we obtain the transformation W = B(ζ ), r = ζ which replaces (4.19) by simply B′ = 0. Case: 4 The subalgebra LA,4 = {V3 , V4 }. From the commutator table, we find that the given generators commute, that is, [V3 , V4 ] = 0. Thus either V3 or V4 can be used to start the reduction procedure. We begin with V4 . In this case (1.16) is reduced to the PDE (4.19). We now express

B. Mayil Vaganan, T. Shanmuga Priya / Nonlinear Analysis: Real World Applications 14 (2013) 1265–1275

1275

V3 in terms of the similarity variables defined in (4.18) as V3∗ = r ∂s .

(5.40)

The associated characteristic equations are dr 0

=

ds r

=

dW 0

.

(5.41)

Integration of (5.41) yields the transformation

ζ (r , s) = r ,

W (r , s) = B(ζ ).

(5.42)

As a result (4.9) reduces to

ζ B′ +

B 2

= 0.

(5.43)

6. Results The (2 + 1)-dimensional generalized KdV–Burgers equation (1.15) is transformed to its canonical form (1.16) via the allowed transformation (2.3) and (2.11). Eq. (1.16) is shown to admit a four-dimensional symmetry group (3.15). It is established that the symmetry generators form a closed Lie algebra. Classification of symmetry algebra of (1.16) into one- and two-dimensional subalgebras is carried out. Systematic reduction to (1 + 1)-dimensional PDEs and then to ODEs are performed using one-dimensional and twodimensional Abelian and non-Abelian subalgebras. Eq. (1.16) is shown to reduce to the PDE (4.3) and then to the ODE (5.3) which is integrated twice to yield (5.6). When λ = ϵ ≪ 1, regular perturbation solution (5.13) of the second order nonlinear ODE (5.3) is derived when n is odd, and initial and boundary value problems are posed to (5.6) and the numerical solutions are shown to decay for a fairly finite value of ζ . The corresponding perturbation solution of the PDE (4.3), when λ = ϵ ≪ 1, is given in (5.34). Further, when n = 1, an exact solution (5.38) to the PDE (4.3) is also obtained. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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