New abundant solutions for the Burgers equation

Computers and Mathematics with Applications 58 (2009) 514–520

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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa

New abundant solutions for the Burgers equation Alvaro H. Salas S. a,b,∗ , Cesar A. Gomez S c , Jairo Ernesto Castillo Hernańdez d,e a

Department of Mathematics, Universidad de Caldas, Manizales-Colombia, Colombia

b

Universidad Nacional de Colombia, Campus la Nubia, Manizales, Colombia

c

Department of Mathematics, Universidad Nacional de, Colombia, Bogota–Colombia, Colombia

d

Departamento de Ciencias, Universidad Central, Bogotá–Colombia, Colombia

e

Department of Mathematics, Universidad Autónoma, Bogotá–Colombia, Colombia

article

abstract

info

Article history: Received 21 July 2008 Received in revised form 17 December 2008 Accepted 12 January 2009 Keywords: Nonlinear evolution equation PDE Burgers equation

This paper applies a new modified Exp-function method to search for exact solitary solutions of the Burgers equation. Using the imaginary transformation suggested by He and Wu, solitary solutions can be easily converted into periodic or compacton-like solutions. Many new solutions are appearing for the first time in the open literature. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The search for exact solutions of nonlinear partial differential equations (NLPDE’s) is of great importance, because these equations appear in complex physics phenomena, mechanics, chemistry, biology and engineering. These solutions may give more insight into the physical aspects of the problem modeled by the NLPDE’s. However, it is well known that apart from a few cases is not easy to obtain solutions by using analytic methods. So, to solve these nonlinear equations, new methods are needed. In recent years, improved semi-exact analytic methods have been used, for instance the inverse scattering method [1], Hirota’s bilinear method [2], homotopy perturbation method [3], variational iteration method [4], non-perturbative methods [5], tanh–coth method [6], and so on. A rational approximation for soliton and soliton-like solutions was first proposed by Hirota [2] and further developed by other authors [7]. However, recently, He and Wu [8] proposed a straightforward and concise method, called the Exp-function method, to obtain generalized solitary and periodic solutions, which is very effective for solving NLPDE’s with strong nonlinearity. The basic idea of the method is as follows: Given the general partial differential equation P (u, ut , ux , uxx , . . .) = 0,

(1.1)

after wave transformation u(x, t ) = u(ξ ),

ξ = x + λt ,

(1.2)

with λ a constant, (1.1) reduces to the ordinary differential equation Q (u, u0 , u00 , . . .) = 0,

∗

(1.3)

Corresponding author at: Department of Mathematics, Universidad de Caldas, Manizales-Colombia, Colombia. E-mail addresses: [email protected] (A.H. Salas S.), [email protected] (C.A. Gomez S), [email protected] (J.E. Castillo Hernańdez).

0898-1221/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2009.01.044

A.H. Salas S. et al. / Computers and Mathematics with Applications 58 (2009) 514–520

515

where the prime denotes derivation with respect to ξ . The Exp-function method is based on the assumption that traveling wave solutions of (1.3) can be expressed in the form [8] d P

an exp (nξ )

n=−c

u(ξ ) =

q P

,

(1.4)

bm exp (mξ )

m=−p

where c, d, p and q are positive integers to be determined. The values of c and p are determined balancing the linear term of highest order in (1.3) with the highest-order nonlinear term. Similarly, the values of d and q are determined balancing the linear term of lowest order in (1.3) with the lowest-order nonlinear term. Substituting (1.4) into (1.3), and after simplifications, an algebraic system in the unknown an , bn , λ is obtained. This system can be solved for an , bn , λ. Reversing, solutions to (1.1) in the form (1.4) are obtained. The method has been employed to solve many nonlinear equations; [8–11] showed that this method is very effective. The method is also used to obtain solitary solutions of difference-differential equations [12]. Some modifications of the method appear in [13]. In this paper, we employ a variant of the Exp-function method to obtain new solitary solutions of the celebrated Burgers equation [14] ut + uux = ν uxx ,

(1.5)

where ν is a constant. For this, we will consider a generalized version of Eq. (1.5), namely ut + α uux + β uxx = 0,

(1.6)

where α and β are constants. Recently, a few solutions of this equation were obtained [15]. 2. Solitary solutions to the generalized Burgers equation (1.6) After wave transformation, (1.2) and (1.6) reduces to

λu0 (ξ ) + α u(ξ )u0 (ξ ) + β u00 (ξ ) = 0.

(2.1)

We shall solve Eq. (2.1) via the ansatz u=p+

a0 + a1 sinh(µξ ) + a2 cosh(µξ )

∆ + b1 sinh(µξ ) + b2 cosh(µξ )

,

(2.2)

where p, a0 , a1 , a2 , b1 , b2 , µ are constants and ∆ = 0 or ∆ = 1. Using the expressions

 exp(µξ ) − exp(−µξ )  sinh(µξ ) = 2

(2.3)

exp(µξ ) + exp(−µξ )  cosh(µξ ) = , 2

we obtain a variant of the Exp-function method. In accord with this method, substituting (2.2) into (2.1), we obtain a polynomial equation in the variable ϕ = exp(µξ ). Equating its coefficients to zero, we obtain an algebraic system. Solving this system and using (2.2), we obtain exact solutions to (2.1). Using the transformation [9]

 µ = iµ, √ −1 sin(µξ ), exp(iµξ ) = cos(µξ ) + √  exp(−iµξ ) = cos(µξ ) − −1 sin(µξ ),

(2.4)

periodic solutions can be obtained. 3. The novel solutions Solving the algebraic system with the aid of Mathematica yields following solutions of Eq. (1.6), where i =

• ∆ = 1, λ = −βµ, a0 =

2βµ

α

, a1 = 0, a2 = 0, b1 = −b2 :

2βµ

. α + b2 α exp(−µ(x − βµt )) 2iβµ u2 = . α + b2 α exp(−µ(ix + βµt )) u1 =

√ −1:

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• ∆ = 1, λ = βµ, a0 = − 2βµ , a1 = 0, a2 = 0, b1 = b2 : α 2βµ . α + b2 α exp(µ(x + βµt )) 2iβµ u4 = − . α + b2 α exp(µ(ix − βµt )) u3 = −

• ∆ = 1, λ = −pα − βµ, a0 =

2βµ

α

, a1 = 0, a2 = 0, b1 = −b2 :

2βµ

. α + b2 α exp(−µ(x − (pα + βµ)t )) 2iβµ u6 = p + . α + b2 α exp(−µ(ix − (ipα − βµ)t )) u5 = p +

b2 • ∆ = 1, λ = − αa2 +βµ , a0 = b2

u7 =

u8 =

a2 b2 a2 b2

+

+

α a2 +2βµb2 , a1 α b2

2βµ

= −a2 , b1 = −b2 :

1

     . α 1 + b exp −µ x − βµ + α a2 t 2 b2

2iβµ

α

1







1 + b2 exp −µi x + iβµ + α b2 a

  . t

2

b2 • ∆ = 1, λ = − αa2 −βµ , a0 = b2

u9 =

u10 =

a2 b2 a2 b2

−

α a2 −2βµb2 , a1 α b2

2βµ

= a2 , b1 = b2 :

1

     . α 1 + b exp µ x + βµ − α a2 t 2 b2

−

2iβµ

α

1







1 + b2 exp iµ x + iβµ − α b2 a

2

• ∆ = 1, λ = − αa2 +pαbb22 +βµb2 , a0 = u11 = p +

u12 = p +

a2 b2 a2 b2

+

+

2βµ

u14 = p +

a2 b2 a2 b2

−

−

t

= −a2 , b1 = −b2 : 1

   . α 1 + b exp −µ x − pα + βµ + α a2 t 2 b2 

2iβµ

α



1

   . a 1 + b2 exp −µ ix − (ipα − βµ + iα b2 )t 2

• ∆ = 1, λ = − αa2 +pαbb22 −βµb2 , a0 = u13 = p +

α a2 +2βµb2 , a1 α b2

  .

α a2 −2βµb2 , a1 α b2

2βµ

= a2 , b1 = b2 : 1

     . α 1 + b exp µ x − pα − βµ + α a2 t 2 b2

2iβµ

α

1







1 + b2 exp iµ x − pα − iβµ + α b2 a

2

• ∆ = 0, λ = −pα, a0 = 0, a1 = 0, a2 =

, b2 = 0 :

2βµ

• ∆ = 0, λ = −pα, a0 = 0, a1 = 2βµ

2βµb2

α

, a2 = 0, b1 = 0 :

tanh(µ(x − pα t )). α 2βµ =p− tan(µ(x − pα t )). α

u17 = p + u18

α

coth(µ(x − pα t )). α 2βµ =p+ cot(µ(x − pα t )). α

u15 = p + u16

2βµb1

  . t

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517

• ∆ = 1, λ = −pα, a0 = 0, a1 = − βµ , a2 = 0, b1 = 0, b2 = −1 : α   1 βµ coth µ(x − pα t ) . u19 = p + α 2   1 βµ cot µ(x − pα t ) . u20 = p + α 2 • ∆ = 1, λ = −pα, a0 = 0, a1 = βµ , a2 = 0, b1 = 0, b2 = 1 : α   βµ 1 u21 = p + tanh µ(x − pα t ) . α 2   βµ 1 u22 = p − tan µ(x − pα t ) . α 2 b1 • ∆ = 0, λ = − αba11 , a0 = 0, a2 = 2βµ , b2 = 0 : α    2βµ α a1 a1 + coth µ x − t . u23 = b1 α b1    2βµ α a1 a1 + cot µ x − t . u24 = b1 α b1 b2 • ∆ = 0, λ = − αba22 , a0 = 0, a1 = 2βµ , b1 = 0 : α    a2 2βµ α a2 u25 = + tanh µ x − t . b2 α b2    2βµ α a2 a2 − tan µ x − t . u26 = b2 α b2 b1 • ∆ = 0, λ = − α(a1b+1pb1 ) , a0 = 0, a2 = 2βµ , b2 = 0 : α      a1 2βµ a1 u27 = p + + coth µ x − p + αt . b1 α b1      a1 2βµ a1 u28 = p + + cot µ x − p + αt . b1 α b1 b2 • ∆ = 0, λ = − α(a2b+2pb2 ) , a0 = 0, a1 = 2βµ , b1 = 0 : α      a2 2βµ a2 u29 = p + + tanh µ x − p + αt . b2 α b2      2βµ a2 a2 − tan µ x − p + αt . u30 = p + b2 α b2 a2 • ∆ = 0, λ = − βµa2a1 , a0 = −a2 , b1 = αβµ , b2 = 0 :     βµ a1 1 u31 = a1 + a2 tanh µ x − βµ t . α a2 2 a2     iβµ a1 1 u32 = a1 + ia2 tan µ x − iβµ t . α a2 2 a2 a2 • ∆ = 0, λ = − βµa2a1 , a0 = a2 , b1 = αβµ , b2 = 0 :     βµ a2 a1 u33 = a1 + a22 coth µ x − βµ t . α a2 2 a2     βµ a2 a1 u34 = ia1 + a22 cot µ x − iβµ t . α a2 2 a2 √ √ √ −p2 α 2 −2pλα−λ2 +β 2 µ2 (pα+λ) −p2 α 2 −2pλα−λ2 +β 2 µ2 −p2 α 2 −2pλα−λ2 +β 2 µ2 • ∆ = 1, a0 = 0, a1 = − , a = , b = 0 , b = − : 2 1 2 α αβµ βµ p β 2 µ2 − (pα + λ)2 ((pα + λ) cosh(µ(x + λt )) − βµ sinh(µ(x + λt )))   u35 = p + . p α βµ − β 2 µ2 − (pα + λ)2 cosh(µ(x + λt ))

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A.H. Salas S. et al. / Computers and Mathematics with Applications 58 (2009) 514–520

(pα + λ)2 + β 2 µ2 ((pα + λ) cos(µ(x + λt )) + βµ sin(µ(x + λt ))) p  . α (pα + λ)2 + β 2 µ2 cos(µ(x + λt )) − βµ √ √ √ −p2 α 2 −2pλα−λ2 +β 2 µ2 (pα+λ) −p2 α 2 −2pλα−λ2 +β 2 µ2 −p2 α 2 −2pλα−λ2 +β 2 µ2 • ∆ = 1, a0 = 0, a1 = , a2 = − , b1 = 0, b2 = : α αβµ βµ p β 2 µ2 − (pα + λ)2 ((−pα − λ) cosh(µ(x + λt )) + βµ sinh(µ(x + λt )))   u37 = p + . p α βµ + β 2 µ2 − (pα + λ)2 cosh(µ(x + λt )) p (pα + λ)2 + β 2 µ2 ((pα + λ) cos(µ(x + λt )) + βµ sin(µ(x + λt )))   u38 = p − . p α βµ + (pα + λ)2 + β 2 µ2 cos(µ(x + λt )) p

u36 = p −

q

q

β 2 µ2 −α 2 a20

β 2 µ2 −α 2 a20

• ∆ = 1, λ = −pα, a1 = − , a2 = 0, b1 = 0, b2 = − α   q βµ α a0 − β 2 µ2 − α 2 a20 sinh(µ(x − pα t ))   . u39 = p + q 2 2 2 2 α βµ − β µ − α a0 cosh(µ(x − pα t ))   q βµ α a0 + β 2 µ2 + α 2 a20 sin(µ(x − pα t ))   . u40 = p + q 2 2 2 2 α βµ − β µ + α a0 cos(µ(x − pα t )) q

q

β 2 µ2 −α 2 a20

βµ

β 2 µ2 −α 2 a20

• ∆ = 1, λ = −pα, a1 = , a2 = 0, b1 = 0, b2 = α βµ   q 2 2 2 2 βµ α a0 + β µ − α a0 sinh(µ(x − pα t ))   . u41 = p + q α βµ + β 2 µ2 − α 2 a20 cosh(µ(x − pα t ))   q βµ α a0 − β 2 µ2 + α 2 a20 sin(µ(x − pα t ))   . u42 = p + q 2 2 2 2 α βµ + β µ + α a0 cos(µ(x − pα t ))

:

:

• ∆ = 1, λ = βµ − pα, a0 = − 2βµ , a1 = 0, a2 = 0, b1 = b2 : α

u44

2βµ

1

. α 1 + b2 exp(µ(x − (pα − βµ)t )) 2iβµ 1 =p− . α 1 + b2 exp(µ(ix − (ipα + βµ)t ))

u43 = p −





iβµ b22 +1

• ∆ = 1, λ = iβµb2 − pα, a0 = 0, a1 = 0, a2 = − , b1 = −i : α
βµ b22 + 1 cosh (µ (x − (pα − iβµb2 )t )) u45 = p + . α (i + sinh (µ (x − (pα − iβµb2 )t )) + ib2 cosh (µ (x − (pα − iβµb2 )t )))
βµ b22 + 1 cos (µ (x − (pα + βµb2 )t )) . u46 = p + α (1 + sin (µ (x − (pα + βµb2 )t )) + b2 cos (µ (x − (pα + βµb2 )t ))) 



iβµ b22 +1

• ∆ = 1, λ = −pα − iβµb2 , a0 = 0, a1 = 0, a2 = , b1 = i : α
iβµ b22 + 1 cosh (µ (x − pα t − iβµb2 t )) u47 = p + . α (i sinh (µ (x − pα t − iβµb2 t )) + b2 cosh (µ (x − pα t − iβµb2 t )) + 1)
βµ b22 + 1 u48 = p − . α (b2 + sec (µ (x − pα t + βµb2 t )) − tan (µ (x − pα t + βµb2 t )))

A.H. Salas S. et al. / Computers and Mathematics with Applications 58 (2009) 514–520

√

β 2 µ2 −λ2

√

λ

519

√

β 2 µ2 −λ2

• ∆ = 1, a0 = 0, a1 = − , a2 = , b1 = 0, b2 = − α αβµ p β 2 µ2 − λ2 (λ cosh(µ(x + λt )) − βµ sinh(µ(x + λt )))   u49 = . p α βµ − β 2 µ2 − λ2 cosh(µ(x + λt )) p λ2 + β 2 µ2 (λ cos(µ(x + λt )) + βµ sin(µ(x + λt )))   u50 = . p α βµ − λ2 + β 2 µ2 cos(µ(x + λt ))

β 2 µ2 −λ2 βµ

:

√ √ √ β 2 µ2 −λ2 β 2 µ2 −λ2 λ β 2 µ2 −λ2 • ∆ = 1, a0 = 0, a1 = , a = − , b = 0 , b = : 2 1 2 α αβµ βµ p β 2 µ2 − λ2 (βµ sinh(µ(x + λt )) − λ cosh(µ(x + λt )))   u51 = . p α βµ + β 2 µ2 − λ2 cosh(µ(x + λt )) p λ2 + β 2 µ2 (λ cos(µ(x + λt )) + βµ sin(µ(x + λt )))   u52 = − . p α βµ + λ2 + β 2 µ2 cos(µ(x + λt )) • ∆ = 1, a1 = −

λ2 +2α a0 λ−β 2 µ2 +α 2 a20 q , a2 α −λ2 −2α a0 λ+β 2 µ2 −α 2 a20

=

  λ λ2 +2α a0 λ−β 2 µ2 +α 2 a20 q , b1 αβµ −λ2 −2α a0 λ+β 2 µ2 −α 2 a20

√ √ αβ a0 µ + βµ R sinh(µ(x + λt )) − λ R cosh(µ(x + λt ))   . u53 = √ α βµ + R cosh(µ(x + λt )) √ √ αβ a0 µ − βµ R0 sin(µ(x + λt )) − λ R0 cos(µ(x + λt )) v54 = . √ α(βµ + R0 cos(µ(x + λt ))) R = β 2 µ2 − (λ + α a0 )2 , R0 = β 2 µ2 + (λ + α a0 )2 . 



iβµ b22 +1

, b1 = −i : • ∆ = 1, λ = iβµb2 , a0 = 0, a1 = 0, a2 = − α
βµ b22 + 1 cosh (µ (x + iβµb2 t )) u55 = . α (i + sinh (µ (x + iβµb2 t )) + ib2 cosh (µ (x + iβµb2 t )))
βµ b22 + 1 u56 = . α (b2 + sec (µ (x − βµb2 t )) + tan (µ (x − βµb2 t ))) 



iβµ b22 +1

• Delta = 1, λ = −iβµb2 , a0 = 0, a1 = 0, a2 = , b1 = i : α
iβµ b22 + 1 cosh (µ (x − iβµb2 t )) u57 = . α (1 + i sinh (µ (x − iβµb2 t )) + b2 cosh (µ (x − iβµb2 t )))
βµ b22 + 1 . u58 = − α (b2 + sec (µ (x + βµb2 t )) − tan (µ (x + βµb2 t ))) • ∆ = 0, λ = −pα, a0 = −ia1 , a2 = 0, b1 = 0, b2 =

α a1 βµ

:

βµ (tanh(µ(x − pα t )) − i sech(µ(x − pα t ))). α βµ =p+ (sec(µ(x − pα t )) − tan(µ(x − pα t ))). α

u59 = p + u60

• ∆ = 0, λ = −pα, a0 = ia1 , a2 = 0, b1 = 0, b2 =

:

βµ (i sech(µ(x − pα t )) + tanh(µ(x − pα t ))). α βµ =p− (sec(µ(x − pα t )) + tan(µ(x − pα t ))). α

u61 = p + u62

α a1 βµ

q

= 0, b2 =

−λ2 −2α a0 λ+β 2 µ2 −α 2 a20 βµ

:

520

A.H. Salas S. et al. / Computers and Mathematics with Applications 58 (2009) 514–520

a1 a2 , a0 = −ia1 , b1 = 0, b2 = αβµ • ∆ = 0, λ = − pαa1 +βµ : a1             a2 a2 βµ a2 + a1 tanh µ x − pα + βµ t − i sech µ x − pα + βµ t . u63 = p + α a1 a1 a1             a2 βµ a2 u64 = p + ia2 + a1 sec µ x − pα + iβµ t − tan µ x − pα + iβµ t . α a1 a1 a1 a1 a2 , a0 = ia1 , b1 = 0, b2 = αβµ : • ∆ = 0, λ = − pαa1 +βµ a1             a2 βµ a2 u65 = p + a2 + a1 i sech µ x − pα + βµ t + tanh µ x − pα + βµ t . α a1 a1 a1              a2 βµ a2 u66 = p − a1 sec µ x − pα + iβµ t + tan µ x − pα + iβµ t − ia2 . α a1 a1 a1 a1 • ∆ = 0, λ = − βµa1a2 , a0 = −ia1 , b1 = 0, b2 = αβµ :         a2 βµ a2 − i sech µ x − βµ t . u67 = a2 + a1 tanh µ x − βµ t α a1 a1 a1         βµ a2 a2 u68 = ia2 + a1 sec µ x − iβµ t − tan µ x − iβµ t . α a1 a1 a1 a1 • ∆ = 0, λ = − βµa1a2 , a0 = ia1 , b1 = 0, b2 = αβµ :         a2 a2 βµ a2 + a1 i sech µ x − βµ t + tanh µ x − βµ t . u69 = α a1 a1 a1          βµ a2 a2 u70 = − a1 sec µ x − iβµ t + tan µ x − iβµ t − ia2 . α a1 a1 a1

4. Conclusions In this work, we have used a variant of the Exp-function method which was developed by He and Wu [8], and with aid of symbolic computation we have obtained solitary and periodic solutions to generalized Burgers equations. Many new solutions have been obtained. Hence, we conclude that the variant of the Exp-method here used is a very powerful mathematical tool for solving other nonlinear equations. Acknowledgements The authors are grateful to the referees for their very constructive comments and recommendations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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