Potential method applied to Boussinesq equation

Applied Mathematics and Computation 215 (2010) 3991–3997

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Potential method applied to Boussinesq equation E.Y. Abu El Seoud, M.M. Kassem * Mathematics and Physics Department, Faculty of Engineering, Zagazig University, Egypt

a r t i c l e

i n f o

Keywords: Shallow water Boussinesq equation Potential method Group similarity method

a b s t r a c t In this paper a solution of the nonlinear Boussinesq equation is presented using the potential similarity transformation method. The equation is first written in a conserved form, a potential function is then assumed reducing it to a system of equations which is further solved through the group transformation method. New transformations are found. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Boussinesq in 1872 [1] described the propagation of solitary waves with small amplitude on the surface of shallow water in the form

utt þ uuxx þ ðux Þ2 þ uxxxx ¼ 0;

ð1Þ

utt þ ðuux Þx þ uxxxx ¼ 0:

ð2Þ

or

These equations attract the attention of searchers due to the travelling property of its solution and its wide range of applications in physics like sound waves in laser beams, electromagnetic waves interacting with transversal optical phonons in nonlinear dielectrics [2], magneto sound waves in plasmas [3]. Eq. (1) was solved by Clarkson and Kruskal [4] using the direct method and lead to new similarity transformations. Levi et al. [5] made the link between the direct method and Lie transformation introducing conditional symmetries. Clarkson et al. [6] non-classical similarity transformation of the generalized Boussinesq equation, lead to a two-soliton solution. Integrable boundary conditions for Boussinesq equation were deduced by Vil’danov [7] through the Lax pair representation of the problem and particular solutions of the boundary problem were found. In 2007 Yan [8] solved a family of higher-dimensional generalized Boussinesq equations using an independent variable form; g = kx + W(y, t). The problem was then solved and reduced to two nonlinear ordinary differential equations whose exact solutions were given for specific forms of W(y, t). In 2008 Wang et al. [9] investigated the existence and uniqueness of the solution to the Cauchy problem for a class of Boussinesq equation, while Wang [10] in 2009 proved the existence and the uniqueness of a global solution for the generalized Boussinesq Cauchy problem and found that the wave amplitude u(x, t) decays to zero as ‘t’ tends to the infinity. Finally Bruzon et al. [11] presented a full analysis of a family of Boussinesq equations and proved that the non-classical method applied to these equations leads to new symmetries, which cannot be obtained by Lie classical method. We here introduce the potential method which consists in the reduction of (2) to a system of equations through the use of an auxiliary function. This system is then solved using the group transformation method [12–14].

*

Corresponding author.

0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.12.005

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E.Y. Abu El Seoud, M.M. Kassem / Applied Mathematics and Computation 215 (2010) 3991–3997

2. Mathematical formulation Consider the Boussinesq equation

  @ 2 u @u @ 4 u þ u þ ¼ 0; 2 @x x @x4 @t

ð3Þ

subjected to initial conditions

u ðx; 0Þ ¼ f ðxÞ;

ð4:aÞ

@u ðx; 0Þ ¼ 0; @t

ð4:bÞ

this equation is written in a conserved form

!   @u ðu Þ2 @ 2 u ¼ : þ 2 @t t 2 @x

ð5Þ

xx

The corresponding potential system is written as

@ 2 w @u ¼ ; @x2 @t " # @w ðu Þ2 @ 2 u ¼ þ 2 ; @t 2 @x

ð6Þ ð7Þ

where w(x, t) is an auxiliary function. A normalization of the initial condition (4.a) is obtained by setting

uðx; tÞ ¼

u ðx; tÞ ; f ðxÞ

ð8Þ

hence (6) and (7) reduce to

wxx ¼ ut f ; 1 2 2 wt ¼ u f  uxx f  2ux fx  ufxx : 2

ð9Þ ð10Þ

This differential system is subjected to the initial conditions

uðx; 0Þ ¼ 1;

ð11Þ

ut ðx; 0Þ ¼ 0:

ð12Þ

2.1. Group formulation A one-parameter group of the form

S ¼ C S ðaÞS þ K S ðaÞ;

ð13Þ

is used to reduce the system of partial equations to a system of ordinary differential equations. S and S refers to variables (x, t; u, w, f) before and after transformation. CS, KS are coefficients function of the group unity parameter ‘a’. First and second order partial derivative are expressed as

 s C Si ; Ci  s  C Sij ¼ Sij ; CiCj Si ¼

ð14Þ ð15Þ

where (i, j) subscripts refer to the variables (x, t). 2.2. Transformation of the system of potential equations The system of Eqs. (9) and (10) is invariantly transformed as follows:

 xx  u t f ¼ H1 ðaÞ½wxx  ut f ; w

  1 2 2 1 xxf þ 2u f xx ¼ H2 ðaÞ wt þ u2 f 2 þ uxx f þ 2ux fx þ ufxx ; xf x þ u  f þu  t þ u w 2 2

ð16Þ ð17Þ

E.Y. Abu El Seoud, M.M. Kassem / Applied Mathematics and Computation 215 (2010) 3991–3997

3993

where H1, H2 are functions in the group parameter ‘a’ and dashes on the system variables (x, t; u, w, f) implies their transformation. From the group definition given in (13)–(15) the left hand side of (16) and (17) reduces to

i Cu h f f ¼ H1 ðaÞ½wxx  ut f ; t ut C f þ K C ðC Þ i2 h i 2 h   Cf Cw 1 u Cu Cu Cf fxx C u þ K u C f f þ K f þ x 2 uxx C f f þ K f þ 2 x ux x fx þ C u u þ K u t wt þ 2 C C C ðC Þ ðC x Þ2 Cw

x 2

wxx 

1 ¼ H2 ðaÞ½wt þ u2 f 2 þ uxx f þ 2ux fx þ ufxx ; 2

ð18Þ

ð19Þ

or

Cw x 2

ðC Þ

wxx 

CuCf ut f þ R1 ðaÞ ¼ H1 ðaÞ½wxx  ut f ; Ct

ð20Þ

  Cw 1 u f 2 2 2 CuCf CuCf Cu Cf 1 2 2 ; u f þ 2 u f þ uf þ R ¼ H ðaÞ w þ f þ u f þ 2u f þ uf u xx x x xx 2 2 t xx x x xx t wt þ ½C C  u f þ 2 2 C ðC x Þ2 ðC x Þ2 ðC x Þ2

ð21Þ

where

R1 ðaÞ ¼  R2 ðaÞ ¼

CuK f ut ; Ct

ð22Þ 

1 u 2 1 ½C u ½2C f K f f þ ðK f Þ2  þ C u K u u þ ðK u Þ2 2 2

h

Cf f þ K f

i2

þ

CuK f x 2

ðC Þ

uxx þ

Cf K u ðC x Þ2

fxx :

ð23Þ

The transformation of (20) and (21) is invariant if R1 and R2 are equal to zero. This implies

K u ¼ K f ¼ 0; C

w

u

ð24Þ

f

C C ; Ct

ð25Þ

Cw Cu Cf u f 2 : t ¼ ðC C Þ ¼ C ðC x Þ2

ð26Þ

ðC x Þ2

¼

The transformation of initial conditions (11) and (12) leads to

C u ¼ 1;

K t ¼ 0:

ð27Þ

Regrouping (25)–(27) results, we obtain a group structure of the form

Cf ¼ Cw ¼

1 1 ¼ ; C t ðC x Þ2

ð28Þ

finally expressed as

8  x ¼ C x x þ K x > > g : > 1 >  > t ¼ ðC x Þ2 t > > < 8  ðx; tÞ ¼ U u > : G: > < > >  x; tÞ ¼ 1x 2 w þ K x wð > : g > Þ ðC > 2 > > > > : : f ðxÞ ¼ 1 f ðxÞ ¼ FðxÞ ðC x Þ2

ð29Þ

3. Invariant transformation of the system variables Morgan basic theorem in group theory state that a function g(x, t; u, w, f) is absolutely invariant if it satisfies the following first order linear differential equations: 5 X

ðai Si þ bi Þ

i¼1

@Si ¼ 0; @Si



Si ¼ x; t; u; w; f ;  ; w;  f ; Si ¼ x; t; u

ð30Þ

where

ai ¼

@C Si ; @a

bi ¼

@K Si : @a

ð31Þ

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E.Y. Abu El Seoud, M.M. Kassem / Applied Mathematics and Computation 215 (2010) 3991–3997

3.1. Transformation of the independent variables The independent variables (x, t) are transformed applying Morgan’s theorem

ða1 x þ b1 Þ

@g @g þ a2 t ¼ 0; @x @t

ð32Þ

where g is the similarity variable. Eq. (32) solution yields

g ¼ tpðxÞ;

ð33Þ

a pðxÞ ¼ ða1 x þ b1 Þm ; m ¼  2 ; a1 – 0: a1

ð34Þ

m will be evaluated later on. 3.2. Transformation of the dependent variables; u(x, t), w(x, t) and f(x) From the group structure described in (29) we obtain the following transformations:

 ðx; tÞ ¼ UðgÞ; u f ðxÞ ¼ FðxÞ:

ð35Þ ð36Þ

The transformation of w(x, t) is then derived through the application of Morgan’s theorem as follows:

@gðx; t; wÞ @gðx; t; wÞ @gðx; t; wÞ þ a2 t þ ða3 w þ b3 Þ ¼ 0; @x @t @w @g @w @ g @g @w @ g @g ¼ 0; ða1 x þ b1 Þ þ a2 t þ ða3 w þ b3 Þ @w @ g @x @ g @ g @t @w @w ½ða1 x þ b1 Þtpx þ a2 t p ¼ ða3 w þ b3 Þ; @g @w @g ¼ ; a3 w þ b3 gðm þ a2 Þ ða1 x þ b1 Þ

ð37Þ ð38Þ ð39Þ ð40Þ

integrating yields

 x; tÞ ¼ GðgÞUðxÞ; wð

ð41Þ

where U(x) is a constant of integration that will be determined analytically later as well as F(x) which represents the initial conditions transformation (8). 3.3. Reduction of the problem to a system of ordinary differential equations  ð   The derivatives of u x; tÞ; wð x; tÞ with respect to ‘x’ and ‘t’ are derived as a function of g and substituted in (9) and (10) differential equations.







Ux p ppxx Uxx p þ 2 þG U px px U px

2

p3 ¼ 0; U p2x    2  2 p p F p F xx p 1 p U p3 g2 U 00 þ gU 0 xx2 þ 2 x þ U2 F þ G0 ¼ 0; þU 2 F px F px F p2x px px g2 G00 þ gG0 2

 xU 0

F

ð42Þ ð43Þ

or in a contracted form as

g2 G00 þ gG0 ½A1 þ A2  þ A3 G  A4 U 0 ¼ 0; g2 U 00 þ gU 0 ½A2 þ A5  þ A6 U þ A7 U 2 þ A8 G0 ¼ 0;

ð44Þ ð45Þ

where

G0 ¼

@G ; @g

G00 ¼

@2G ; @ g2

U0 ¼

@U ; @g

Fx p A5 ¼ 2 ; F px  2 ppxx F xx p A2 ¼ 2 ; A 6 ¼ ; px F px  2  2 Uxx p 1 p A3 ¼ ; A7 ¼ F ; 2 px U px A1 ¼ 2

A4 ¼

@2U ; @ g2

Ux p ; U px

p3 U p3 ; A8 ¼ : 2 U px F p2x F

U 00 ¼

ð46Þ

E.Y. Abu El Seoud, M.M. Kassem / Applied Mathematics and Computation 215 (2010) 3991–3997

3995

In order to reduce (44) and (45) to a system of ordinary differential equations, the coefficients Ai (i = 1, 2, . . . , 8) must be constants or functions of g. Their evaluation follows:

ppxx m  1 ¼ ; m p2x  2 1 p 1 FðxÞða1 x þ b1 Þ2 ¼ ; A7 ¼ FðxÞ 2 2 px a21 m2 A2 ¼

ð47Þ ð48Þ

for A7 to be constant, let F(x) = (a1x + b1)2, hence A7 reduces to

A7 ¼

1 1 ; 2 a21 m2

ð49Þ

and invoking F(x) form we obtain

A5 ¼ 2

Fx F



p 4 F xx p ¼ ; A6 ¼ F px px m

2 ¼

6 : m

ð50Þ

Let in (46) A4/A8 = 1

A4 ¼1¼ A8



FðxÞ UðxÞ

FðxÞ ¼ UðxÞ ¼

2 ð51Þ

; 1

ða1 x þ b1 Þ2

ð52Þ

;

  where U(x) in (41) is part of the auxiliary function wð x; tÞ and A4 and A8 simplify to

A4 ¼ A 8 ¼

p3 ða1 x þ b1 Þmþ2 ¼ ¼ 1; 2 a21 m2 ð px Þ

ð53Þ

the above equation is satisfied if m = 2 and a1 m = 1. Hence a1 = 1/2, a2 = 1 (m = a2/a1). Then the rest of the constants are evaluated

A1 ¼ 2

Ux p ¼ 2; U px



A3 ¼

Uxx p U px

A5 ¼ 2 A2 ¼ A3 ¼ A6 ¼ 3=2;

2 ¼

3 ; 2

A4 ¼ A8 ¼ 1 A7 ¼ 1=2:

ð54Þ ð55Þ

Substituting for them in (44) and (45) we obtain

7 3 2 2 7 0 3 1 2 00 g U þ gU þ U þ U 2 þ G0 ¼ 0: 2 2 2

g2 G00 þ gG0 þ G  U 0 ¼ 0;

ð56Þ ð57Þ

This system is subjected to the initial conditions (11) and (12)

Uð0Þ ¼ 1;

U 0 ð0Þ ¼ 0:

ð58Þ

4. Results and discussion 4.1. Analytical results The group method used here leads to a similarity variable g = t/(a1 x + b1)2 which is new if compared to the classical transformation g = x  b t usually used for the transformation of wave equation and the exact solution of Boussinesq equation. The similarity variable found here is also totally different from Clarkson et al. [4] results using the direct transformation method. 4.2. Numerical results Eqs. (56) and (57) being nonlinear differential equations are numerically solved using the shooting method under the 0 assumption that U (1) = 0 = G(1). As more conditions are needed to solve (56) we did solve it asymptotically for g ? 0. This resulted in

Gð0Þ ¼ 0:

ð59Þ

The results obtained numerically for U(g) are depicted in Fig. 1. The wave form is similar in shape to half a soliton that remains in this shape all the time as g contains t. In this figure the soliton form can travel in either positive and negative directions as p(x) has an inverse parabolic form; (a1 x + b1)2 which is not affected by a change of direction of the travelling wave.

3996

E.Y. Abu El Seoud, M.M. Kassem / Applied Mathematics and Computation 215 (2010) 3991–3997 1.2 1

u(η)

0.8 0.6 0.4 0.2 0

0

1

2

3

4

5

η

6

7

8

9

10

Fig. 1. Boussinesq half wave form (numerical solution).

Fig. 2. The sech2 (x, t) soliton wave, moving at a speed b, Debnath [15].

0.5

u(η)

0.4 0.3 0.2 0.1

2

4

η

6

8

10

Fig. 3. Boussinessq analytical solution in the form of a Hermite function Hn, n < 0.

A comparison of our results with the analytically solution of Boussinesq equation u(x, t) = a sech2 [(x  b t)], where a is the wave height and b is the wave speed [15] is carried. This comparison is realized through the plot of u(x, t) which shows in Fig. 2 a visual similarity with u(x, t) profile depicted in Fig. 1. We then compare our numerical results with Clarkson’s [4] where the direct transformation of Boussinessq equation results in a; ‘‘Painleve fourth differential equation whose solution is expressed in terms of Weber–Hermite (parabolic cylinder) functions” i.e. in terms of Hermite polynomials Hn(g). We did for the comparison plot H1.8(g) in Fig. 3. This plot if compared with Fig. 1 show a similarity of soliton shapes. References [1] J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles a la surface au fond, J. Math. Pures Appl. Ser. 217 (1872) 55–108. [2] L. Xu, D.H. Auston, A. Hasegawa, Propagation of electromagnetic solitary waves in dispersive nonlinear dielectrics, Phys. Rev. A 45 (1992) 3184–3193. [3] V.I. Karpman, Nonlinear Waves in Dispersive Media, Pergamon, New York, 1975. [4] P.A. Clarkson, M.D. Kruskal, New similarity solutions of the Boussinesq equation, J. Math. Phys. (1989) 2201–2213. [5] D. Levi, P. Winternitz, Non classical symmetry reduction: example of the Boussinesq equation, J. Phys. A 22 (1989) 2915–2924. [6] P.A. Clarkson, E.L. Mansfield, Algorithms for the non classical method of symmetry reductions, SIAM J. Appl. Math. 54–56 (1994) 1693–1719. [7] A.N. Vil’danov, Integrable boundary value problem for the Boussinesq equation, Theor. Math. Phys. 141–142 (2004) 1494–1508.

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