# New Multiple Solution to the Boussinesq Equation and the Burgers-Like Equation

Authors: Hasan Bulut; Münevver Tuz; Tolga Akturk
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

By considering an improved tanh function method, we found some exact solutions of Boussinesq and Burgers-like equations. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions. We found some exact solutions of the Boussinesq equation and the Burgers-like equation.


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## Body

## 1. Introduction

In recent years, nonlinear phenomena play a crucial role in applied mathematics and physics. Directly searching for exact solutions of nonlinear partial differential equations (PDEs) has become more and more attractive partly due to the availability of computer symbolic systems like Mathematica or Maple that allow us to perform some complicated and tedious algebraic calculation on a computer as well as help us to find exact solutions of PDEs [1–5] now.Many explicit exact methods have been introduced in the literature [6–14]. Some of them are Painlevé method, homogeneous balance method, similarity reduction method, sine-cosine method, Darboux transformation, Cole-Hopf transformation, generalized Miura transformation, tanh method, Backlund transformation, and others methods [15, 16].One of the most effectively straightforward methods constructing exact solution of PDEs is the extended tanh function method [17]. Let us simply describe the tanh function. For doing this, one can consider in two variables general form of nonlinear PDE as follows:
(1)H(u,utt,ux,uxx,…)=0
and transform (1) with
(2)u(x,t)=u(ξ),ξ=k(x-λt),
where k and λ are the wave number and wave speed, respectively. After the transformation, we get a nonlinear ODE for u(ξ) as follows:
(3)H′(u′,u′′,u′′′,…)=0.The fact that the solutions of many nonlinear equations can be expressed as a finite series oftanh functions that motivates us to seek for the solutions of (3) in the form
(4)u(x,t)=u(ξ)=∑i=0maitanhi(ξ)=∑i=0maiFi,
where Fi=tanhi(ξ), an equation for F(ξ), is obtained. m is a positive integer that can be determined by balancing the linear term of highest order with the nonlinear term in (1), and k,λ,a1,a2,…,am are parameters to be determined.Substituting solution (4) into (3) yields a set of algebraic equations for Fi, then all coefficients of Fi have to vanish. From these relations k,λ,a1,a2,…,am can be determined.In this work, we will consider to solve general Boussinesq equation and the Burgers-like equation by using the improvedtanh function method which is introduced by Chen and Zhang [18].


## 2. Method and Its Applications

The main idea of this method is to take full advantage of the Riccati equation thattanh function satisfies and uses its solutions F to replace tanhξ. The required Riccati equation is given as
(5)F′=A+BF+CF2,
where F=(dF/dξ) and A,B, andC are constant. In the following, Chen and Zhang [18] have given several cases to get the solution of (5) in the form of finite series of tanh functions (4).Case 1.
IfC=0,B≠0, then (5) has the solutions
(6)exp(Bξ)-AB.Case 2.
IfA=1/2,B=0, and C=-1/2, then (5) has the solutions cothξ±cschξ, tanhξ±isechξ(i2=-1).Case 3.
IfA=C=±1/2,B=0, then (5) has the solutions secξ±tanξ, cscξ±cotξ.Case 4.
IfA=1, B=0, andC=-1, then (5) has the solutions tanhξ, cothξ.Case 5.
IfA=C=1, B=0, then (5) has the solutions tanξ.Case 6.
IfA=C=-1, B=0, then (5) has the solutions cotξ.Case 7.
IfA=B=0, C≠0, then (5) has the solutions -1/(cξ+c0).The solutions of (1) can be expressed in the form
(7)u(x,t)=u(ξ)=∑i=0maiFi,
where ξ=kx-kωt, k and ω are the wave number and the wave speed, respectively, n is a positive integer that can be determined by balancing the linear term of highest order with the nonlinear term in (1), and a0,a1,…,an are parameters to be determined.Introducing the similarity variableξ=kx-kωt, the travelling wave solutions u(ξ) satisfy
(8)H′(u′,u′′,u′′′,…)=0.Balancing the highest order of linear term with nonlinear term in (3), we can determine n in (4).We illustrate the method by considering the Boussinesq equation and Burgers-like equtions.Example 1 (The Boussinesq equation).
Let’s consider(9)utt=(c1u+c2un)xx+uxxtt=0.
If we accept thatc1≥0, c2≤0 and n=2, c=-1, we conclude (10) by (9) as follows:
(10)utt-uxx+2u·uxx-uxxtt=0
for doing this example. We could use transformation with (1) for the Boussinesq equation. Let us consider the Boussinesq equation as follows:
(11)u(x,t)=u(ξ),ξ=kx-kwt.
Substituting (11) into (10), we get
(12)w2u′′-u′′+2(u′)2+2u·u′′-k2w2u4=0,
and integrating (12) we deduce the following equation:
(13)w2u′-u′+2u·u′-k2w2u′′′=0.
Balancing(u·u′) with (u′′′) gives m=2. Therefore, we may choose
(14)u=a0+a1F+a2F2.
Substituting (14) into (13) along with (5) and using Mathematica yield a system of equations w,t, and FM. Setting the coefficients of FM in the obtained system of equations to zero, we can deduce the following set of algebraic polynomials with respect to unknowns a0, a1, and a2(15)u′=a1A+a1BF+a1CF2+2a2AF+2a2BF2+2a2CF3=0,u·u′=a0a1A+a0a1BF+a0a1CF2+2a0a2AF+2a0a2BF2+2a0a2CF3+a12AF+a12BF2+a12CF3+2a1a2AF2+2a1a2BF3+2a1a2CF4+a1a2AF2+a1a2BF3+a1a2CF4+2a22AF3+2a22BF4+2a22CF5,u′′′=a1B2A+a1B3F+7a1B2CF2+8a1ABCF+12a1BC2F3+2a1A2C+8a1AC2F2+6a1C3F4+6a2A2B+14a2AB2F+52a2ABCF2+16a2A2CF+40a2AC2F3+8a2B3F2+38a2B2CF3+54a2BC2F4+24a2C3F5,F0:w2a1A-a1A+2a0a1A-k2w2a1B2A-2k2w2a1A2C-6k2w2a2A2B=0,F1:w2a1B+2w2a2A-a1B-2a2B+2a0a1B+4a0a2A+2a12A-k2w2a1B3-8k2w2a1ABC-14k2w2a2AB2-16k2w2a2A2C=0,F2:w2a1C+2w2a2B-a1C-2a2B+2a0a1C+4a0a2B+2a12B+6a1a2A-7k2w2a1B2C-8k2w2a1AC2-52k2w2ABC-8k2w2a2B3=0,F3:2w2a2C-2a2C+4a0a2C+2a12C+4a1a2B+2a1a2B+4a22A-12k2w2a1BC2-40k2w2a2AC2-38k2w2B2C=0,F4:-6a1a2C+4a22B-6k2w2a1C3-54k2w2a2BC2=0,F5:4a22C-24k2w2a2C3=0.
From the solutions of the system, we can find
(16)B=0,a0=4k2w2AC-12w2+12,a1=0,a2=6k2w2C2,w=14k2AC+1,
and with the aid of Mathematica, we find the following.
(i) When we chooseA=1, B=0, andC=1 in (16), then we can deduce
(17)a0=4k2w2-12w2+12,a2=6k2w2.
Therefore, the solution can be found as(18)u(x,t)=(4k2w2-12w2+12)+6k2w2[tan2(kx-kwt)].
(ii) In this case, if we takeA=-1, B=0, and C=-1 in (16), then we have
(19)a0=4k2w2-12w2+12,a2=6k2w2,u(x,t)=(4k2w2-12w2+12)+6k2w2cot[kx-kwt].
(iii) Again, when we chooseA=1, B=0, C=-1 then from (16) is
(20)a0=-4k2w2-12w2+12,a2=6k2w2,u(x,t)=(-4k2w2-12w2+12)+6k2w2tanh2(kx-kwt),u(x,t)=(-4k2w2-12w2+12)+6k2w2coth2(kx-kwt).
(iv) When we chooseA=(1/2), B=0, and C=(1/2), then we can find the coefficients of (16) as
(21)a0=k2w2-12w2+12,a2=32k2w2,
and using the coefficients, the solutions can be found as
(22)u(x,t)=(k2w2-12w2+12)+32k2w2×[sec(kx-kwt)+tan(kx-kwt)]2,u(x,t)=(k2w2-12w2+12)+32k2w2×[csc(kx-kwt)+cot(kx-kwt)]2.
(v) When we chooseA=-(1/2), B=0, andC=-(1/2), then we can find the coefficients of (16) as follows:
(23)a0=k2w2-12w2+12,a2=32k2w2,u(x,t)=(k2w2-12w2+12)+32k2w2×[sec(kx-kwt)+tan(kx-kwt)]2,u(x,t)=(k2w2-12w2+12)+32k2w2×[csc(kx-kwt)+cot(kx-kwt)]2.
(vi) When we chooseA=(1/2), B=0, and C=-(1/2), then we can find the coefficients of (16) as follows:
(24)a0=-k2w2-12w2+12,a2=32k2w2,u(x,t)=(-k2w2-12w2+12)+32k2w2×[coth(kx-kwt)±csch(kx-kwt)]2,u(x,t)=(-k2w2-12w2+12)+32k2w2×[tanh(kx-kwt)±isech(kx-kwt)]2.
(vii) When we chooseA=0, B=0, and C≠0, then we can find the coefficients of (16) as follows:
(25)a0=-12w2+12,a2=6k2w2C2,u(x,t)=(-12w2+12)+32k2w2C2(-1(kx-kwt)+C0)2.
Figure1 gives to us 2D and 3D graphics for (25).The solution (25) is shown at k=w=1, C=0.5, and C0=0.1, and the second graph represents the exact analytical solution of (25) for t=0.5.
(a)
(b)Example 2 (Burger-like equations).
Let’s consider(26)ut+ux+uux+λuxx=0,
where λ=1, and in order to obtain Burger-like solution of equation, we get
(27)u(x,t)=u(ξ),ξ=k(x-wt).
Substituting (27) into (26), we get
(28)-wu′+u′+uu′+ku′′=0.
Balancing(uu′) with u′′ gives m=1. Therefore, we may choose the following ansatz:
(29)u=a0+a1F.
Substituting (29) into (28) along with (5) and using Mathematica yield the following set of algebraic polynomials with respect to unknowns a0, a1, a2, and a3(30)u′=a1A+a1BF+a1CF2+2a2AF+2a2BF2+2a2CF3=0,(31)uu′=(a0+a1F)(a1A+a1BF+a1CF2),u′′′=a1BA+a1B2F+a1CAF+3a1BCF2+2a1C2F3,F0:w2a1A-a1A+2a0a1A-k2w2a1B2A-2k2w2a1A2C-6k2w2a2A2B=0,F1:w2a1B+2w2a2A-a1B-2a2B+2a0a1B+4a0a2A+2a12A-k2w2a1B3-8k2w2a1ABC-14k2w2a2AB2-16k2w2a2A2C=0,F2:w2a1C+2w2a2B-a1C-2a2B+2a0a1C+4a0a2B+2a12B+6a1a2A-7k2w2a1B2C-8k2w2a1AC2-52k2w2a2B3=0,F3:2w2a2C-2a2C+4a0a2C+2a12C+4a1a2B+2a1a2B+4a22A-12k2w2a1BC2-40k2w2a2AC2-38k2w2B2C=0.
From the solutions of the system, we can find
(32)a0=w-1,a1=-2kc,
and we obtain the following multiple solution and triangular periodic solutions of (26).(i)
When we chooseA=C=1, B=0 in (31), then
(33)a1=-2k,a0=w-1.
Therefore, the solution can be found as
(34)u(x,t)=(w-1)-2ktan[k(x-wt)].(ii)
In the case if we takeA=C=-1, B=0 in (31), then we have
(35)a1=2k,a0=w-1,u(x,t)=(w-1)+2kcot[k(x-wt)].(iii)
Again, when we chooseA=1, B=0, and C=-1,
(36)a0=w-1,a1=-2k,u(x,t)=(w-1)+2ktanh[k(x-wt)],u(x,t)=(w-1)+2kcoth[k(x-wt)].(iv)
When we chooseA=(1/2), B=0, and C=-(1/2),
(37)a0=w-1,a1=-k,u(x,t)=(w-1)-k[sec[k(x-wt)]+tan[k(x-wt)]],u(x,t)=(w-1)-k[cosec[k(x-wt)]+cot[k(x-wt)]].(v)
When we chooseA=C=-(1/2), B=0, then we can find the coefficients of (31) as follows:
(38)a0=k,a0=w-1,u(x,t)=(w-1)+k[sec[k(x-wt)]-tan[k(x-wt)]],u(x,t)=(w-1)+k[cosec[k(x-wt)]-cot[k(x-wt)]].(vi)
When we chooseA=(1/2), B=0, and C=-(1/2), then we can find the coefficients of (31) as follows:
(39)u(x,t)=(w-1)+k[coth[k(x-wt)]±cosech[k(x-wt)]],u(x,t)=(w-1)+k[tanh[k(x-wt)]±isech[k(x-wt)]].Figure 2 gives to us 2D and 3D graphics for (39).The solution (39) is shown at k=w=1, and the second graph represents the exact analytical solution of (39) for t=0.5.
(a)
(b)


## 3. Conclusion

We have presented a generalizedtanh function method and used it to solve the Boussinesq equation and the Burgers-like equation. In fact this method is readily applicable to a large variety of nonlinear PDEs.Firstly, all the nonlinear PDEs which can be solved by other tanh function method can be solved easily by this method. Secondly we have used only the special solutions of (4). If we use only the special solutions of (4), we can obtain more solutions. We are also aware of the fact that not all fundamental equations can be treated with the method.We also obtain some new and more general solutions at the same time. Furthermore, this method is also computerizable, which allows us to perform complicated and tedious algebraic calculation on a computer.



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