Accepted Manuscript

New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water

Mostafa M.A. Khater, Dipankar Kumar

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S2468-0133(17)30052-9 10.1016/j.joes.2017.07.001 JOES 47

To appear in: Journal of Ocean Engineering and Science

Received date: 2 June 2017 Accepted date: 21 July 2017

Please cite this article as: Mostafa M.A. Khater, Dipankar Kumar, New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water, Journal of Ocean Engineering and Science (2017), doi: 10.1016/j.joes.2017.07.001

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New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water

Mostafa M.A. Khater 1,2 and Dipankar Kumar 3 4 department of Mathematics, Faculty of Science, Jiangsu University, China.

Faculty of Science, Department of Mathematics, Mansoura University, 35516 Mansoura,

Egypt.

Doctoral Student, Division of Engineering Mechanics and Energy, Graduate School of Systems and Information Engineering, University of Tsukuba, Tennodai 1-1-1, Tsukuba,

Ibaraki, Japan.

Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science, and Technology University, Gopalganj-8100, Bangladesh.

Corresponding Author :( Mostafa M.A. Khater): Email: mostafa.khater2024@yahoo.com

Phone Number: 00201149206914 & 00201061355409

Abstract

The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq-Burger and approximate long water wave equations by using the generalized Kudryashov method. The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann-Liouville derivative sense. Applying the generalized Kudryashov method through with symbolic computer maple package, numerous new exact solutions are successfully obtained. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.

Keywords: The generalized Kudryashov method; The time fractional coupled Boussinesq-Burger equation; The time fractional approximate long water wave equation; Exact solutions.

1. Introduction

Fractional order differential equations are the generalized form of an integer order differential equations. Last few years, there has been the great attention of fractional calculus and the fractional differential equation for the used to physical processes model. The fractional differential equation is also widely applied in various physical and engineering fields models like as physics, mathematical biology, fluid mechanics, plasma physics, optical fibers, quantum field theory, neural physics, solid state physics and etc [1-3]. Due to the rapid development of artificial intelligence-based symbolic computation software like as Maple, Mathematica, and MATLAB, various powerful analytical approaches have been built and efficiently accomplished for searching more general and new exact solutions of nonlinear partial differential equations (NPDEs) such as the fractional sub-equation method [4,5], the improved fractional sub-equation method [6,7], the EXP-function method [8,50], the first integral method [9,10], the (G'/G)-expansion method [11,12], the improved (G'/G)-expansion method [13,14], the (G'/G,1/G)-expansion method [15,16], the modified simple equation method [17-19], the Darboux transform [36-38], the generalized algebraic method [39], the modified mapping method [40], the extended homogeneous balance method [41], Jacobi elliptic function method [42], the Hirota bilinear method [44,48], the optimal homotopy asymptotic method [46], Exponential rational function method [20,21], the exp( exp hod [22-24], Kudryashov method [25] and modified Kudryashov method [26-

The generalized Kudryashov method [29-32] is a new solution technique to seek the exact solutions of nonlinear partial differential equations (PDEs) in mathematical physics. Recently, this method has received considerable attention due to its capabilities in extracting more and new exact solutions of PDEs rather than Kudryashov method. For example,

Demiray et al. [29] have been implemented the generalized Kudryashov method to attain new exact solutions of the time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time fractional generalized third-order KdV equation and obtained some new hyperbolic function solutions. Shafiqul et al. [30] applied the generalized Kudryashov method for solving some (3+1)-dimensional nonlinear evolution equations. The generalized Kudryashov method is used to four nonlinear PDEs namely, the Burgers equation, the modified Benjamin-Bona-Mahony (mBBM) equation, the Potential Kadomtsev-Petviashvili (PKP) equation and the Cahn-Hilliard equation. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, hyperbolic function solutions, and rational solutions [31]. Kaplan at al. [32] has obtained exact solutions of the nonlinear Jaulent-Miodek hierarchy and (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation by using the generalized Kudryashov method.

In the case of fractional order differential equations, all fractional order differential equation

converted into integer order ordinary differential equation by using the fractional complex transformation and the modified Riemann-Liouville derivative sense. Finally, we attempt to obtain a variety of exact solutions of the integer-order ordinary differential equations aided by Maple.

Based on the recent cited article [29-32] of the generalized Kudryashov method, the objective of this article is applied the generalized Kudryashov method to construct the new exact solutions for the time fractional coupled Boussinesq-Burger and approximate long water wave equation by using fractional complex transform and Jumarie's modified Riemann-Liouville derivatives sense.

The rest of this paper is arranged as follows: In Section 2, the description of the Conformable fractional derivative and some of its properties are discussed. In Section 3, we elaborate the generalized Kudryashov method in details. We have shown two application of the

generalized Kudryashov method for finding the new exact solutions of the nonlinear time fractional equations in Section 4. Finally, we briefly make a conclusion to the obtained results in Section 5.

2. Description of the Conformable fractional derivative and some of its properties

In this section, we give a definition and some important features of the Jumarie modified Riemann- Liouville derivative which is used in this paper.

The conformable fractional derivative of order (et) is defined by the following expression [33]:

Da№ = lira r—sa

Chain Rule [34]:

Assume f, g: (a,co) ->■ K be (left) a-differentiable functions, where (0 < a < l).Let MO = f(g(t)). Then h(t) is (left) K-diffe rentiable functions for all t with t.ia and g(t) =£0,

(00)= {t:n (V)(t)5(t>al-

We list some important properties for the conformable fractional derivative as follows:

3. Description of the generalized Kudryashov method

In this section, we will describe the algorithm of the generalized Kudryashov method for finding traveling wave solutions of nonlinear fractional differential equations (FDEs). Let us consider general nonlinear FDEs in the form

P(u, Dy, и, м , м u , Д2Ч w ..........) = 0, 0 < « < 1, (7)

where, u = u(t, xx, x2,....., xm), is an unknown function, P is a ^polynomial in

u = u(t, Xj, x2,....., xm) and its various partial derivatives, in which the nonlinear terms and

highest order derivatives are involved. The fractional complex transforms [35] and with the help of the Conformable fractional derivative the nonlinear PDEs of the fractional order have converted into ordinary differential equations (ODEs), which can be solved easily using simple calculus [1-3]. The main steps of this method are detailed in the article [29-32].

Step 1: Combine the real variables x, y, z and t by a compound variable £

U(t, X2,....., Xm ) = u(£), £ = KX1 + k2X2 +......+ kmXm ± ~ (8)

where, K, K, K3.... km and V are arbitrary constants. The traveling wave transformation (7), converts Eq. (8) into an ordinary differential equation (ODE) for u = u(£):

F (u, u', u", u ",....................) = 0, (9)

where, F is a polynomial of u and its derivatives and the superscripts indicate the ordinary derivatives with respect to £. If possible, we should integrate Eq. (9) term by term one or more times.

Step 2: Suppose the traveling wave solution of Eq. (9) can be expressed as follows:

,1X1 + 12 x

S aQ(ï)]

u(£) = jM-, (10)

S bQ(&

where the coefficients aj (0 < j < N, N e N) and b (0 < i < M,M e N)are constants to be determined and a^ may not be zero. The positive integer N and M can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq. (9). Moreover, we define the degree of u(£) as D(u(£)) = N - M, which gives rise to a degree of another expression as follows:

v d£q J

= (N - M) + q, D

vd£qJ j

= (N - M)p + s(N - M + q).

) as D( ) as D(

-s(N - M + q

M in Eq. (9

Therefore, we can find the value of N and M in Eq. (9), where Q = Q(£) satisfies the following ordinary differential equation:

Q'(t) = Q 2(£) - Q(£). (11)

The general solution of the Eq. (11) is Q(£) =-1-. (12)

1 ± A exp(£)

Step 3: After we determine the index parameter N and M, we substitute Eq. (10) along Eq.

(11) into Eq. (9) and collecting all the terms of the same power (Q(£))', i = 0,1,2,..... and

equating them to zero, we obtain a system of algebraic equations, which can be solved by Maple to get the values of a 's, b 's, k1, k2, k3,..., km, vand constant of integration.

Substituting the values of a s, b s, k1, k2, k3,..., km, vand other values into Eq. (10) along with general solutions of Eq. (11) completes the determination of the solution of Eq. (9).

4. Applications of the generalized Kudryashov method

In this section, we will implement the method described in section 3 to look for the new exact solutions for the nonlinear time fractional coupled Boussinesq-Burger equation and the time fractional approximate long water wave equation in shallow water. 4.1 The nonlinear coupled time fractional Boussinesq-Burger equation The Boussinesq-Berger equation is an interesting mathematical model that arises in the study of fluids flow in a dynamic system and describes the propagation of shallow water waves. The shallow water equations are a system of partial differential equations that describe the flow below a pressure surface in a fluid, motion of water bodies and flow in vertically well-mixed water bodies. A good understanding of its solutions is very helpful for coastal and civil engineers to apply the nonlinear water wave model to the harbor and coastal designs [38,44,48].

The soliton and multi-soliton solutions of the Boussinesq-Burger have been discussed in [36,37,43]. Wang and Chen [38] applied Darboux transformation with multi-parameters for the Boussinesq-Burgers (BB) equation and obtained some explicit solutions, including 2N-soliton solution and periodic solution. Gao et al. [39] solved the Boussinesq-Burger equation and (3+1)-dimensional Kadomtsev Petviashvili equation by using a generalized algebraic method. As a result, a variety of explicit exact traveling wave solutions, including solitary wave, Jacobi and Weierstrass elliptic function periodic, triangular periodic and rational solutions, are obtained. The exact traveling solutions of Boussinesq-Burger equations have been studied in [40,41]. Redy and Khalfallah [42] investigated the multi-phase periodic solutions for Boussinesq-Burgers equations are obtained by using of Jacobi elliptic function method. Jin-Ming and Yao-Ming [44] and Wazwaz A. M. [48] examine the Boussinesq-Burgers equation and the higher-order Boussinesq- Burgers equation and obtained the multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions by using the simplified Hirota's method. Wang et al. [45] investigated the Boussinesq-Burgers (BB)

equations based on the binary Bell polynomials, Hirota method, and symbolic computation. The binary-Bell-polynomial and bilinear forms for the BB equations are derived aided by Backlund transformations. Gupta and Saha Ray [46] studied the homotopy perturbation method (HPM) and optimal homotopy asymptotic method (OHAM) for finding approximate solutions of the Boussinesq-Burger equation. The approximate solutions of the Boussinesq-Burger equation are compared with the OHAM as well as with HPM. Gupta and Saha Ray [46] shown that the OHAM is accurate with less number of iterations in compared to HPM. Ravi et al. [47] build the new analytical exact solutions of the Boussinesq-Burgers equations, by means of Exp-function method.

In this part, we consider the time fractional coupled Boussinesq-Burger equation [49].

D"u -1 vx + 2uux = 0 -1 Uxx + 2(uv)x = 0

ussinesq-B ussinesq-B

y field and

t > 0,

where u(x, t)is the horizontal velocity field and v(x, t)is the height of the water surface above a horizontal level at the bottom. a is a parameter describing the order of the fractional time derivative and 0<a< 1. Sunil et al. [49] solved the time fractional Boussinesq-Burger's equation by using the residual power series method (RPSM) and found the good accuracy with the modified homotopy analysis transform method results.

For our goal, we present the traveling wave solutions for Eq. (13), and we perform the fractional complex transformation like as section 3.

u(x, t) = u(4), 4 = x--, (14)

where, c are nonzero constants.

Then Eq. (14) can be reduced to the following ordinary differential equations is in the form

- cu' -1 v' + 2uu' = 0

- cv' -1 u " + 2(uv) ' = 0

Integrating Eq. (15) once with respect to %, choosing the constant of integration as zero and we obtain the following ordinary differential equations respectively:

- cu -1 v + u2 = 0

- cv -1 u" + 2(uv) = 0

From first equation of (16), we get v = 2(u2 - cu)

Substituting Eq. (17) into the second equation of (16), we obtai

-1 u " + 4u3 + 2c 2u - 6cu2 = 0 w N (18)

respect to term u

where primes denote differentiation with respect to %. By balancing the highest order

derivative term u" with the nonlinear term u in Eq. (18), gives N = M +1. Therefore, the generalized Kudryashov me thod a llows us to use the solution in the following form:

u(%)=+a2Q, (19)

b0 + b,!

Now substituting Eq. (19) along with Eq. (11) into Eq. (18), we obtain a polynomial of Qk,(k = 0,1,2,...). Equating the coefficients of this polynomial of the same powers of Qto zero, we obtain a system of algebraic equations. This system of equations yields the values for c, a, a, a, \ and b values with the aid of symbolic computer software Maple, Mathematica or MATLAB.

Set-1: c = ±1, a = 0, a = 0, a = ±0.5b, b = -0.5b and bx = bx.

Set-2: c = ±1, a = +0.5^, a = ±b, a = +0.5^, b0 = -0.5b and b = b.

Set-3: c = ±0.5, a0 = 0, a = ±0.5b0, a2 = ±0.5b,b = b and b = b.

Set-4: c = ±0.5, a = ±0.5b0, a = ±0.5(b -b), a = +0.5^,b = b and b = b.

Set-5: c = , a0 = + ^ b, a =1

± -1 lb, a = 0.5b, b = -0.5b and b = b.

V2 ' 4V2 2 V" V2 J1

Set-6: c = , a = , a = 1 ±+1 Jjb, a = -0.5b, b = -0.5b and b = b.

Substituting Eq. (12) into Eq. (19) along with set-1 to set-6, we get the following families of

solutions.

Set-1 corresponds to the following solutions for the time fractional coupled Boussinesq-Burger equation.

u(*0 = + A2 exp(2£) -1 (20>

V( * ( ) 2 ^^ (21) (A exp(2£) -1)

where, £ = x + -

T(1 + a)

Set-2 corresponds to the following solutions for the time fractional coupled Boussinesq-Burger equation.

u(x, t) = ± A2exp(2£) (22)

(,) A exp(2£)-1 ( )

, , 2A2 exp(2£)

v(x,t) = —-Xp( £)X2 (23)

exp(2£) -1)

where, £ = x + -

T(1 + a)

Set-3 corresponds to the following solutions for the time fractional coupled Boussinesq-Burger equation.

"( X ' > = ± 2(1 + aU—)) <24)

v( .X, t ) = —

1 ( A exp (—) ^

(1 + A exp (—)) 0.5ta

where, E = x +

T(1 + a)

Set-4 corresponds to the following solutions for the time fractional coupled Boussinesq-Burger equation.

u(*, t) = ± AeXP(E^ (26)

2 (1 + A exp(E))

v( X, t ) = - -

« Boissii

Aexp(E) 1

olutions for the

(1 + A exp (—)) 0.5ta

where, e = x +

T(1 + a)

Set-5 corresponds to the following solutions for the time fractional coupled Boussinesq-

Burger equation.

-1)+ 4 A e

u( x, t ) = _ ± (A2exp(E ^ y ■ ■ a exp — (28)

v( *,t )=41 A4exp^Eexp62E)T/l 1 (29)

Set-6 corresponds to the following solutions for the time fractional coupled Boussinesq-Burger equation.

u(X, t) = — 4

:lV2(a2 exp(2—) -1)- 4A exp (—)

± l „ 2.A exp, 2— 1 j , A ex,p — ,

±1 „2 — - ■ a — — (30)

A2exp(2£) -1

v(X, t) = 4

^ a PYn^/- i -+- (-> A~ PYn \ //- i -+- i

1 A4 exp (4£) + 6A2 exp (2£) +1

(a2 exp (2£)-1)2

where, £ = x + —¡= I x-

42 T(1 + a)

The nonlinear time fractional approximate long water wave equation Consider the coupled Whitham-Broer-Kaup (WBK) equations which have been introduced by Whitham [50], Broer [51] and Kaup [52]. The equations describe the propagation of shallow water waves, with different dispersion relations. Consider the following the time fractional coupled Whitham-Broer-Kaup (WKB) equations is of the form (for a = 1, see [53,54]):

Dau + uu + v + bu = 0 I

t X X XX I, t > 0, ^ (32)

Dav + (uv)x + au^ - bvxx = °J

where, a is a parameter describing the order of the fractional time derivative and 0<a< 1. In WKB equations (32), the field of horizontal field of horizontal velocity is represented by

u = u( x, t), v = v( x, t) which is the height that deviates from equilibrium position of liquid, and the constants a, b are represented in different diffusion power.

If a = 0 and b = 1, the fractional coupled WKB equations reduces to the nonlinear time

fractional approximate long wave (ALW) equations which are a special case of WKB equations.

In this part, we consider the following nonlinear time fractional approximate long wave (ALW) equations [5, 55,56]:

Dau + uu + v +1 u = 0

t XX ^^ XX

D^v + (uv)x -1 vxx = 0

t >0, (33)

where, a is a parameter describing the order of the fractional time derivative and 0 < a < 1.

Yan [5] applied the fractional sub-equation method for constructing new exact traveling solutions of the fractional ALW equations. As a result, three types of traveling wave solutions are found and expressed by generalized hyperbolic function solutions, generalized trigonometric function solutions, and rational solutions. Guner and his collogues [55] executed the G'/G-expansion method to obtain some new solutions of the fractional ALW equations. Saha Ray [56] solved the time fractional ALW equations using coupled fractional reduced differential transform method. The obtained results are compared with Adomian decomposition method (ADM) and variational iteration method (VIM), he concludes that the implemented coupled fractional reduced differential transform method is superior to others. For our purpose, we introduce the same transformations of Eq. (18) and same procedure we have the ODE from Eq. (33),

- cu' + uu' + v' +1 u" = 0

- cv' + (uv) " -1 v " = 0

ns of Eq. (18)

pect to ifferent

Integrating Eq. (34) once with respect to £, choosing the constant of integration as zero and

we obtain the following ordinary differential equations respectively: we obtain the following ordinary differential equations respectively:

1 2 1 f ~

- cu + — u + v + — u = 0

- cv + (uv) -1 v' = 0

From first equation of (35), we get

v = cu -1 u2 - — u' (36)

Substituting Eq. (36) into the second equation of (35), we obtain 11 3

-- u" + -u3 + c2u -- cu2 = 0 (37)

where primes denote differentiation with respect to £. By balancing the highest order

derivative term u" with the nonlinear term u3 in Eq. (37), gives N = M +1. Therefore, the generalized Kudryashov method allows us to use the solution in the following form:

= ao + aQ + a2Q2

b + bQ

Now substituting Eq. (38) along with Eq. (11) into Eq. (37), we obtain a polynomial of Qk,(k = 0,1,2,...). Equating the coefficients of this polynomial of the same powers of Qto zero, we obtain a system of algebraic equations. This system of equations yields the values for c, a0, a, a2, b and b values with the aid of symbolic computational software Maple, Mathematica or MATLAB.

Set-1: c = ±1, a0 = 0, a = 0, a = ±b, b = -0.5b and b = b. Set-2: c = ±1, a0 = +b, a = ±2b, a2 = +b, b = -0.5b and b = b. Set-3: c = ±0.5, a = 0, a = ±b, a = ±b, b = b and b = b. Set-4: c = ±0.5, a0 = ±b, a = ±(b -b), a = +b,b = b and b = b.

Set-5: c = I, a

Ib, a =

- I -1 |b, a = b, b = -0.5b and b = b •

V21 ' a* + 2V2b a

Set-6: c = I, a = Ib, a = - I + l||b, a = -b, b = -0.5b and b = b •

Substituting Eq. (12) into Eq. (37) along with set-1 to set-6, we get the following families of solutic

Set-1 corresponds to the following solutions for the time fractional approximate long wave

equation.

u( x, t) = +

A exp(2£) -1

x,,) 4A2 exp(2£) (40)

[A exp(2£) -1)2

where, E = x +-

T(1 + a)

Set-2 corresponds to the following solutions for the time fractional approximate long wave equation.

„( x, t) ^JA^esM.

A2exp(2£) -1

4A2 exp(2£) (a2 exp(2£) -1)2

where, E = x + -

v( x, t ) = -

T(1 + a)

Set-3 corresponds to the following solutions for the time fractional approximate long wave equation.

u( x, t ) = ±-1--(43)

v( x, t ) = ■ (44)

where, E = x +-

, E —+a)

x • £

orrespo uation.

Set-4 corresponds to the following solutions for the time fractional approximate long wave equat

u( x, t ) = ± A exp (E) (45)

1 + A exp(E)

v(x, t) = ■ A exp(E)

(1 + A exp(E))2 (46)

where, £ = x +

Set-5 corresponds to the following solutions for the time fractional approximate long wave

equation.

u( x, t ) = 1

lV2(a2 exp(2£) -1)+ 4A exp (£)

A2 exp(2£) -1

f A2 exp (2£)- 2A exp (£) +1 ' (A exp(£) +1)2 _

v( x, t ) = — 4

where, £ = x + —I x — V2 a

Set-6 corresponds to the following solutions for the time equation.

u( xt ) =1

v( x, t ) = -

1v2 (a2 exp(2£) -1)+ 4 A exp (£)~ A2 exp(2£) -1

^A2 exp (2£)+ 2 A exp (£) +1

(A exp(£) -1)

where, £ = x + —;= I V2

me fractional )

approximate long wave

5. Conclusions

The generalized Kudryashov method has been successfully implemented to seek exact solutions for the coupled time fractional Boussinesq-Burger equation and approximate long water wave equation. The results demonstrate that generalized Kudryashov method is straightforward and concise mathematical tool to establish the exact analytical solutions of nonlinear time fractional equations. Finally, we conclude that the studied method can be more effectively applied to investigate other nonlinear fractional partial differential equations which frequently arise in mathematical physics and other scientific application fields.

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