A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines

Chaos, Solitons and Fractals 78 (2015) 148–155

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines E.M.E. Zayed∗, K.A.E. Alurrfi Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt

a r t i c l e

i n f o

Article history: Received 28 June 2015 Accepted 16 July 2015 Available online 1 September 2015 Keywords: New Jacobi elliptic function expansion method Exact solutions Nonlinear low-pass electrical lines Kirchhoff’s laws

a b s t r a c t The first elliptic function equation is used in this article to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing the nonlinear low-pass electrical lines are obtained with the aid of computer algebraic system Maple. Based on Kirchhoff’s law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction In the recent years, investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena in such as fluid mechanics, hydrodynamics, optics, plasma physics, solid state physics, biology and so on. Several methods for finding the exact solutions to nonlinear equations in mathematical physics have been presented, such as the inverse scattering method [1], the Hirota bilinear transform method [2], the truncated Painlevé expansion method [3–6], the Bäcklund transform method [7,8], the exp-function method [9–11], the tanhfunction method [12,13], the Jacobi elliptic function expan sion method [14–16], the ( GG )-expansion method [17–22], 



the modified ( GG )-expansion method [23], the ( GG , G1 )expansion method [24–27], the modified simple equation method [28–30], the multiple exp-function algorithm method [31,32], the transformed rational function method [33], the local fractional series expansion method [34],

∗

Corresponding author. E-mail addresses: [email protected] (E.M.E. Zayed), alurrfi@ yahoo.com (K.A.E. Alurrfi). http://dx.doi.org/10.1016/j.chaos.2015.07.018 0960-0779/© 2015 Elsevier Ltd. All rights reserved.

the first integral method [35,36], the generalized Riccati equation mapping method [37,38] and so on. The objective of this article is to use a new Jacobi elliptic function expansion method to construct the exact solutions of the following nonlinear PDE governing wave propagation in nonlinear low-pass electrical transmission lines [39]:

∂ 2V (x, t ) ∂ 2V 2 (x, t ) ∂ 2V 3 (x, t ) −α +β 2 2 ∂t ∂t ∂t2 2 4 4 ∂ V (x, t ) δ ∂ V (x, t ) − δ2 − = 0, 12 ∂ x2 ∂ x4

(1.1)

where α , β and δ are constants, while V(x, t) is the voltage in the transmission lines. The variable x is interpreted as the propagation distance and t is the slow time. The physical details of the derivation of Eq. (1.1) using the Kirchhoff’s laws are given in [39], which are omitted here for simplicity. Note that Eq. (1.1) has been discussed in [39] using an auxiliary equation method [40] and its exact solutions have been found. This paper is organized as follows: In Section 2, the description of a new Jacobi elliptic function expansion method is given. In Section 3, we use the given method described in Section 2, to find exact solutions of Eq. (1.1). In Section 4, we solve Eq. (1.1) using a direct method. In Section 5, physical

E.M.E. Zayed, K.A.E. Alurrfi / Chaos, Solitons and Fractals 78 (2015) 148–155

explanations of some results are presented. In Section 6, some conclusions are obtained. 2. Description of a new Jacobi elliptic function expansion method Consider a nonlinear PDE in the form

P (V, Vx , Vt , Vxx , Vtt , . . . ) = 0,

(2.1)

where V = V (x, t ) is a unknown function, P is a polynomial in V(x, t) and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. Let us now give the main steps of the Jacobi elliptic function expansion method [41]: Step 1. We look for the voltage V(x, t) in the traveling form

V (x, t ) = V (ξ ),

ξ=



k(x − λt ),

(2.2)

where k and λ are undetermined positive parameters, and λ is the velocity of propagation, to reduce Eq. (2.1) to the following nonlinear ordinary differential equation (ODE):

H (V, V  , V  , . . . ) = 0,

(2.3)

where H is a polynomial of V(ξ ) and its total derivatives d . V  (ξ ), V  (ξ ), . . . and  = dξ Step 2. We suppose that the solution of Eq. (2.3) has the form: N 

V (ξ ) = g0 +

i=1

 + fi



z(ξ ) 1 + z2 (ξ )

1 − z (ξ ) 1 + z2 (ξ ) 2

i−1   gi



z(ξ ) 1 + z2 (ξ )



,

(2.4)

2 z (ξ ) = a + bz2 (ξ ) + cz4 (ξ ),

(2.5)

where a, b, c, g0 , gi , fi (i = 1, 2, . . . , N) are constants to be determined later, such that gN = 0 or fN = 0. Step 3. We determine the positive integer N in (2.4) by balancing the highest-order derivatives and the nonlinear terms in Eq. (2.3). Step 4. Substituting (2.4) along with Eq. (2.5) into Eq. (2.3) and collecting all the coefficients of zi (ξ ) (i = 0, 1, 2, . . . ), then setting these coefficients to zero, yield a set of algebraic equations, which can be solved by using the Maple or Mathematica to find the values of g0 , gi , fi , λ, k, a, b, c. Step 5. It is well-known [41] that Eq. (2.5) has families of Jacobi elliptic function solutions as follows: c

z(ξ )

−(1 + m2 )

m2

snξ

2m2 − 1

−m2

cnξ

−(1 + m2 )

1

−m

2m − 1

1−m

5

1 4

1 − 2m 2

1 4

nsξ ± csξ

6

1 − m2 4

1 + m2 2

1 − m2 4

ncξ ± scξ or

No.

a

b

1

1

2

1 − m2

3

m2

4

2

Note that there are other Jacobi elliptic function solutions of Eq. (2.5) which are omitted here for simplicity. In this table, snξ = sn(ξ , m), cnξ = cn(ξ , m), dnξ = dn(ξ , m), nsξ = ns(ξ , m), csξ = cs(ξ , m), dsξ = ds(ξ , m), scξ = sc(ξ , m), sdξ = sd(ξ , m) are the Jacobi elliptic function with modulus m, where 0 < m < 1. These functions degenerate into hyperbolic functions when m → 1 as follows: snξ → tanh ξ , cnξ → sechξ , dnξ → sechξ , nsξ = coth ξ , csξ = cschξ , dsξ = cschξ , scξ = sinh ξ , sdξ = sinh ξ , ncξ = cosh ξ . and into trigonometric functions when m → 0 as follows: snξ → sin ξ , cnξ → cos ξ , dnξ → 1, nsξ → csc ξ , csξ → cot ξ ,dsξ → csc ξ , scξ → tan ξ , sdξ → sin ξ , ncξ → sec ξ . Also, these functions satisfy the following formulas: sn2 ξ + cn2 ξ = 1, dn2 ξ + m2 sn2 ξ = 1, and sn ξ =   2 cnξ dnξ , cn ξ = −snξ dnξ , dn ξ = −m snξ cnξ , cd ξ = −(1 − m2 )sdξ ndξ , ns ξ = −csξ dsξ , dc ξ = (1 − m2 )ncξ scξ , nc ξ = scξ dcξ , nd ξ = m2 cdξ sdξ , sc ξ = dcξ ncξ , cs ξ = −nsξ dsξ , ds ξ = −csξ nsξ , sd ξ = ndξ cdξ , where  = ddξ . Step 6. Substituting the values of g0 , gi , fi , k, λ, a, b, c as well as the solutions of Eq. (2.5) obtained in Step 5, into (2.4) we have the exact solutions of Eq. (2.1).

3. Exact solutions of Eq. (1.1) using the proposed method of Section 2 In this section, we apply the Jacobi elliptic function expansion method of Section 2 to find families of new Jacobi elliptic function solutions of Eq. (1.1). To this end, we use the transformation (2.2) to reduce Eq. (1.1) to the following nonlinear ODE:

d2 dξ 2



k2 δ 4 d2V + (kδ 2 − kλ2 )V + α kλ2V 2 − β kλ2V 3 12 dξ 2



= 0. (3.1)

where z(ξ ) satisfies the Jacobi elliptic equation:



149

2

2

Integrating Eq. (3.1) twice and vanishing the constants of integration, we find the following ODE:

K 2 d 2V + (K − U )V + αUV 2 − β UV 3 = 0. 12 dξ 2

where K = kδ 2 and U = kλ2 . 2 Balancing ddξV2 with V3 gives N = 1. Therefore, (2.4) reduces to

V (ξ ) = g0 + g1

ncξ = (cnξ )



z(ξ ) 1 + z2 (ξ )



 + f1



1 − z2 (ξ ) , 1 + z2 (ξ )

(3.3)

where g0 , g1 and f1 are constants to be determined such that g1 = 0 or f1 = 0. Substituting (3.3) along with Eq. (2.5) into Eq. (3.2) and collecting all the coefficients of zi (ξ ), (i = 0, 1, . . . , 6) and setting them to zero, we have the following algebraic equations:

z6 : 4cK 2 f1 − 12K f1 + 12Kg0 + 12U β f13 − 36U β f12 g0 + 12U α f12 + 36U β f1 g20 − 24U α f1 g0 + 12U f1

nsξ = (snξ )−1 2

(3.2)

− 12U β g30 + 12U α g20 − 12Ug0 = 0,

−1

z : 12Kg1 − 12Ug1 + K 2 bg1 − 6K 2 cg1 − 36U β f12 g1 5

cnξ 1±snξ

− 36U β g20 g1 −24U α f1 g1 +24U α g0 g1 +72U β f1 g0 g1 =0, 4

z : 36Kg0 − 12K f1 + 12U f1 − 36Ug0 + 8K 2 b f1 − 12K 2 c f1

150

E.M.E. Zayed, K.A.E. Alurrfi / Chaos, Solitons and Fractals 78 (2015) 148–155

− 12U α f12 + 36U α g20 + 12U α g21 − 36U β f13 − 36U β g30 + 36U β f1 g20 + 36U β f12 g0 + 36U β f1 g21 − 36U β g0 g21 − 24U α f1 g0 = 0, z3 : 24Kg1 − 24Ug1 + 2K 2 ag1 − 6K 2 bg1 + 2K 2 cg1 −12U β g31

m=

1 4

2

2

z : 12K f1 + 36Kg0 − 12U f1 − 36Ug0 + 12K a f1 − 8K b f1 − 12U α f12 + 36U α g20 + 12U α g21 + 36U β f13 − 36U β g30

⎡

V (ξ ) =

13α ⎣ 110β

− 36U β f1 g20 + 36U β f12 g0 − 36U β f1 g21 − 36U β g0 g21 + 24U α f1 g0 = 0, z : 12Kg1 −12Ug1 −6K 2 ag1 +K 2 bg1 −36U β f12 g1 −36U β g20 g1 + 24U α f1 g1 + 24U α g0 g1 − 72U β f1 g0 g1 = 0, z0 : 12K f1 − 4aK 2 f1 + 12Kg0 − 12U β f13 − 36U β f12 g0 + 12U α − 12U β

f12 g30

− 36U β

α f1 g0 − 12U f1

+ 12U α

= 0.

f1 g20 + 24U g20 − 12Ug0

With reference to Step 5, of Section 2, we have the following results. Result 1. If we substitute a = 1, b = −(1 + m2 ), c = m2 into the algebraic equations (3.4) and solve them by Maple 14, we have the following two cases: Case 1.

where ξ =



> 0. Case 2.

− 2δ 2 (23αα2 −9β) x − 2

27α 2 β t, 2(2α 2 −9β)2

(3.5)

2α 2 < 9β , β

48α 2 3888α 2 β , U =− , 2 16α − 81β (16α 2 − 81β)2 4α 4α , g1 = 0, f1 = . m = 0, g0 = 9β 9β

(3.7)

In this case, we deduce that z(ξ ) = sn(ξ , 0) → sin ξ . Now, Eq. (3.2) has the trigonometric solution





8α 1 , 9β 1 + sin2 ξ

where ξ =

β < 0.



48α 2 x− δ 2 (16α 2 −81β)

(3.8)



α β − (163888 t, 16α 2 > 81β , α 2 −81β)2 2

Result 2. If we substitute a = 1 − m2 , b = 2m2 − 1, c = −m2 into the algebraic equations (3.4) and solve them by Maple 14, we have the following two cases: Case 1.

K =

14352α 13892736α β ,U = − , 5(1027α 2 − 4840β) (1027α 2 − 4840β)2 2

2

 67  ⎤ 23

 67  ⎦, 1

ξ, 4

(3.10)

23

14352α 2 x− 5δ 2 (1027α 2 −4840β)



13892736α β − (1027 t, β < α 2 −4840β)2 2

(3.11)

In this case, we deduce that z(ξ ) = cn(ξ , 1) → sechξ . Now, Eq. (3.2) has the hyperbolic solution



V (ξ ) =



8α sech ξ , 9β 1 + sech2 ξ



2

(3.12)

α − δ 2 (1648 x− α 2 −81β) 2



3888α 2 β t, β > 0, 16α 2 < (16α 2 −81β)2

Result 3. If we substitute a = m2 , b = −(1 + m2 ), c = 1 into the algebraic equations (3.4) and solve them by Maple 14, we have the following two cases Case 1.

3α 2 27α 2 β ,U = , 2 2(2α − 9β) 2(2α 2 − 9β)2 2α α , g1 = , f1 = 0. m = 1, g0 = 3β 3β K =−

(3.13)

In this case, we deduce that z(ξ ) = ns(ξ , 1) → coth ξ . Now, Eq. (3.2) has the hyperbolic solution

V (ξ ) =

K =

V (ξ ) =



ξ , 14

48α 2 3888α 2 β , U= , 16α 2 − 81β (16α 2 − 81β)2 4α 4α , g1 = 0, f1 = − . m = 1, g0 = 9β 9β

81β .

(3.6)



1 + cn2



0, 1027α 2 > 4840β . Case 2.

where ξ =

In this case, we deduce that z(ξ ) = sn(ξ , 1) → tanh ξ . Now, Eq. (3.2) has the hyperbolic solution

  α (1 + tanh ξ )2 V (ξ ) = , 3β 1 + tanh2 ξ

where ξ =



7 + cn2

K =−

(3.4)

3α 2 27α 2 β K =− , U= , 2 2(2α − 9β) 2(2α 2 − 9β)2 α 2α , g1 = , f1 = 0. m = 1, g0 = 3β 3β

(3.9)

In this case, we deduce that z(ξ ) = cn(ξ , m) where 0 < m < 1. Now, Eq. (3.2) has the Jacobi elliptic solution

+ 72U β f12 g1 − 72U β g20 g1 + 48U α g0 g1 = 0, 2

26α 39α 67 , g1 = 0, f1 = . , g0 = 23 55β 110β

  α (1 + coth ξ )2 , 3β 1 + coth2 ξ

where ξ =



9β . Case 2.

− 2δ 2 (23αα2 −9β) x − 2

(3.14)



27α 2 β t, 2(2α 2 −9β)2

48α 2 3888α 2 β ,U = − , (16α 2 − 81β) (16α 2 − 81β)2 4α 4α , g1 = 0, f1 = − . m = 0, g0 = 9β 9β

β > 0, 2α 2 <

K =

(3.15)

In this case, we deduce that z(ξ ) = ns(ξ , 0) → csc ξ . Now, Eq. (3.2) has the trigonometric solution





8α csc2 ξ V (ξ ) = , 9β 1 + csc2 ξ where ξ = 81β .



48α 2 x− δ 2 (16α 2 −81β)

(3.16)



α β − (163888 t, β < 0, 16α 2 > α 2 −81β)2 2

E.M.E. Zayed, K.A.E. Alurrfi / Chaos, Solitons and Fractals 78 (2015) 148–155

Result 4. If we substitute a = −m2 , b = 2m2 − 1, c = 1 − m2 into the algebraic equations (3.4) and solve them by Maple 14, we have the following three cases: Case 1.

14352α 13892736α β K = ,U = − , 5(1027α 2 − 4840β) (1027α 2 − 4840β)2 2

1 m= 4

2

26α 39α 67 , g0 = , g1 = 0, f1 = − . 23 55β 110β

(3.17)

 67  ⎤ ξ , 14 23 13α ⎣  V (ξ ) =  67  ⎦, 110β 1 + nc2 ξ , 14 23 where ξ =



1 + 5nc2



14352α 2 x− 5δ 2 (1027α 2 −4840β)

0, 1027α 2 > 4840β . Case 2.



In this case, we deduce that z(ξ ) = ns(ξ , m) ± cs(ξ , m) where 0 < m < 1 . Now, Eq. (3.2) has the Jacobi elliptic solution

α 2α m V (ξ )= ± 3β 3β

(3.18)

13892736α β − (1027 t, β < α 2 −4840β)2 2

48α 2 3888α 2 β , U =− , 2 16α − 81β (16α 2 − 81β)2 4α 4α m = 0, g0 = , g1 = 0, f1 = − . 9β 9β

(3.19)

(3.20) α β − (163888 t, 16α 2 > 81β , α 2 −81β)2 2

Case 3.

48α 2 3888α 2 β , U= , 16α 2 − 81β (16α 2 − 81β)2 4α 4α m = 1, g0 = , g1 = 0, f1 = . 9β 9β

where ξ =

where ξ = 81β .



α2 x− − δ 2 (1648 α 2 −81β)

(3.22)



> 0,

16α 2

<

2

Result 5. If we substitute a = 14 , b = 1−2m , c = 14 into the al2 gebraic equations (3.4) and solve them by Maple 14, we have the following two cases: Case 1.

K =−

−

12α 2 108α 2 β ,U = , (m2 + 1)(2α 2 − 9β) (m2 + 1)(2α 2 − 9β)2

2α m α , g1 = ± m = m, g0 = 3β 3β

2

9β 2 f12

2(α − 9β 2 f12 ) 2



54α 2 β t, β > 0, 2α 2 < 9β . (2α 2 −9β)2

2 , f1 = 0. m2 + 1

(3.23)

, g0 =

α

3β

, g1 = 0, f1 = f1 , (3.27)

where α 2 < 9β 2 f12 .In order that 0 < m < 1, we have to choose the condition 2α 2 < 9β 2 f12 . In this case, we deduce that the Jacobi elliptic solution of Eq. (3.2) is as follows:

α

3β

    2 ⎤ β 2 f12 9β 2 f12 ± cs , − ξ , − 2(α92 −9 ξ 2 2 2 2 2 β f1 ) 2(α −9β f1 ) ⎢ ⎥ + f1 ⎣     2 ⎦, 9β 2 f12 9β 2 f12 1 + ns ξ , − 2(α 2 −9β 2 f 2 ) ± cs ξ , − 2(α 2 −9β 2 f 2 ) 1 − ns

1

1

(3.28) where ξ =



9β . 3888α 2 β t, β (16α 2 −81β)2

− δ 2 (26αα2 −9β) x −

(3.26)

12(α 2 − 9β 2 f12 ) 108(α 2 − 9β 2 f12 )β ,U = , 2 2α − 9β (2α 2 − 9β)2



8α 1 , 9β 1 + cosh2 ξ





⎡

(3.21)

In this case, we deduce that z(ξ ) = nc(ξ , 1) → cosh ξ . Now, Eq. (3.2) has the hyperbolic solution

V (ξ ) =

β >

In this case, we deduce that z(ξ ) = ns(ξ , m) ± cs(ξ , m) → cothξ ± cschξ . Now, Eq. (3.2) has the hyperbolic solution

V (ξ ) =

K =−



108α 2 β t, (m2 +1)(2α 2 −9β)2

6α 2 54α 2 β α ,U = , m = 1, g0 = , 2 3β (2α − 9β) (2α 2 − 9β)2 2α g1 = ± , f1 = 0. (3.25) 3β

m=

β < 0.

2

Case 2.





12α − δ 2 (m2 +1 x− )(2α 2 −9β)



K =−

8α sec2 ξ V (ξ ) = , 9β 1 + sec2 ξ 48α 2 x− δ 2 (16α 2 −81β)



0, 2α 2 < 9β . Note that, if m → 1 in (3.23), then we have

K =−





where ξ =

In this case, we deduce that z(ξ ) = nc(ξ , 0) → sec ξ . Now, Eq. (3.2) has the trigonometric solution

where ξ =



2 ns(ξ , m) ± cs(ξ , m) , m2 + 1 1 + (ns(ξ , m) ± cs(ξ , m))2

  α (1 ± (cothξ + cschξ ))2 V (ξ ) = , 3β 1 + (cothξ ± cschξ )2

K =



(3.24)

In this case, we deduce that z(ξ ) = nc(ξ , m) where 0 < m < 1. Now, Eq. (3.2) has the Jacobi elliptic solution

⎡

151

12(α 2 −9β 2 f 2 )

− δ 2 (2α 2 −9β)1 x −



108(α 2 −9β 2 f12 )β (2α 2 −9β)2 t,

2

2

β < 0, 2α 2 > 2

1+m 1−m into Result 6. If we substitute a = 1−m 4 ,b = 2 ,c = 4 the algebraic equations (3.4) and solve them by Maple 14, we have the following two cases: Case 1.

K =−

6α 2 54α 2 β , U = , 2α 2 − 9β (2α 2 − 9β)2

m = 1, g0 =

α

3β

, g1 = 0, f1 = −

α

3β

,

(3.29)

In this case, we deduce that z(ξ ) = nc(ξ , 1) ± sc(ξ , 1) → cn(ξ , 1) sechξ → . Now, cosh ξ ± sinh ξ or z(ξ ) = 1 ± tanh ξ 1 ± sn(ξ , 1)

152

E.M.E. Zayed, K.A.E. Alurrfi / Chaos, Solitons and Fractals 78 (2015) 148–155

Eq. (3.2) has the hyperbolic solutions



V (ξ ) =

96αU (K − U )csch2



2α (cosh ξ ± sinh ξ ) , 3β 1 + (cosh ξ ± sinh ξ )2 2

(3.30)

V2 (ξ ) =

 1 2

) − 12(KK−U ξ 2



 64α

2U 2

+ 72β U (K − U ) 1 + ε coth

1 2



) − 12(KK−U 2

2 , ξ

or

(4.3)

  2 α sech ξ V (ξ ) = , 3β 1 ± tanh ξ where ξ =



−

Case 2.

(3.31)

6α 2 x− δ 2 (2α 2 −9β )



54α 2 β t, (2α 2 −9β)2

β > 0, 2α 2 < 9β .

m=

9β 2 f12

, g0 = 2

2α 2 − 9β 2 f 1

(3.32)

where 2α 2 > 9β 2 f12 . In order that 0 < m < 1, we have to choose the condition α 2 > 9β 2 f12 . In this result, we deduce that the Jacobi elliptic solutions of Eq. (3.2) are as follows:

α 3β ⎡     2 ⎤ 9β 2 f 2 9β 2 f 2 1 − nc ξ , 2α 2 −9β12 f 2 ± sc ξ , 2α 2 −9β12 f 2 ⎢ ⎥ 1 1 + f1 ⎣     2 ⎦, 9β 2 f12 9β 2 f12 1 + nc ξ , 2α 2 −9β 2 f 2 ± sc ξ , 2α 2 −9β 2 f 2

V (ξ ) =

1

(3.33) or

V (ξ ) =

α

± f1 sn(ξ ,

ξ=

where

9β

2 f2 1

2α 2 − 9β 2 f12

6(9β 2 f12 −2α 2 )

δ 2 (2α 2 −9β )

9β , β > 0.

V4 (ξ ) = 8αU + ε

.

2 8αU K2

+

288β U (K−U ) K2

α 2U 2

>

− 92 β U (K

− U ), we have

−24(K − U )



64α

2U 2



+ 288β U (K − U ) cosh

) − 12(KK−U ξ 2

.

(4.5) (3) If K > U, ε = ±1, α 2U 2 > − 92 β U (K − U ), we have V5 (ξ ) = 8αU + ε



−24(K − U ) 64α 2U 2 + 288β U (K − U ) cos



12(K−U ) K2

. ξ

(4.6) (4) If K < U, ε = ±1, V6 (ξ ) =

8αU + ε

x−

−

),

(3.34)



α 2U 2

<

− 92 β U (K

− U ), we have

−24(K − U )



−(288β U (K − U ) + 64α 2U 2 ) sinh

(5) If K > U, ε = ±1, V7 (ξ ) =

54(9β 2 f12 −2α 2 )β

(2α 2 −9β)2

t, 2α 2 <

8αU + ε

) − 12(K−U ξ 2

.

K



α 2U 2

>

− 92 β U (K

− U ), we have

−24(K − U ) 64α

2U 2

+ 288β U (K − U ) sin



(6) If K < U, ε = ±1, β U > 0, we have 12(K − U )sech

Let us now solve Eq. (3.2) using a direct method. To this end, we multiply Eq. (3.2) by V (ξ ) and integrate with zero constant of integration, we get

(V  (ξ ))2 = a1V 2 (ξ ) + b1V 3 (ξ ) + c1V 4 (ξ ),

(1) If K < U, ε = ±1, we have −96αU (K − U )sech



2

1 2

) − 12(KK−U ξ 2



64α 2U 2 + 72β U (K − U ) 1 + ε tanh

1 2

V8 (ξ ) = −8αU + 2ε

2

 1 2

−72β U (K − U ) tanh

−12(K − U )csch V9 (ξ ) = −8αU + 2ε

2

 1 2

 1 2

. ξ

2 ,



) − 12(KK−U ξ 2

−72β U (K − U ) coth

 1 2

,

(4.9)



) ξ − 12(KK−U 2

.

(4.10)

12(K − U ) sec2 V10 (ξ ) = −8αU + 2ε

(4.2)



) ξ − 12(KK−U 2

(7) If K > U, ε = ±1, β U > 0, we have



) − 12(KK−U ξ 2

) ξ − 12(KK−U 2



(4.1)

) , b1 = − 8Kα2U , c1 = 6Kβ2U . where a1 = − 12(KK−U 2 It is well-known [42] that Eq. (4.1) has many solutions. With the aid of these solutions, we have the following solutions of Eq. (3.2):



12(K−U ) K2

(4.8)

4. Further results

V1 (ξ ) =

  ) ξ + ε − 12(KK−U 2

(4.7)



3β

 K 2 exp

(2) If K < U, ε = ±1,

α , g1 = 0, f1 = f1 , 3β

1

V3 (ξ ) =

(4.4)

6(9β 2 f12 − 2α 2 ) 54(9β 2 f12 − 2α 2 )β K = ,U = − , 2 2α − 9β (2α 2 − 9β)2



  ) −48(K − U ) exp ε − 12(KK−U ξ 2



 1 2

12(K−U ) K2

72β U (K − U ) tan

ξ  1 2



12(K−U ) K2

, ξ (4.11)

E.M.E. Zayed, K.A.E. Alurrfi / Chaos, Solitons and Fractals 78 (2015) 148–155

153

Fig. 1. The plot of solution (3.6) when α = 1, β = 1, δ = 1.

Fig. 4. The plot of solution (3.20) when α = 3, β = −1, δ = 1.

Fig. 2. The plot of solution (3.12) when α = 3, β = 2, δ = 1.

Fig. 5. The plot of solution (3.26) when α = 2, β = 1, δ = 24.

V11 (ξ ) =

12(K − U ) csc2 ( 12 −8αU + 2ε





12(K−U ) K2

72β U (K − U ) cot ( 12

ξ)

12(K−U ) K2

ξ)

.

(4.12) (8) If K < U, ε = ±1, α 2U = − 29 β(K − U ), we have



−3(K − U ) 1 V12 (ξ ) = 1 + ε tanh ( 2αU 2



12(K − U ) − ξ) , K2 (4.13)



−3(K − U ) 1 V13 (ξ ) = 1 + ε coth ( 2αU 2 Fig. 3. The plot of solution (3.14) when α = 2, β = 1, δ = 1.



12(K − U ) − ξ) . K2 (4.14)

154

E.M.E. Zayed, K.A.E. Alurrfi / Chaos, Solitons and Fractals 78 (2015) 148–155

Fig. 6. The plot of solution (3.30) when α = 2, β = 3, δ = 2.

Fig. 8. The plot of solution (4.11) when α = 3, β = 4, δ = 1, K = 1, U = 2 , ε = 1. 3

5. Physical explanations of some results In this section, we have presented some graphs of the exact solutions (3.6), (3.12), (3.14), (3.20), (3.26), (3.30), (4.2) and (4.11) constructed by taking suitable values of involved unknown parameters to visualize the mechanism of the original equations (1.1) (Figs. 1–8). These solutions are kink, singular kink-shaped soliton solution, bell-shaped soliton solutions, singular bell-shaped soliton solutions, hyperbolic solutions and trigonometric periodic solutions. For more convenience the graphical representations of these solutions are shown in the following figures: 6. Conclusions

Fig. 7. The plot of solution (4.2) when α = 3, β = 1, δ = 1, K = 1, U = 4 , ε = 1. 3

(9) If K < U, ε = ±1, α = 0, we have

V14 (ξ ) = ∓

48(K − U ) exp (ε K2 +

288β U (K−U ) K2



) − 12(KK−U ξ) 2

exp (2ε



) − 12(KK−U ξ) 2

, (4.15)

ξ=



√ K x − Ut, K, U > 0. δ2

The new Jacobi elliptic function expansion method described in Section 2 of this article has been applied to construct many new exact solutions of the nonlinear PDE (1.1) which describes the nonlinear low-pass electrical transmission lines with the aid of Maple 14. On comparing our results obtained in Section 3 of this article with the results obtained in [39] using the auxiliary equation method, we conclude that our results are different and new. So, the advantage of the method used in this article over the method used in [39] is that the first method yields results in terms of Jacobi elliptic functions which are more generalization than the results obtaining using the second method. Also, the results obtained in Section 4 using a direct method and the results obtained in [39] using an auxiliary equation method are different. In Section 5, we have presented some graphs of some exact solutions of Section 3 by choosing suitable values of parameters. Finally, all solutions obtained in this article have been checked with the Maple 14 by putting them back into the original Eq. (1.1).

E.M.E. Zayed, K.A.E. Alurrfi / Chaos, Solitons and Fractals 78 (2015) 148–155

Acknowledgment The authors wish to thank the referee for his comments on this paper. References [1] Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering transform. New York, NY, USA: Cambridge University Press; 1991. [2] Hirota R. Exact solutions of the KdV equation for multiple collisions of solutions. Phys Rev Lett 1971;27:1192–4. [3] Weiss J, Tabor M, Carnevale G. The Painlevé property for partial differential equations. J Math Phys 1983;24:522–6. [4] Kudryashov NA. Exact soliton solutions of a generalized evolution equation of wave dynamics. J Appl Math Mech 1988;52:361–5. [5] Kudryashov NA. Exact solutions of the generalized KuramotoSivashinsky equation. Phys Lett A 1990;147:287–91. [6] Kudryashov NA. On types of nonlinear nonintegrable equations with exact solutions. Phys Lett A 1991;155:269–75. [7] Miura MR. Bäcklund transformation. Berlin, Germany: Springer; 1978. [8] Rogers C, Shadwick WF. Bäcklund transformations and their applications. New York, NY, USA: Academic Press; 1982. [9] He J-H, Wu X-H. Exp-function method for nonlinear wave equations. Chaos, Solitons Fractals 2006;30:700–8. [10] Yusufoglu E. New solitary for the MBBM equations using exp-function method. Phys Lett A 2008;372:442–6. [11] Zhang S. Application of Exp-function method to high-dimensional nonlinear evolution equations. Chaos, Solitons Fractals 2008;38:270–6. [12] Fan EG. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 2000;277:212–18. [13] Zhang S, Xia TC. A further improved tanh-function method exactly solving the (2+1)-dimensional dispersive long wave equations. Appl Math E-Notes 2008;8:58–66. [14] Chen Y, Wang Q. Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1+1)-dimensional dispersive long wave equation. Chaos, Solitons Fractals 2005;24:745–57. [15] Liu S, Fu Z, Liu S, Zhao Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys Lett A 2001;289:69–74. [16] Lu D. Jacobi elliptic function solutions for two variant Boussinesq equations. Chaos, Solitons Fractals 2005;24:1373–85. [17] Zayed EME. New traveling wave solutions for higher dimensional non linear evolution equations using a generalized ( GG )-expansion method. J Phys A: Math Theor 2009;42:195202.[13 pages  [18] Wang ML, Li X, Zhang J. The ( GG )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys Lett A 2008;372:417–23.  [19] Zhang S, Tong JL, Wang W. A generalized ( GG )-expansion method for the mKdV equation with variable coefficients. Phys Lett A 2008;372:2254–7.  [20] Zayed EME, Gepreel KA. The ( GG )-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. J Math Phys 2009;50:013502–12.  [21] Kudryashov NA. A note on the ( GG ) -expansion method. Appl Math Comput 2010;217:1755–8. [22] Zayed EME. Traveling wave solutions for higher dimensional nonlinear  evolution equations using the ( GG )-expansion method. J Appl Math Informatics 2010;28:383–95.

155 

[23] Zhang S, SUN YN, B JM, Dong L. The modified ( GG )-expansion method for nonlinear evolution equations. Z Naturforsch 2011;66a:33–9.  [24] Li Lx, Li QE, Wang LM. The ( GG , G1 )-expansion method and its application to traveling wave solutions of the Zakharov equations. Appl Math J Chin Uni 2010;25:454–62.  [25] Zayed EME, Abdelaziz MAM. The two variables ( GG , G1 )-expansion method for solving the nonlinear KdV-mKdV equation. Math Prob Eng 2014;2012:725061.  [26] Zayed EME, Alurrfi KAE. The ( GG , G1 )-expansion method and its applications to find the exact solutions of nonlinear PDEs for nanobiosciences. Math Prob Eng 2014;2014:521712.  [27] Zayed EME, Alurrfi KAE. The ( GG , G1 )-expansion method and its applications for solving two higher order nonlinear evolution equations. Math Prob Eng 2014;2014:746538. [28] Jawad AJM, Petkovic MD, Biswas A. Modified simple equation method for nonlinear evolution equations. Appl Math Comput 2010;217:869– 77. [29] Zayed EME. A note on the modified simple equation method applied to Sharma-Tasso-Olver equation. Appl Math Comput 2011;218:3962–4. [30] Zayed EME, Ibrahim SAHoda. Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method. Chin Phys Lett 2012;29:060201–4. [31] Ma WX, Zhu Z. Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl Math Comput 2012;218:11871–9. [32] Ma WX, Huang T, Zhang Y. A multiple exp-function method for nonlinear differential equations and its application. Phys Script 2010;82:065003. [33] Ma WX, Lee JH. A transformed rational function method and exact solutions to the (3+1) dimensional Jimbo-Miwa equation. Chaos, Solitons Fractals 2009;42:1356–63. [34] Yang AM, Yang XJ, Li ZB. Local fractional series expansion method for solving wave and diffusion equations on cantor sets. Abstr Appl Anal 2013;2013:351057. [35] Taghizadeh N, Mirzazadeh M, Farahrooz F. Exact solutions of the nonlinear Schrödinger equation by the first integral method. J Math Anal Appl 2011;374:549–53. [36] Lu BHQ, Zhang HQ, Xie FD. Traveling wave solutions of nonlinear partial differential equations by using the first integral method. Appl Math Comput 2010;216:1329–36. [37] Zayed EME, Amer YA, Shohib RMA. The improved Riccati equation mapping method for constructing many families of exact solutions for a nonlinear partial differential equation of nanobiosciences. Int J Phys Sci 2013;8:1246–55. [38] Zhu SD. The generalized Riccati equations mapping method in nonlinear evolution equation: application to (2+1)-dimensional Boiti-LionPempinelle equation. Chaos, Solitons and Fractals 2008;37:1335–42. [39] Abdoulkary S, Beda T, Dafounamssou O, Tafo EW, Mohamadou A. Dynamics of solitary pulses in the nonlinear low-pass electrical transmission lines through the auxiliary equation method. J Mod Phys Appl 2013;2:69–87. [40] Sirendaoreji. Exact traveling wave solutions for four forms of nonlinear Klein–Gordon equations. Phys Lett A 2007;363:440–7. [41] Hong-cai MA, Zhang Zhi-Ping, Deng Ai-ping. A New periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation. Acta Math Appl Sin 2012;28:409–15. [English series]. [42] Hong-Cai Ma, Yu Yao-Dong, Ge Dong-Jie. The auxiliary equation method for solving the Zakharov–Kuznetsov (ZK) equation. Comput Math Appl 2009;58:2523–7.