New Applications of the Homotopy Analysis Method

New Applications of the Homotopy Analysis Method

Elsayed Abd Elaty Elwakil^aand Mohamed Aly Abdou^a,b

aTheoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt

bFaculty of Education for Girls, Physics Department, King Kahlid University, Bisha, Saudi Arabia Reprint requests to M. A. A.; E-mail: m abdou eg@yahoo.com

Z. Naturforsch.63a,385 – 392 (2008); received January 4, 2008

An analytical technique, namely the homotopy analysis method (HAM), is applied using a com- puterized symbolic computation to find the approximate and exact solutions of nonlinear evolution equations arising in mathematical physics. The HAM is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The validity and reliability of the method is tested by application on three nonlinear problems, namely the Whitham-Broer-Kaup equations, coupled Korteweg-de Vries equation and coupled Burger’s equations. Comparisons are made between the results of the HAM with the exact solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics.

Key words:Homotopy Analysis Method; Nonlinear Evolution Equations; Approximate and Exact Solutions.

1. Introduction

Since the world around us is inherently nonlinear, nonlinear evolution equations are widely used to de- scribe complex phenomena in various fields of sci- ences, especially in physics, such as plasma physics, fluid mechanics, optical fibers, solid state physics, non- linear optics. Exact travelling wave solutions of non- linear evolution equations are one of the fundamental objects of study in mathematical physics. These ex- act solutions, when they exist, can help to well under- stand the mechanism of complicated physical phenom- ena and dynamical processes modelled by these non- linear evolution equations. In the past several decades, many significant methods [1 – 9], namely the extended tanh function method, F-expansion method, extended mapping method, Jacobi elliptic function method, variational iteration method, Adomian decomposition method, generalized auxiliary equation method have been applied.

In 1992, Liao [10 – 18] employed the basic ideas of homotopy in topology to propose a general analyti- cal method for nonlinear problems, namely the homo- topy analysis method (HAM). Based on homotopy of topology, the validity of the HAM is independent of whether or not small parameters exist in the consid- ered equation. Therefore, the HAM can overcome the foregoing restrictions and limitations of perturbation

0932–0784 / 08 / 0700–0385 $ 06.00 c2008 Verlag der Zeitschrift f¨ur Naturforschung, T ¨ubingen·http://znaturforsch.com

techniques [19]. The HAM also avoids discretization and provides an efficient numerical solution with high accuracy, minimal calculation, and avoidance of phys- ically unrealistic assumption. Furthermore, the HAM always provides us with a family of solution expres- sions in the auxiliary parameterh; the convergences region and rate of each solution might be determined conveniently by the auxiliary parameterh.

The present paper is arranged as follows. In Sec- tion 2, we simply provide the mathematical framework of the HAM. In Section 3, in order to illustrate the method, three nonlinear evolution equations are inves- tigated. Also, a comparison is made with the excat so- lutions obtained by other methods. Finally, conclusions are given in Section 4.

2. Homotopy Analysis Method

To illustrate the methodology of the homotopy anal- ysis method [20 – 23], we consider a differential equa- tion in the form

N[z(x,t)] =0,

whereN is a nonlinear operator,xandt denote inde- pendent variables,z(x,t)is a unknown function. For simplicity, we ignore all boundary or initial condi- tions, which can be treated in the similar way. By

means of the generalizing homotopy method, Liao [18]

constructed the so-called zero-order deformation equa- tions

(1−p)L[φ(x,t;p)−z₀(x,t)] =phN[φ(x,t;p)], (1) wherep∈(0,1)is the embedding parameter,h=0 a nonzero auxiliary parameter,Lan auxiliary linear op- erator,z₀(x,t)an initial guess ofz(x,t), andφ(x,t;p) a unknown function. It is important, that one has great freedom to choose auxiliary items in the HAM. Obvi- ously, whenp=0 andp=1, it holds

φ(x,t; 0) =z₀(x,t), φ(x,t; 1) =z(x,t).

Expandingφ(x,t;p)in a Taylor series with respect to pit follows

φ(x,t;p) =z₀(x,t) +

∑

m=1

z_m(x,t)p^m, (2)

z_m(x,t) = 1 m!

∂^mφ(x,t;p)

∂^{pm |p=0. (3)} The series (2) converges atp=1 and one has

z(x,t) =z₀(x,t) +

∑

m=1

z_m(x,t), (4) which must be one of the solutions of the original non- linear equation [18]. As long ash=−1, (1) reduces to (1−p)L[φ(x,t;p)−z₀(x,t)]+pN[φ(x,t;p)] =0, (5) which is used in the HAM [18], whereas the solution obtained directly, without using a Taylor series [9].

From (3), the governing equation can be deduced from the zero-order deformation (1). Define the vector

Z_n= [z₀(x,t),z₁(x,t),z₂(x,t),... ,z_n(x,t)], differentiate (1)mtimes with respect to the embedding parameterp, and then setp=0 and finally divide by m!, one has the so-calledmth-order deformation equa- tion

L[z_m(x,t)−χmz_m−1(x,t)] =hR_m(z_m−1), (6) R_m(z_m−1) = 1

(m−1)!

∂^m−1N[φ(x,t;p)]

∂^{pm−1 |p=0. (7)} The solution of themth-order deformation (6) is read- ily found to be

zm(x,t) =χmz_m−1(x,t) +hL⁻¹[Rm(z_m−1)] (8)

withχm=0 form<1, andχm=1 form>1.

3. New Applications

To illustrate effectiveness of the HAM three non- linear evolution equations arising in physics are cho- sen, namely the coupled Burger’s equations, Whitham- Broer-Kaup equations, and coupled Korteweg-de Vries (KdV) equations.

3.1. Example 1: The Coupled Burger’s Equations Let us first consider the coupled Burger’s equa- tions [24]

u_t−u_xx−2uu_x+uv_x+vu_x=0,

v_t−v_xx−2vv_x+uv_x+vu_x=0 (9) with the initial condition [24]

u(x,0) =sin(x), v(x,0) =sin(x). (10) To investigate the travelling wave solutions of (9), we choose the linear operator

L[φ(x,t;p)] =∂φ(x,t;p)

∂^t

with the property L[c] =0, where c is constant. By means of (9), we define a system of nonlinear opera- tors as follows:

N₁[φ1(x,t;p),φ2(x,t;p)] =

∂φ1(x,t;p)

∂^{t −}∂²φ1(x,t;p)

∂^x2

−2φ1(x,t;p)∂φ1(x,t;p)

∂^{x +} ∂

∂^x[φ1(x,t;p)φ2(x,t;p)], N₂[φ1(x,t;p),φ2(x,t;p)] =

∂φ2(x,t;p)

∂^{t −}

∂²φ2(x,t;p)

∂^x2

−2φ2(x,t;p)∂φ2(x,t;p)

∂^{x +} ∂

∂^x[φ1(x,t;p)φ2(x,t;p)].

With the aid of the above definition, we construct the zero-order deformation equations

(1−p)L[φi(x,t;p)−z_i,0(x,t)]

=phiNi[φ1(x,t;p),φ2(x,t;p)], i=1,2.

Obviously, in case ofp=0 andp=1, φ1(x,t; 0) =z_1,0(x,t) =u(x,0), φ1(x,t; 1) =u(x,t),

φ2(x,t; 0)(x,t) =z_2,0(x,t) =v(x,0), φ2(x,t; 1) =v(x,t).

Therefore, as the embedding parameter p increases from 0 to 1, φi(x,t;p) varies from the initial guess z_i,0(x,t)to the solutionu(x,t)andv(x,t), fori=1,2, respectively. Expanding φi(x,t;p) in a Taylor series with respect topfori=1,2 admits

φi(x,t;p) =z_i,0(x,t) +

∑

m=1

z_i,m(x,t)p^m, (11)

z_i,m(x,t) = 1 m!

∂^mφi(x,t;p)

∂^{pm |p=0. (12)} If the auxiliary linear parameter, the initial conditions, and the auxiliary parametersh_iare chosen, the above series converges atp=1, and one has

u(x,t) =z_1,0(x,t) +

∑

m=1

z_1,m(x,t), (13)

v(x,t) =z_2,0(x,t) +

∑

m=1

z_2,m(x,t), (14) which must be one of the solutions of original nonlin- ear equations [18]. Defining the vectors

Z_i,n= [z_i,0(x,t),z_i,1(x,t),...,z_i,n(x,t)],i=1,2, we obtain themth-order deformation equations

L[z_i,m(x,t)−χmz_i,m−1(x,t)]

=h_iR_i,m(z_1,m−1,z_2,m−1), i=1,2, (15) R_1,m(z_1,m−1,z_2,m−1) =

∂^z1,m−1(x,t)

∂^{t −}

∂^2z1,m−1(x,t)

∂^x2

−2

m−1

∑

n=0

z_1,n(x,t)∂^z1,m−1−n(x,t)

∂^x + ∂

∂^x m−1

n=0

∑

z_1,n(x,t)z_2,m−1−n(x,t)

,

R_2,m(z_1,m−1,z_2,m−1) =

∂^z2,m−1(x,t)

∂^{t −}

∂^2z2,m−1(x,t)

∂^x2

−2

m−1

∑

n=0

z_2,n(x,t)∂^z2,m−1−n(x,t)

∂^x +∂

∂^x m−1

n=0

∑

z_1,n(x,t)z_2,m−1−n(x,t)

.

Then the solution of themth-order deformation equa- tions (15) form>1 admits

z_i,m(x,t) =χmz_i,m−1(x,t)+h_iL⁻¹[R_i,m(z_1,m−1,z_2,m−1)].

(16) Knowing the zeroth initial conditions (10) and the re- cursive relationship (16), the rest of components are:

z_1,1(x,t) =hsin(x)t, z_1,2(x,t) =1

2hsin(x)t(2+2h+ht), z_1,3(x,t) =1

6hsin(x)t

6+12h+6ht+6h²+6h²t +h²t²−2h²sin(x)t²−6hsin(x)t−6h²sin(x)t +2h²cos(x)t²+6hcos(x)t+6h²cos(x)t

,

z_2,1(x,t) =hsin(x)t, z_2,2(x,t) =1

2hsin(x)t(2+2h+ht), z_2,3(x,t) =1

6hsin(x)t(6+12h+6ht +6h²+6h²t+h²t²).

In the same manner the rest of the components can obtained. Using a Taylor series with the initial condi- tions (10), we obtain the closed form solutions of (9) as follows:

u(x,t) =sin(x−t)e^t,

v(x,t) =sin(x−t)e^t, (17) which are exactly the same as obtained by the varia- tional iteration method [24] with the fixed valueh=

−1/101.

3.2. Example 2: The Whitham-Broer-Kaup Equations A second instructive model are the Whitham-Broer- Kaup equations [25]

u_t+uu_x+v_x+β^uxx=0,

v_t+ (vu)x+α^uxxx−β^vxx=0, (18) which is a complete integrable model and describes the dispersive long wave in shallow water. In system (18), α,β are real constants that represent different disper- sive powers. Ifα=0,β=0, system (18) becomes the approximate equations for the long wave equation. If α=1,β=0, system (18) becomes the variant Boussi- nesq equation. According to the HAM, we choose the initial approximation [25]

u(x,0) =2R

α²+β^2tanh(ξ)−λ, v(x,0) =−

2R²βα²+β²+2R²

α+β^2sec2(ξ), ξ=x+λ^t,

whereλ,Rare constants, and the linear operator L[φ(x,t;p)] =∂φ(x,t;p)

∂^{t .}

Furthermore, via (18), we define the nonlinear opera- tors

N₁[φ1(x,t;p),φ2(x,t;p)] =

∂φ1(x,t;p)

∂^{t +}β∂²φ1(x,t;p)

∂^{x2 +}

∂φ2(x,t;p)

∂^x +φ1(x,t;p)∂

∂^xφ1(x,t;p), N₂[φ1(x,t;p),φ2(x,t;p)] =

∂φ2(x,t;p)

∂^{t −}β∂²φ2(x,t;p)

∂^{x2 +}α∂³φ1(x,t;p)

∂^x3 + ∂

∂^x[φ1(x,t;p)φ2(x,t;p)].

With the assumptionh=1, we construct the zero-order deformation equations

(1−p)L[φi(x,t;p)−z_i,0(x,t)]

=ph_iN_i[φ1(x,t;p),φ2(x,t;p)], i=1,2. As long asp=0 andp=1,

φ1(x,t; 0) =z_1,0(x,t) =u(x,0), φ1(x,t; 1) =u(x,t), φ2(x,t; 0) =z_2,0(x,t) =v(x,0), φ2(x,t; 1) =v(x,t).

Expandingφi(x,t;p)in a Taylor series with respect to pfori=1,2 admits

u(x,t) =z_1,0(x,t) +

∑

m=1

z_1,m(x,t), (19)

v(x,t) =z_2,0(x,t) +

∑

m=1

z_2,m(x,t). (20) Themth-order deformation equations read

L[z_i,m(x,t)−χmz_i,m−1(x,t)]

=h_iR_i,m[z_1,m−1,z_2,m−1], i=1,2, (21) R_1,m(z_1,m−1,z_2,m−1) =

∂^z1,m−1(x,t)

∂^{t +}β∂^2z1,m−1(x,t)

∂^{x2 +}∂^2z2,m−1(x,t)

∂^x +^m−1

∑

n=0

z_1,n(x,t)∂^z1,m−1−n(x,t)

∂^{x ,} R_2,m(z_1,m−1,z_2,m−1) =

∂^z2,m−1(x,t)

∂^{t −}β∂^2z2,m−1(x,t)

∂^{x2 +}α∂^3z1,m−1−n(x,t)

∂^x3 + ∂

∂^x m−1

n=0

∑

z_1,n(x,t)z_2,m−1−n(x,t)

.

Now, the solution of themth-order deformation equa- tions (21) form>1 becomes

z_i,m(x,t) =χmz_i,m−1(x,t)

+h_iL⁻¹[R_i,m(z_1,m−1,z_2,m−1)], i=1,2.

(22)

With the knowledge of the recursive relationship (22) and knowing the zeroth components, the explicit forms of[z_1,m,z_2,m(i=1,2,3,m=1,2,3,m>1)]are written in Appendix A. For simplicity we omit them here.

The numerical behaviour of the approximate solu- tions using the HAM is shown in Figs. 1a and 2a for different values ofxandt.

It should be noted that the HAM solutions are equiv- alent to the exact solution [25]

u(x,t) =2R

α²+β^2tanh(x+λ^t)−λ, v(x,t) =−[2R²βα²+β²+2R²

α+β²]

·sec²(x+λ^t).

(23)

The behaviour of the exact solutions of u(x,t) and v(x,t)is shown in Figs. 1b and 2b.

(a) (b)

Fig. 1. (a) Approximate solution ofu(x,t)of Whitham-Broer-Knaup equations using the HAM with fixed values ofh=

−1/101,α=1.0,β=1.0 andλ=0.1 for different values ofxandt. (b) Exact solution ofu(x,t)of Whitham-Broer-Knaup equations with fixed values ofh=−1/101,α=1.0,β=1.0 andλ=0.1 for different values ofxandt.

(a) (b)

Fig. 2. (a) Approximate solution ofv(x,t)of Whitham-Broer-Knaup equations using the HAM with fixed values ofh=

−1/101,α=1.0,β=1.0 andλ=0.1 for different values ofxandt. (b) Exact solution ofv(x,t)of Whitham-Broer-Knaup equations with fixed values ofh=−0.01,α=1.0,β=1.0 andλ=0.1 for different values ofxandt.

3.3. Example 3: The Coupled KdV Equations

In this case, the coupled KdV equations [26] read u_t=a(u_xxx+6uu_x) +2bvv_x,

v_t=−v_xxx−3uv_x, (24)

wherea=−(1/2),ab<0, andM= (−24a/b)^1/2k², wherekis an arbitrary constant. Proceeding as before to solve (24) by means of the HAM, we choose the initial approximation [26]

u(x,0) =−1+a

3+6ak²+4k² e^kx (1+e^kx)², v(x,0) = Me^kx

(1+e^kx)², M=

−24a b k².

(25)

Following the HAM to solve (24), we define the two operatorsLandN_i(i=1,2)as follows:

L[φ(x,t;p)] =∂φ(x,t;p)

∂^{t ,}

N₁[φ1(x,t;p),φ2(x,t;p)] =∂φ1(x,t;p)

∂^t

−a∂³φ1(x,t;p)

∂^{x3 −6a}φ1(x,t;p)∂φ1(x,t;p)

∂^x

−2bφ2(x,t;p)∂φ2(x,t;p)

∂^{x ,} N₂[φ1(x,t;p),φ2(x,t;p)] =

∂φ2(x,t;p)

∂^{t +}∂³φ2(x,t;p)

∂^{x3 +3}φ1(x,t;p)∂φ2(x,t;p)

∂^{x .} Themth-order deformation equations yield

L[z_i,m(x,t)−χmz_i,m−1(x,t)]

=h_iR_i,m(z_1,m−1,z_2,m−1), i=1,2 (26) R_1,m(z_1,m−1,z_2,m−1) =

∂^z1,m−1(x,t)

∂^{t −a}∂^3z1,m−1(x,t)

∂^x3

−6a

m−1

∑

n=0

z_1,n(x,t)∂^z1,m−1−n(x,t)

∂^x

−2b

m−1

∑

n=0

z_2,n(x,t)∂^z2,m−1−n(x,t)

∂^{x ,}

(a) (b)

Fig. 3. (a) Approximate solution ofu(x,t)of coupled KdV equations using the HAM with fixed values ofh=−0.01,a=−1.5, c=0.1,b=0.1 andκ=0.1 for different values ofxandt. (b) Exact solution ofu(x,t)of coupled KdV equations with fixed values ofh=−0.01,a=−1.5,c=0.1,b=0.1 andκ=0.1 for different values ofxandt.

(a) (b)

Fig. 4. (a) Approximate solution ofv(x,t)of coupled KdV equations using the HAM with fixed values ofh=−0.01,a=−1.5, c=0.1,b=0.1 andκ=0.1 for different values ofxandt. (b) Exact solution ofv(x,t)of coupled KdV equations with fixed values ofh=−0.01,a=−1.5,c=0.1,b=0.1 andκ=0.1 for different values ofxandt.

R_2,m(z_1,m−1,z_2,m−1) =

∂^z2,m−1(x,t)

∂^{t +}∂^3z2,m−1(x,t)

∂^x3 +3

m−1

∑

n=0

z_1,n(x,t)∂^z2,m−1−n(x,t)

∂^{x .}

The solution of themth-order deformation becomes z_i,m(x,t) =χmz_i,m−1(x,t)

+h_iL⁻¹[R_i,m(z_1,m−1,z_2,m−1)], i=1,2. (27) By means of the recursive relationships (27) and (25), the rest of the components of [z_1,m,z_2,m (i=1,2,3, m=1,2,3,m>1)]can be directly evoluted (see Ap- pendix B). The behaviour of the approximate solutions ofu(x,t)and v(x,t) with the fixed valuesa=−1.5, c=0.1,b=0.1,k=0.1 andh=−1/101 for different values ofxandt are shown in Figs. 3a and 4a. Also it is shown from Figs. 3b and 4b that the HAM solutions agree with the exact solution [26]

u(x,t) =− 1+a

3+6ak²+4k² e^k(x+ct) (1+e^k(x+ct))²,

v(x,t) = Me^k(x+ct)

(1+e^k(x+ct))². (28)

4. Conclusion

In this paper, the homotopy analysis method (HAM) is applied for constructing the approximate and exact solutions of three nonlinear evolution equations arising in mathematical physics, namely the coupled Burger’s equations, Whitham-Broer-Kaup equations, and cou- pled KdV equations.

The advantages of the HAM with respect to the ho- motopy perturbation method are illustrated. The HAM provides us with a convenient way to control the con- vergence of approximation series, which is a funda- mental qualitative difference in analysis between the HAM and other methods.

It is worthwhile to mention that the HAM is straight- forward and concise, and it can also be applied to other nonlinear problems in science and engineering. This is our task in the future.

Appendix A z_1,0:=2

α+β^2tanh(x)−λ, z_2,0:=− 2βα+β²+2

α+β^2sec(x)2, z_1,1:=2ht

−2 sinh(x)cos(x)³βα+β²+2 sinh(x) cos(x)³α+2 sinh(x)cos(x)³β²

−2

α+β^2sin(x)β ^cosh(x)³−2

α+β^2sin(x)β^cosh(x)³−

α+β²λ^cosh(x)cos(x)³ cosh(x)³ cos(x)³

, z_2,1:=−4ht

2 cosh(x)³cos(x) sin(x) sinh(x)βα+2 cosh(x)³cos(x)sin(x) sinh(x)β³ +2 cosh(x)³cos(x)sin(x)sinh(x)α+2 cosh(x)³cos(x)sin(x) sinh(x)β²

−

α+β^2cos(x) sin(x)λβ^cosh(x)⁴−

α+β^2cos(x) sin(x)λ^cosh(x)⁴

−3

α+β²β^cosh(x)⁴+2

α+β^2cos(x)²β^2cosh(x)⁴+2

α+β^2cos(x)²β^cosh(x)⁴

−3

α+β²β^2cosh(x)⁴+cos(x)²cosh(x)²β²−2

α+β²α^cos(x)⁴ cosh(x)² +cos(x)²cosh(x)²βα+cos(x)²cosh(x)²β³+cos(x)²cosh(x)²α+3

α+β²α^cos(x)⁴ cosh(x)⁴ cos(x)⁴

, z_1,2:=2h t

−6h tcosh(x)cos(x)⁴λα+6h tcosh(x)⁵

α+β²λβ−hcosh(x)³cos(x)⁴

α+β²λ +2 cosh(x)²cos(x)⁴ sinh(x)β²+2 cosh(x)²cos(x)⁴sinh(x)α−2 cosh(x)⁵cos(x)

α+β^2sin(x)

−cosh(x)³ cos(x)⁴

α+β²λ+8h tcosh(x)⁴cos(x)² sinh(x)β³−2hcosh(x)⁵ cos(x)

α+β^{2 sin}(x)

−2hcosh(x)² cos(x)⁴sinh(x)βα+β²+8hβ^3tcosh(x)² cos(x)⁴sinh(x) +8hβ^tcosh(x)²cos(x)⁴sinh(x)α−8hβ^2tcosh(x)²cos(x)⁴ sinh(x)

α+β²

−20hβ^tsinh(x) cos(x)⁴α+20hβ^2tsinh(x)cos(x)⁴

α+β²−4hβ^tcosh(x)³cos(x)⁴

α+β²λ +6hβ^tcosh(x)

α+β²λ ^cos(x)⁴−12h tcosh(x)⁴sinh(x)β³−h tcosh(x)² cos(x)⁴

α+β²λ^2sinh(x) +4h tcosh(x)³cos(x)⁴λα−6h tcosh(x) cos(x)⁴λβ²−8h tcosh(x)² cos(x)⁴

α+β²α^sinh(x)

−4h tcosh(x)⁵cos(x)²

α+β²λβ+4h tcosh(x)³ cos(x)⁴λβ²−2hcosh(x)⁵cos(x)

α+β²β^sin(x)

−2 cosh(x)²sinh(x)βα+β²−2 cosh(x)⁵ cos(x)

α+β²β ^sin(x) +2h tcosh(x)² cos(x)²β^3sinh(x)

−12h tcosh(x)⁴ sinh(x)α+2h tcosh(x)²cos(x)²βα^sinh(x) +6h tcosh(x)⁵

α+β²λ

−6h tcosh(x)³cos(x)α^sin(x) +2h tcosh(x)²cos(x)²β^2sinh(x)−6h tcosh(x)³cos(x)β^2sin(x)

−6h tcosh(x)³cos(x)βα^sin(x) +8h tcosh(x)⁴cos(x)² sinh(x)βα−12h tcosh(x)⁴sinh(x)βα

−6h tcosh(x)³cos(x)β^3sin(x) +2h tcosh(x)²cos(x)²α^sinh(x) +8h tcosh(x)⁴cos(x)²sinh(x)α +8h tcosh(x)⁴ cos(x)²sinh(x)β²−4h tcosh(x)⁵cos(x)²

α+β²λ+20h tsinh(x)cos(x)⁴

α+β²α

−12h tcosh(x)⁴ sinh(x)β²−20hβ^3tsinh(x) cos(x)⁴+2hcosh(x)²cos(x)⁴ sinh(x)β² +2hcosh(x)² cos(x)⁴sinh(x)α ^cos(x)⁴ cosh(x)⁵

.

Appendix B

z_1,0:=−(1+a)k²

3+6a + 4k²e^{(k x)}

1+e^{(k x)}2, z_2,0:= Me^{(k x)} 1+e^{(k x)}2, z_1,1:=−2hke^{(k x)}

−2a k⁴+22a k⁴e^{(k x)}+48a²k⁴e^{(k x)}−22a k⁴e^{(2k x)}−48a²k⁴e^{(2k x)} +2a k⁴e^{(3k x)}+b M²e^{(k x)}+2b M²e^{(k x)}a−b M²e^{(2k x)}−2b M²e^{(2k x)}a

t

(1+2a)(1+e^{(k x)})⁵ , z_2,1:=−t(−1+e^{(k x)})ae^{(k x)}k³M h

(1+e^{(k x)})³(1+2a) , z_2,2:=−1

2te^{(k x)}k²M h

−48he^{(3k x)}t b M²a−h t k⁴a²−2h a k−4h a²k+576he^{(4k x)}ta³k⁴+2a ke^{(6k x)} +4a²ke^{(6k x)}+16a²ke^{(5k x)}−1152he^{(3k x)}ta³k⁴+24he^{(4k x)}t b M²a+4h a²ke^{(6k x)}−8a ke^{(k x)}−16a²ke^{(k x)}

−48he^{(3k x)}t b M²a²−8h a ke^{(k x)}−16h a²ke^{(k x)}−10a ke^{(2k x)}−20a²ke^{(2k x)}t b M²+6he^{(4k x)}t b M² +10a ke^{(4k x)}+20a²ke^{(4k x)}+24he^{(2k x)}t b M²a²+144h t k⁴ae^{(2k x)}+20h a²ke^{(4k x)}+8h a ke^{(5k x)} +16h a²ke^{(5k x)}−10h a ke^{(2k x)}+8a ke^{(5k x)}+585h t k⁴a²e^{(2k x)}−20h a²ke^{(2k x)}+10h a ke^{(4k x)} +6he^{(2k x)}t b M²−2a k−4a²k−288h t k⁴ae^{(3k x)}−1136h t k⁴a²e^{(3k x)}−12he^{(3k x)}t b M² +144h t k⁴ae^{(4k x)}+585h t k⁴a²e^{(4k x)}−h t k⁴a²e^{(6k x)}+24he^{(4k x)}t b M²a²+576he^{(2k x)}ta³k⁴ +24he^{(2k x)}t b M²a+2h a ke^{(6k x)}

(1+e^{(k x)})⁸(1+2a)² .

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