on using the he’s polynomials for solving the nonlinear ... · on using the he’s polynomials...

On using the He’s polynomials for solving the nonlinear coupledevolution equations in mathematical physics

E. M. E. ZayedFaculty of Science , Zagazig University

Department of MathematicsEGYPT

[email protected]

H. M. Abdel RahmanTenth Of Ramadan City

Higher Institute of TechnologyEGYPT

hanan [email protected]

Abstract: In this article, we apply the modified variational iteration method for solving the (1+1)- dimensional Ra-mani equations and the (1+1)-dimensional Joulent Moidek (JM) equations together with the initial conditions. Theproposed method is modified the variational iteration method by the introducing He’s polynomials in the correctionfunctional. The analytical results are calculated in terms of convergent series with easily computated components.

Key–Words: Variational iteration method, Homotopy perturbation methods, Coupled nonlinear evaluation equa-tions, Exact solutions

1 IntroductionThe nonlinear coupled evolution equations have manywide array of applications of many fields, which de-scribed the motion of the isolated waves, localized ina small part of space, in many fields such as physics,mechanics, biology, hydrodynamics, plasma physic-s, etc.. To further explain some physical phenom-ena, searching for exact solutions of nonlinear par-tial differential equations is very important. Up tonow, many researches in mathematical physics havepaid attention to these topics, and a lot of power-ful methods have been presented such as the modi-fied extended tanh-function method [7,10,12,38], gen-eralized F-expansion method [34], Adomian decom-position method [1-3,39], homotopy analysis method[5,8,35], Jacobi elliptic function method [36], thetanh-hyperbolic function method [25-26], the extend-ed F-expansion method [23]. He [15-22] developedthe variational iteration method and homotopy pertur-bation method for solving linear and nonlinear initialand boundary value problems. It is worth mentioningthat the origin of the variational iteration method canbe traced by Inokuti [24], but the real potential of thismethod was explored by He [15]. Moreover, He real-ized the physical significance of the variational itera-tion method, its compatibility with the physical prob-lems and applied this promising technique to a wideclass of linear and nonlinear, ordinary, partial, deter-ministic or stochastic differential equation. The ho-motopy perturbation method [9,13,17,19-22] was alsodeveloped by He by merging two techniques, the stan-dard homotopy and the perturbation. The homotopyperturbation method was formulated by taking full ad-

vantage of the standard homotopy and perturbationmethods. The variational iteration and homotopy per-turbation methods have been applied to a wide classof functional equations. In these methods the solutionis given in an infinite series usually converging to anaccurate solution. In a later work Ghorbani et.al. [14]splited the nonlinear term into a series of polynomialscalling them as the He’s polynomials. Most recently,Noor and Mohyud- Din used this concept for solvingnonlinear boundary value problems (see Ref.[29-31])and anther authors [4]. The basic motivation of thispaper is an extension of the modified variational iter-ation method which is formulated by the coupled ofvariational iteration method and He’s polynomials forsolving the (1+1)-dimensional Ramani equations andthe (1+1)-dimensional Jaulent-Miodek (JM) equation-s. The MVIM provides the solution in a rapid conver-gent series which may lead the solution to a closedform. In this method, the correct functional is de-veloped [15,31-33] and the Lagrange multipliers arecalculated optimally via variational theory. The useof Lagrange multipliers reduce the successive appli-cation of the integral operator and the cumbersomeof huge computational work while still maintaininga very high level of accuracy. Finally, He’s polyno-mials are introduced in the correction functional andthe comparison of like powers of p gives solutions ofvarious orders. In this paper, we use the modified vari-ational iteration method (MVIM) to solve

WSEAS TRANSACTIONS on MATHEMATICS E. M. E. Zayed, H. M. Abdel Rahman

E-ISSN: 2224-2880 294 Issue 4, Volume 11, April 2012

the (1+1)-dimensional Ramani equations [37]

u6x + 15uxxu3x + 15uxu4x + 45u2xuxx −5(u3xt + 3uxxut + 3uxuxt)− 5utt + 18vx = 0,

vt − v3x − 3vxux − 3vuxx = 0, (1)

and the (1+1)-dimensional Jaulent-Miodek (JM) e-quations[11]

ut + uxxx +3

2vvxxx +

9

2vxvxx −

6uux − 6uvvx −3

2uxv

2 = 0,

vt + vxxx − 6uxv − 6uvx −15

2vxv

2 = 0. (2)

2 Variational iteration method

To illustrate the basic concept of the technique, weconsider the following general differential equation

Lu+Nu = g. (3)

where L is a linear operator, N is a nonlinear operatorand g is the forcing term. According to variational it-eration method [15,31-33], we can construct a correctfunctional as follows

un+1(x, t) = un(x, t) +

∫ t

0λ(τ)[Lun(x, τ) +

Nun(x, τ)− g]dτ, (n ≥ 0), (4)

where λ is a Lagrange multiplier [15], whichcan be identified optimally via variational iterationmethod. The subscript n denotes the nth approxi-mation, u is considered as restricted variation. i.e.δu = 0. Eq.(4) is called as a correct functional. Thesolution of the linear problem can be solved in a sin-gle iteration step due to the exact identification of theLagrange multiplier. The principals of variational it-eration method and its applicability for various kindsof differential equations are given in [31-33]. In thismethod, it is required first to determine the Lagrangemultiplier λ optimally. The successive approximationun+1, n ≥ 0 of the solution u will be readily obtainedupon using the determined Lagrange multiplier andany selective function u0. Consequently, the solutionis given by

u = limn→∞

un. (5)

3 Homotopy perturbation method

The homotopy perturbation method is considered asspacial case of homotopy analysis method. To illus-trate the homotopy perturbation method [9,13,27], we

consider a general equation of the type,

L(u) = 0, (6)

where L is any integral or differential operator. Wedefine a convex homotopy H(u, p) by

H(u, p) = (1− p)F (u) + pL(u), (7)

where F (u) is a functional operator with known solu-tion v0, which can be obtained easily. It is clear that,for

H(u, p) = 0, (8)

we have

H(u, 0) = F (u), H(u, 1) = L(u). (9)

This shows that H(u, p) continuously traces an im-plicitly defined curve from a starting point H(v0, 0)to a solution function H(f, 1). The embedding pa-rameter monotonically increases from zero to unit asthe trivial problem F (u) = 0 is continuously deform-s the original problem L(u) = 0. The embeddingparameter p ∈ [0, 1] can be considered as an expand-ing parameter [9,13,27]. The homotopy perturbationmethod uses the homotopy parameter p as an expand-ing parameter to obtain

u =∞∑i=0

piui = u0 + pu1 + p2u2 + p3u3 + .... (10)

If p → 1, then (10) corresponds to (7) and be-comes the approximate solution of the form,

f = limp→1

u =∞∑i=0

ui. (11)

It is well known that the series (10) is convergentfor most of the cases and also the rate of convergenceis depending on L(u); (see[19-22]). we assume that(10) has a unique solution. The comparisons of likepowers of p give solutions of various orders.

4 Modified variational iterationmethod (MVIM) with He’s poly-nomials

The modified variational iteration method is obtainedby the elegant coupling of the correction functionalformula (4) with the He’s polynomial [16-22]. Ac-cording to [16-22] He has been considered the solu-tion u of the homptopy equation as a series of p which



obtained in Eq. (10), and the method considered thenonlinear term N(u)as

N(u) =∞∑i=0

piHi = H0 + pH1 + p2H2 + ..., (12)

where H ′ns are the so-called He’s polynomials [16-

22], which can be calculated by using the formula

Hn(u0, u1, ..., un) =1

n!

∂n

∂pn

{N(n∑

i=0

piui)}p=0, n = 0, 1, 2, .. (13)

The modified variational iteration method is obtainedby the elegant coupling of correction functional (4) ofVIM with He’s polynomial [16-22] and is given by

∞∑n=0

pnun = u0 + p

∫ t

0λ(τ)[

∞∑n=0

pnL(un) +

∞∑n=0

pnN(un)]dτ −∫ t

0λ(τ)gdτ. (14)

Comparisons of like powers of p give solutions of var-ious orders.

5 Applications

The modified variational iteration method is used tosolve the (1+1)-dimensional Ramani equations (1),and the (1+1)-dimensional Jaulent-Miodek (JM) e-quations (2).

5.1 Solving the (1+1)-dimensional Ramaniequations using MVIM

In this subsection, we find the solutions u(x, t) andv(x, t) satisfying the coupled nonlinear Ramani equa-tions (1) with the following initial conditions [37]:

u(x, 0) = a0 + 2α coth(αx),

ut(x, 0) + 2tβα2csch2(αx),

v(x, 0) = −4

9βα4 − 16

27α6 +

5

9β2α2 − 5

54β3 +

(20

9βα4 +

16

9α6 − 5

9β2α2) coth2(αx). (15)

These initial conditions follow by setting t = 0 in thefollowing exact solutions of Eqs. (1):

u(x, t) = a0 + 2α coth(αζ),

v(x, t) = −4

9βα4 − 16

27α6 +

5

9β2α2 −

5

54β3 + (

20

9βα4 +

16

9α6 −

5

9β2α2) coth2(αζ), (16)

where ζ = (x − βt), a0, β and α are arbitrary con-stants. These exact solutions have been derived byYusufoglu et al. [37] using the tanh method. Let usnow solve the initial value problem (1) with the ini-tial conditions (15) using the MVIM. To this end, weconvey the basic idea of the modified variational iter-ation method for the equations (1), let us consider thefunctional iteration formula

un+1(x, t) = un +

∫ t

0λ1(τ)[−5(un)ττ + (un)6x +

15(un)xx(un)3x + 15(un)x(un)4x +

45(un)2x(un)xx − 5{(un)3xτ + 3(un)xx(un)τ +

3(un)x(un)xτ}+ 18(vn)x]dτ,

vn+1(x, t) = vn +

∫ t

0λ2(τ)[(vn)τ − (vn)3x −

3(vn)x(un)x − 3(vn)(un)xx]dτ. (17)

Making the correct functional stationary, the Lagrangemultipliers can be identified as λ1(τ) = t−τ

5 andλ2(τ) = −1, consequently, we have

un+1(x, t) = un +

∫ t

0

t− τ

5[−5(un)ττ + (un)6x +

15(un)xx(un)3x + 15(un)x(un)4x +

45(un)2x(un)xx − 5{(un)3xτ +

3(un)xx(un)τ + 3(un)x(un)xτ}+ 18(vn)x]dτ,

vn+1(x, t) = vn −∫ t

0[(vn)τ − (vn)3x −

3(vn)x(un)x − 3vn(un)xx]dτ. (18)

Applying the modified variational iteration method,hence

∞∑n=0

pnun = u0 + p

∫ t

0

t− τ

5[−5(

∞∑n=0

pnun)ττ +

(∞∑n=0

pnun)6x + 15(∞∑n=0

pnun)xx(∞∑n=0

pnun)3x

+ 15(∞∑n=0

pnun)x(∞∑n=0

pnun)4x +

45(∞∑n=0

pnun)2x(

∞∑n=0

pnun)xx −

5[(∞∑n=0

pnun)(3x)τ +

3(∞∑n=0

pnun)xx(∞∑n=0

pnun)τ +

3(∞∑n=0

pnun)x(∞∑n=0

pnun)xτ ] +

18(∞∑n=0

pnvn)x]dτ, (19)



and∞∑n=0

pnvn = v0 − p

∫ t

0[(

∞∑n=0

pnvn)τ −

(∞∑n=0

pnvn)3x − 3(∞∑n=0

pnvn)x(∞∑n=0

pnun)x −

3(∞∑n=0

pnvn)(∞∑n=0

pnun)xx]dτ, (20)

using Eqs. (19) and (20) to compare the coefficientsof like powers of p then we have

u0(x, t) = u(x, 0) + ut(x, 0),

u1(x, t) =

∫ t

0

t− τ

5[(u0)6x + 15(u0)xx(u0)3x +

15(u0)x(u0)4x + 45(u0)2x(u0)xx −

5{(u0)(3x)τ + 3(u0)xx(u0)τ +

3(u0)x(u0)xτ}+ 18(v0)x]dτ,

u2(x, t) =

∫ t

0

t− τ

5[−5(u1)ττ + (u1)6x +

15{(u0)xx(u1)3x + (u1)xx(u0)3x}+15{(u0)x(u1)4x + (u1)x(u0)4x}+45{(u0)2x(u1)xx + 2(u1)x(u0)x(u0)xx} −5{(u1)(3x)τ + 3{(u0)xx(u1)τ +(u1)xx(u0)τ}+ 3{(u0)x(u1)xτ +(u1)x(u0)xτ}} + 18(v1)x]dτ, (21)

v0(x, t) = v(x, 0),

v1(x, t) =

∫ t

0[(v0)3x + 3(v0)x(u0)x +

3v0(u0)xx]dτ,

v2(x, t) = −∫ t

0[(v1)τ − (v1)3x −

3{(v0)x(u1)x + (v1)x(u0)x} −3{v0(u1)xx + v1(u0)xx}]dτ, (22)

the other components can be found similarly. Aftersome reduction, we have

u0(x, t) = a0 + 2α coth(αx) + 2tβα2csch2(αx),u1(x, t) = −2α3β2t2 coth(αx)csch2(αx)

u2(x, t) =2

3α4β3t3{2 coth2(αx)csch2(αx) +

csch4(αx)}, (23)

v0(x, t) = −4

9βα4 − 16

27α6 +

5

9β2α2 −

5

54β3 + (

20

9βα4 +

16

9α6 − 5

9β2α2) coth2(αx),

v1(x, t) = 2tαβ(20

9βα4 +

16

9α6 −

5

9β2α2) coth(αx)csch2(αx),

v2(x, t) = −(20

9βα4 +

16

9α6 − 5

9β2α2)t2

×{2α2β2 coth2(αx)csch2(αx) + csch4(αx)}. (24)

Therefore, using the Eq.(10), then approximate solu-tions of the system of equations of Eqs. (1) take thefollowing forms:

u(x, t) = a0 + 2α coth(αx) + 2tβα2csch2(αx)−

2α3β2t2 coth(αx)csch2(αx) +2

3α4β3t3

{2 coth2(αx)csch2(αx) + csch4(αx)}+ ...,(25)

v(x, t) = −4

9βα4 − 16

27α6 +

5

9β2α2 −

5

54β3 + (

20

9βα4 +

16

9α6 − 5

9β2α2)

× {coth2(αx) + 2tαβ coth(αx)csch2(αx)−2t2α2β2 coth2(αx)csch2(αx)−t2csch4(αx)}+ ... (26)

The accuracy of the modified variational iterationmethod for the Eqs. (1) under conditions (15) is con-trollable and the absolute errors are very small withthe present choice of x, t. These results are listed inTables 1, 2 and Figures 1-4. It is also clear that whenmore terms for MVIM are computed, the numericalresults are much more closer to the corresponding ex-act solution.

Table 1. The MVIM results of u(x, t) for the firstthree approximation in comparison with the exact so-lution if a0 = 1, β = α = .01,and t = 20 for thesolution of the system (1) with the initial condition-s(15).

x uExact

uMV IM

|uExact

− uMV IM |-50 0.956868 0.956869 1.27251607E-6-40 0.947597 0.9476 2.48972102E-6-30 0.931774 0.93178 5.90319012E-6-20 0.899647 0.899667 1.98912333E-5-10 0.803242 0.8034 1.58430305E-410 1.20473 1.20457 1.58430305E-420 1.10233 1.10231 2.00989070E-530 1.06909 1.06908 5.94270457E-640 1.05288 1.05287 2.50222741E-650 1.04343 1.04343 1.27764171E-6



Figure 1: The exact solution of u(x,t) for the equations(1) if a0 = 1, β = α = .01.

Figure 2: The approximate solution of u(x,t) for thefirst three approximation of the equations (1) if a0 =1, β = α = .01.

Table 2. The MVIM results of v(x, t) for the firstthree approximation in comparison with the analyticalsolution if a0 = 1, β = α = .01,and t = 20 for the so-lution of the system (1) with the initial conditions(15).

x vExact vMV IM |vExact−vMV IM |-50 -1.1204856E-7 -1.12232E-7 1.853690E-7-40 -1.2401512E-7 -1.24375E-7 1.855450E-7-30 -1.4991042E-7 -1.50764E-7 1.860388E-7-20 -2.2395349E-7 -2.26833E-7 1.880649E-7-10 -6.2401173E-7 -6.46982E-7 2.081564E-710 -6.2401173E-7 -6.00746E-7 1.619202E-720 -2.2395349E-7 -2.21054E-7 1.822860E-730 -1.4991042E-7 -1.49052E-7 1.843273E-740 -1.2401512E-7 -1.23653E-7 1.848238E-750 -1.1204856E-7 -1.11863E-7 1.850006E-7

5.2 Solving the (1+1)-dimensional Jaulent-Miodek (JM) equations using MVIM

In this subsection, we find the solutions u(x, t),v(x, t) satisfying the (1+1)-dimensional Jaulent-Miodek (JM) equations with the following initial con-

Figure 3: The exact solution of v(x,t) for the equations(1) if a0 = 1, β = α = .01.

Figure 4: The approximate solution of v(x,t) for thefirst three approximation for the equations (1) if a0 =1, β = α = .01.

ditions [11]:

u(x, 0) =1

4(c− b2)− 1

2b√c sech(

√c x)−

3c

4sech2(

√c x),

v(x, 0) = b+√c sech(

√c x). (27)

These initial conditions follow by setting t = 0 in thefollowing exact solutions of eqs. (2):

u(x, t) =1

4(c− b2)− 1

2b√c

sech(√c (x+

(6b2 + c)t

2))−

3c

4sech2(

√c (x+

(6b2 + c)t

2)),

v(x, t) = b+√c

× sech(√c (x+

(6b2 + c)t

2)), (28)

where c and b are arbitrary constants. These exac-t solutions have been derived by Fan [11] using theunified algebraic method. Let us now solve the initialvalue problem (2) and (27) using the MVIM. To this



end, we convey the basic idea of the modified vari-ational iteration method for the equations (2), let usconsider the functional iteration formula

un+1(x, t) = un(x, t) +

∫ t

0λ1(τ)[(un)τ +

(un)xxx +3

2(vn)(vn)xxx +

9

2(vn)x(vn)xx −

6{(un)(un)x + (un)(vn)(vn)x} −3

2(un)x(vn)

2]dτ,

vn+1(x, t) = vn(x, t) +

∫ t

0λ2(τ)[(vn)τ +

(vn)xxx − 6{(un)x(vn) +

(un)(vn)x} −15

2(vn)x(vn)

2]dτ. (29)

Making the correct functional stationary, the Lagrangemultipliers can be identified as λ1(τ) = −1=λ2(τ) =−1, consequently

un+1(x, t) = un(x, t)−∫ t

0[(un)τ + (un)xxx +

3

2(vn)(vn)xxx +

9

2(vn)x(vn)xx −

6{un)(un)x + (un)(vn)(vn)x} −3

2(un)x(vn)

2]dτ,

vn+1(x, t) = vn(x, t)−∫ t

0[(vn)τ + (vn)xxx −

6{(un)x(vn) + (un)(vn)x} −15

2(vn)x(vn)

2]dτ. (30)

Applying the modified variational iterationmethod, we have

∞∑n=0

pnun = u0 − p

∫ t

0[(

∞∑n=0

pnun)τ +

(∞∑n=0

pnun)xxx +3

2(

∞∑n=0

pnvn)(∞∑n=0

pnvn)xxx +

9

2(

∞∑n=0

pnvn)x(∞∑n=0

pnvn)xx −

6{∞∑n=0

pnun)(∞∑n=0

pnun)x +

(∞∑n=0

pnun)(∞∑n=0

pnvn)(∞∑n=0

pnvn)x} −

3

2(

∞∑n=0

pnun)x(∞∑n=0

pnvn)2]dτ,

∞∑n=0

pnvn = v0(x, t)− p

∫ t

0[(vn)τ + (vn)xxx −

6{(un)x(vn) + (un)(vn)x} −15

2(

∞∑n=0

pnvn)x(∞∑n=0

pnvn)2]dτ. (31)

Using eqs. (31) to compare the coefficient of like pow-ers of p then we have

u0(x, t) = u(x, 0),

u1(x, t) = −∫ t

0[(u0)xxx +

3

2(v0)(v0)xxx +

9

2(v0)x(v0)xx − 6{u0)(u0)x +

(u0)(v0)(v0)x} −3

2(u0)x(v0)

2]dτ,

u2(x, t) = −∫ t

0[(u0)τ (u1)xxx +

3

2{(v1)(v0)xxx +

(v0)(v1)xxx}+9

2{(v0)x(v1)xx + (v1)x(v0)xx} −

6{{u0)(u0)x + u1)(u1)x}+ {(u1)(v0)(v0)x +(u0)(v1)(v0)x + (u0)(v0)(v1)x}} −3

2{(u1)x(v0)2 + 2(u0)x(v0)(v1)}]dτ, (32)

v0(x, t) = v(x, 0),

v1(x, t) = −∫ t

0[(v0)xxx − 6{(u0)x(v0) +

(u0)(v0)x} −15

2(v0)x(v0)

2]dτ,

v2(x, t) = −∫ t

0[(v1)τ − v1)xxx − 6{{(u0)x(v1) +

(u1)x(v0)}+ {(u0)(v1)x + (u1)(v0)x}} −15

2{(v1)x(v0)2 + 2(v0)xv1v0}]dτ, (33)

The other components can be found similarly. Aftersome reduction, we have

u0(x, t) =1

4(c− b2)− 1

2b√c sech(

√c x)−

3c

4sech2(

√c x),

u1(x, t) = − t

4(c+ 6b2){bc sech(

√c x) tanh(

√c x)

+ 3c3/2 sech2(√c x) tanh(

√c x)},

u2(x, t) = − t2

8(c+ 6b2)2{bc3/2 (sech3(

√c x) +

sech(√c x) tanh2(

√c x)) + 3c2(sech4(

√c x)

+ 2sech2(√c x) tanh2(

√c x))},

(34)



v0(x, t) = b+√c sech(

√c x),

v1(x, t) =c t

2(c+ 6b2)sech(

√c x) tanh(

√c x),

v2(x, t) =c3/2t2

4(c+ 6b2)2{sech3(

√c x) +

sech(√c x) tanh2(

√c x)}, (35)

In this manner the other components can be eas-ily obtained. We construct the solutions u(x, t) andv(x, t) as follows:

u(x, t) =1

4(c− b2)− 1

2b√c sech(

√c x)−

3c

4sech2(

√c x)− t

4(c+ 6b2){bc sech(

√c x)

× tanh(√c x) + 3c3/2 sech2(

√c x)

tanh(√c x)} − c3/2t2

8(c+ 6b2)2{bc3/2

( sech3(√c x) + sech(

√c x) tanh2(

√c x)) +

3c2(sech4(√c x) + 2sech2(

√c x)

× tanh2(√c x))}+ ..., (36)

v(x, t) = b+√c sech(

√c x) +

ct

2(c+ 6b2)

× sech(√c x) tanh(

√c x) +

c3/2 t2

4(c+ 6b2)2

× {sech3(√c x) + sech(

√c x)

× tanh2(√c x)}+ ... (37)

The accuracy of the MVIM for the eqs. (2) under con-ditions (27) are controllable and the absolute errorsare very small with the present choice of x, t. Theseresults are listed in Tables 3, 4 and Figures 5-8. It is al-so clear that when more terms for homotopy analysismethod are computed, the numerical results are muchmore closer to the corresponding exact solution.

Figure 5: The exact solution of u(x,t) for the equations(2) if b = .1, c = .01.

Figure 6: The approximate solution of u(x,t) for thefirst three approximation for the equations (2) if b =.1, c = .01.

Table 3. The MVIM results of u(x, t) for the firstthree approximation in comparison with the exact so-lution if b = .1, c = .01 and t = .2 for the solution ofthe Eqs. (2) with the initial conditions (27).

x uExact

uMV IM

|uExact

− uMV IM

|-50 -0.00006878 -0.000068786 1.03236E-9-40 -0.00019329 -0.000193286 7.98438E-9-30 -0.00057108 -0.000571017 6.75572E-8-20 -0.00186051 -0.00185984 6.65449E-700 -0.00639517 -0.00638898 6.18910E-600 -0.0125 -0.0124967 3.29510E-610 -0.00638499 -0.00639128 6.28541E-620 -0.00185728 -0.00185806 7.81587E-730 -0.000570186 -0.00057026 8.12470E-840 -0.00019301 -0.00019302 9.60074E-950 -0.000068689 -0.00006869 1.22044E-9

Table 4. The MMVIM results of v(x, t) for thefirst three approximation in comparison with the exactsolution if b = .1, c = .01 and t = .2 for the solutionof the eqs. (2) with the initial conditions (27).

x vExact

vMV IM

|vExact

− vMV IM

|-50 0.101348 0.101347 1.6994901E-6-40 0.103664 0.10366 4.8481756E-6-30 0.10994 0.109915 2.5172504E-5-20 0.126598 0.126426 1.7214243E-4-10 0.16484 0.163784 1.0560166E-300 0.2 0.199999 1.1865000E-610 0.164771 0.165828 1.0567760E-320 0.126562 0.126735 1.7238910E-430 0.109926 0.109951 2.5211635E-540 0.103659 0.103664 4.8552982E-650 0.101347 0.101348 1.7027722E-6

6 ConclusionIn the present paper, the modified variational itera-tion method (MVIM) is used to find the solutions of



Figure 7: The exact solution of v(x,t) for the equations(2) if b = .1, c = .01.

Figure 8: The approximate solution of v(x,t) for thefirst three approximation for the equations (2) if b =.1, c = .01.

the nonlinear coupled equations in the mathematicalphysics via the (1+1)-dimensional Ramani equation-s and the (1+1)-dimensional Jaulent-Miodek (JM) e-quations together with the initial conditions. It canbe concluded that the MVIM is very powerful and ef-ficient in finding the exact solutions for wide classesof problems. It is worth pointing out that the MVIMpresents rapid convergence solutions.

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