Homotopy perturbation

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ISSN 1749-3889 (print), 1749-3897 (online)

International Journal of Nonlinear Science

Vol.12(2011) No.4,pp.485-497

Approximate Solutions to System of Nonlinear Partial Differential

Equations Using Homotopy Perturbation Method

Marwan Alquran1 , Mahmoud Mohammad1 Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan

(Received 13 September 2011, accepted 26 November 2011)

Abstract: In this paper, the homotopy perturbation method (HPM) is applied to obtain approximate solutions

to three systems of nonlinear wave equations, namely, two component evolutionary system of a homogeneous

KdV equations of order 3 (type I) as well as (type II), and the generalized coupled Hirota Satsuma KdV. The

numerical results show that this method is a powerful tool for solving systems of nonlinear PDEs.

Keywords: Homotopy perturbation method (HPM); systems of nonlinear PDEs

1 Introduction

Homotopy perturbation method was established in 1999 by J. H. He [1], and was further developed and improved by

him [24]. The method is a powerful and efficient technique to solve many types of linear and nonlinear problems. The

coupling of the perturbation method and homotopy method is called the homotopy perturbation method. This method can

take the advantages of the perturbation method while eliminating its restrictions.

HPM has been applied by many authors [5, 6, 13, 14], to solve many types of linear and non-linear equations in

science and engineering. The investigation of exact or approximate solutions to partial differential equations help us to

understand physical phenomena better. In this method, the solution is considered as the sum of an infinite series, whichrapidly converges to accurate solutions. Using the homotopy technique in topology, a homotopy is constructed with an

embedding parameter p [0, 1], which is considered as a small parameter.In this work, we solve three systems of nonlinear wave equations, these systems can be seen in [8]. In mathematical

physics, they play a major role in various fields, such as plasma physics, fluid mechanics, optical fibers, solid state physics,

geochemistry, and so on. One of these systems is the generalized coupled Hirota Satsuma KdV system given by

u

t=

1

2

3u

x3 3u u

x+ 3

x

x;

t=

3

x3+ 3u

x;

t=

3

x3+ 3u

x,

subject to u(x, 0) = 13

+ 2tanh2(x), (x, 0) = tanh(x), (x, 0) = 83

tanh(x). The generalized coupled HirotaSatsuma KdV system investigated by many authors using different methods such as the extended tanh method [9, 11],

differential transform method [7] and trig-hyperbolic function method [10].

The other two systems are called component evolutionary systems of homogeneous KdV equations of order 3 (type

I), (type II) respectively given by

u

t

3u

x3 u u

x

x= 0,

Corresponding author. E-mail address: [email protected]

Copyright cWorld Academic Press, World Academic Union

IJNS.2011.12.31/566


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486 International Journal of NonlinearScience,Vol.12(2011),No.4,pp. 485-497

t+ 2

3

x3+ u

x= 0,

subject to

u(x, 0) = 3

6 tanh2(

x

2

), (x, 0) =

3

2tanh2(x

2

),

andu

t

3u

x3 2 u

x u

x= 0,

t u u

x= 0,

subject to

u(x, 0) = tanh( x3

), (x, 0) = 16 1

2tanh2(

x3

).

In [9, 11] the extended tanh method is used to obtain soliton solutions of the above two systems.

2 Analysis of hes homotopy perturbation methodTo illustrate the basic idea of this method, we consider the following nonlinear differential equation

A(u) f(r) = 0, r , (1)with the boundary conditions

B(u,u

) = 0, r , (2)

where A is a general differential operator, B is a boundary operator, f(r) is a known analytical function, and is theboundary of the domain . The operator A can be divided into two parts, which are L and N, where L is the linear, andN is the nonlinear operator. Therefore, (1) can be written as follows:

L(u) + N(u) f(r) = 0, r . (3)By the homotopy technique, we construct a homotopy (r, p) : [0, 1] R, which satisfies

H(, p) = (1 p)[L() L(u0)] + p[A() f(r)] = 0, (4)which is equivalent to,

H(, p) = L() L(u0) + p[L(u0) + N() f(r)] = 0, (5)where p [0, 1] is an embedding parameter, u0 is an initial approximation solution of (1), which satisfies the boundaryconditions. Obviously, from (4) and (5) we obtain

H(, 0) = L() L(u0) = 0,H(, 1) = A()

f(r) = 0.

Changing the process of p from zero to unity is just a change of (r, p) from u0(r) to u(r). In topology, this is calledhomotopy. According to the HPM, we can first use the embedding parameter p as a small parameter, and assume that the

solutions of equations (4) and (5) can be written as a power series in p:

= 0 + p1 + p22 + p

33 + . (6)Setting p = 1, gives the solution of (1)

u = limp 1

= 0 + 1 + 2 + 3 + . (7)

To study the convergence of the method [12], we rewrite the equation (5) in the following form

L() = L(u0) + p[f(r)

L(u0)

N()], (8)

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M. Alquran, M. Mohammad: Approximate Solutions to System of Nonlinear Partial Differential Equations 487

applying the inverse operator L1, to both sides of equation (8), we obtain

= u0 + p[L1f(r) L1N() u0], (9)

substituting (6) into the right-hand side of equation (9), we get the following form

= u0 + p[L1f(r) (L1N)(

i=0

pii) u0].

The exact solution may be obtained by using (7)

u = limp 1

= L1f(r) (L1N)(i=0

i)

= L1f(r)

i=0

(L1N)(i).

The series (7) is convergent for most cases. However the following suggestions has been made by He [1] to find theconvergence rate on nonlinear operator:

(1) The second derivative of N() with respect to must be small because the parameter may be relatively large, i.e.,p 1.

(2) The norm ofL1 N

must be smaller than one so that the series converges.

3 Systems of nonlinear wave equations

In this section, we apply the HPM to the proposed systems of interest

3.1 System I

Consider a two component evolutionary system of a homogeneous KdV equations of order 3 (type I)

u

t

3u

x3 u u

x

x= 0, (10)

t+ 2

3

x3+ u

x= 0, (11)

subject to

u(x, 0) = 3 6tanh2

(

x

2 ), (x, 0) = 3

2tanh

2

(

x

2 ). (12)By HPM, the homotopy construction of system (10), (11) is

u

t+ p(

3u

x3 u u

x

x) = 0, (13)

t+ p(2

3

x3+ u

x) = 0. (14)

Suppose that the solution of system (10), (11) has the form

u = u0 + pu1 + p2u2 + p

3u3 + (15)

= 0 + p1 + p22 + p

33 +

. (16)

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Substituting (15), (16) in (13), (14), respectively; and comparing the coefficients of identical degrees of p, we obtain

p0 :u0

t= 0,

p0

:0

t = 0, (17)

p1 :u1

t u0 u0

x

3u0

x3 0 0

x= 0,

p1 :1

t+ u0

0

x+ 2

30

x3= 0, (18)

p2 :u2

t u1 u0

x u0 u1

x 1 0

x 0 1

x

3u1

x3= 0,

p2 :2

t+ u1

0

x+ u0

1

x+ 2

31

x3= 0. (19)

The solutions of (17) can be obtained by using (12)

u0 = u(x, 0) = 3 6tanh2( x2

)

0 = (x, 0) = 3

2tanh2(x

2)

Hence, the solution of (18) is

u1 =

t0

u0u0

x+

3u0

x3+ 0

0

xdt

= 48t csch3(x)(sinh4( x2

)

1 =

t

0

u0 0x

2 30

x3dt

= t(9

2sech2(x

2) tanh(

x

2) 12

2sech4(

x

2) tanh(

x

2)

12

2sech2(x

2)tanh3(

x

2)).

Considering the first eight terms of (15), (16), then the approximate solution of the system (10), (11) by setting p = 1 is

uapp(x, t) =

7i=0

ui

= 3

6tanh2(

x

2

)

48t csch3(x)(sinh4(

x

2

) +

app(x, t) =

7i=0

i

= 3

2 tanh2(x

2) + t(9

2sech2(

x

2) tanh(

x

2)

12

2sech4(x

2) tanh(

x

2) 12

2sech2(

x

2)tanh3(

x

2)) + .

The exact solution of (10), (11) is

u(x, t) = 3 6tanh2( t + x2

)

(x, t) =

3

2 tanh2(t + x

2

). (20)



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Table 1: Absolute error | u(x, t) uapp(x, t) | for system I with n=7t

x 0.05 0.1 0.15 0.2 0.25 0.3

-6 8.9 1016 1.8 1015 6.6 1014 6.8 1013 4.2 1012 1.9 1011-4 1.3 1015 3.5 1013 9 1012 9.1 1011 5.5 1010 2.4 109-2 2.7 1014 6.9 1012 1.8 1010 1.7 109 1 108 4.4 1082 2.8 1014 6.9 1012 1.8 1010 1.8 109 1 108 4.5 1084 4.4 1016 3.4 1013 8.7 1012 8.7 1011 5.1 1010 2.2 1096 8.9 1016 1.8 1015 5.2 1014 4.8 1013 2.7 1012 1.1 1011

The absolute errors with respect to uapp(x, t), app(x, t), are shown respectively in Table 1 and Table 2.

Table 2: Absolute error | (x, t) app(x, t) | for system I with n=7t

x 0.05 0.1 0.15 0.2 0.25 0.3

-6 8.9 1016 8.9 1016 4.5 1014 4.8 1013 3 1012 1.3 1011-4 8.9 1016 2.4 1013 6.4 1012 6.4 1011 3.9 1010 1.7 109-2 1.9 1014 4.9 1012 1.2 1010 1.2 109 7.3 109 3.1 1082 2 1014 4.9 1012 1.2 1010 1.2 109 7.4 109 3.2 1084 4.4 1016 2.4 1013 6.2 1012 6.1 1011 3.6 1010 1.6 1096 8.9 1016 8.9 1016 3.6 1014 3.4 1013 1.9 1012 7.9 1012

Also, the absolute errors with respect to uapp(x, t), app(x, t) are shown respectively in Figure 1 in the region 10 x 10 and 0 t 0.5.

3.2 System II

Consider a two component evolutionary system of a homogeneous KdV equations of order 3 (type II)

u

t

3u

x3 2 u

x u

x= 0, (21)

t u u

x= 0, (22)

subject to

u(x, 0) = tanh( x3

), (x, 0) = 16 1

2tanh2(

x3

). (23)

The homotopy construction of (21), (22) is

u

t+ p(

3u

x3 2 u

x u

x) = 0, (24)

t+ p(u u

x) = 0. (25)

Suppose that the solution of system (21), (22) has the form

u = u0 + pu1 + p2u2 + p

3u3 +

(26)



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Figure 1: The absolute errors with respect to uapp(x, t), app(x, t) are shown respectively, 10 x 10, and 0 t 0.5, for system I with n=7.

= 0 + p1 + p22 + p

33 + . (27)Substituting (26), (27) in (24), (25), respectively; and comparing the coefficients of identical degrees of p, we obtain

p0 :u0

t= 0,

p0 :0

t= 0, (28)

p1 :u1

t 20 u0

x u0 0

x

3u0

x3= 0,

p1

:

1

t u0u0

x = 0, (29)

p2 :u2

t 21 u0

x 20 u1

x u1 0

x u0 1

x

3u1

x3= 0,

p2 :2

t u1 u0

x u0 u1

x= 0.

(30)

The solutions of (28) can be obtained by using (23)

u0 = u(x, 0) = tanh( x3

)

0 = (x, 0) = 1

6 1

2 tanh2

(

x

3 ),then we derive the solution of equations (29) as

u1 =

t0

20u0

x+ u0

0

x+

3u0

x3dt

=1

3t sech2(

x3

)

1 =

t0

u0u0

xdt

=1

3t tanh(

x

3)sech2(

x

3).



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Considering the first nine terms of (26), (27), then the approximate solution of (21), (22) by setting p = 1 is

uapp(x, t) =8

i=0

ui

= tanh( x3

) +1

3t sech2(

x3

) +

app(x, t) =8

i=0

i

= 16 1

2tanh2(

x3

) +1

3t tanh(

x3

)sech2(x

3) + . (31)

The exact solution of (21), (22) is

u(x, t) = tanh(t + x3

)

(x, t) = 16 1

2tanh2(t + x

3). (32)

Behavior ofu(x, t), uapp(x, t), are shown in Figures 2, 4 and the behavior of (x, t), app(x, t) are shown in Figures 3, 5in the regions 10 x 10 and 0 t 0.5.

Figure 2: u(x, t), uapp(x, t) are shown respectively, 10 x 10, and 0 t 0.5, for system II with n=8.

3.3 System III

Consider the generalized coupled Hirota Satsuma KdV system

u

t 1

2

3u

x3+ 3u

u

x 3

x( ) = 0, (33)

t+

3

x3 3u

x= 0, (34)

t

+3

x3

3u

x

= 0, (35)



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Figure 3: (x, t), app(x, t) are shown respectively,

10

x

10, and 0

t

0.5, for system II with n=8.

Figure 4: u(x, ti), uapp(x, ti) are shown respectively, 5 x 5, for different value of t, t = 0.1, 0.2, 0.3, 0.4, 0.5,for system II with n=8.

Figure 5: (x, ti), app(x, ti) are shown respectively, 5 x 5, for different value of t, t = 0.1, 0.2, 0.3, 0.4, 0.5,for system II with n=8.



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subject to

u(x, 0) =13

+ 2 tanh2(x),

(x, 0) = tanh(x),

(x, 0) =8

3tanh(x).

By HPM, the homotopy construction of system (33)-(35) is

u

t+ p(1

2

3u

x3+ 3u

u

x 3

x( )) = 0, (36)

t+ p(

3

x3 3u

x) = 0, (37)

t+ p(

3

x3 3u

x) = 0. (38)

Suppose that the solution of (33)-(35) has the form

u = u0 + pu1 + p2u2 + p

3u3 + (39)

= 0 + p1 + p22 + p

33 + (40) = 0 + p1 + p

22 + p33 + . (41)

Substituting (39) - (41) in (36) - (38), respectively; and comparing the coefficients of identical degrees of p, we obtain

p0 :u0

t= 0,

p0 :0

t= 0,

p0 : 0t

= 0, (42)

p1 :u1

t+ 3u0

u0

x 30 0

x 30 0

x 1

2

3u0

x3= 0,

p1 :1

t 3u0 0

x+

30

x3= 0,

p1 :1

t 3u0 0

x+

30

x3= 0, (43)

p2 :u2

t+ 3u1

u0

x+ 3u0

u1

x 31 0

x 30 1

x

310

x 301

x 1

2

3u1

x3 = 0,

p2 :2

t 3u1 0

x 3u0 1

x+

31

x3= 0,

p2 :2

t 3u1 0

x 3u0 1

x+

31

x3= 0.

The solution of (42) can be obtained by using

u0 = u(x, 0) =13

+ 2 tanh2(x)

0 = (x, 0) = tanh(x)

0 = (x, 0) =8

3

tanh(x),



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and hence the solution of equations in (43) is

u1 =

t0

3u0 u0x

+ 300

x+ 30

0

x+

1

2

3u0

x3dt

= 4tsech2

(x) tanh(x)

1 =

t0

3u00

x

30

x3dt

= tsech 2(x)

1 =

t0

3u00

x

30

x3dt

=8

3tsech 2(x).

Considering the first seven terms of (39)-(41), then the approximate solution of (33)-(35) by setting p = 1 is

uapp(x, t) =6

i=0

ui

=13

+ 2 tanh2(x) + 4tsech 2(x) tanh(x) +

app(x, t) =

6i=0

i

= tanh(x) + tsech 2(x) +

app(x, t) =6

i=0

i

= 83

tanh(x) + 83

tsech 2(x) + .

The exact solution of (33)-(35) is

u(x, t) =13

+ 2 tanh2(t + x)

(x, t) = tanh(t + x)

(x, t) =8

3tanh(t + x). (44)

Behavior of u(x, t), uapp(x, t) are shown in Figures 6, 9, behavior of (x, t), app(x, t), are shown in Figures 7, 10 andthe behavior of(x, t), app(x, t), are shown in Figures 8, 11 in the regions 10 x 10, and 0 t 0.5.

4 Conclusion

Homotopy perturbation method has widely been used in solving nonlinear problems. It requires small size of compu-

tations and rapid convergence, the homotopy perturbation method is more reliable and confirms the power and ability

as an easy device for computing the solution of partial differential equations PDEs. In this work, HPM was successful

implemented in approximating the solutions of nonlinear systems of PDEs.

References

[1] J. H. He. Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178(1999):257-262.



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Figure 6: u(x, t), uapp(x, t) are shown respectively, 10 x 10, and 0 t 0.5, for system III with n=6.

Figure 7: (x, t), app(x, t) are shown respectively, 10 x 10, and 0 t 0.5, for system III with n=6.

Figure 8: (x, t), app(x, t) are shown respectively, 10 x 10, and 0 t 0.5, for system III with n=6.



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Figure 9: u(x, ti), uapp(x, ti) are shown respectively, 5 x 5, for different value of t, t = 0.1, 0.2, 0.3, 0.4, 0.5,for system III with n=6.

Figure 10: (x, ti), app(x, ti) are shown respectively, 5 x 5, for different value of t, t = 0.1, 0.2, 0.3, 0.4, 0.5,for system III with n=6.

Figure 11: (x, ti), app(x, ti) are shown respectively, 5 x 5, for different value of t, t = 0.1, 0.2, 0.3, 0.4, 0.5,for system III with n=6.



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[2] J.H. He. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Non-

linear Mech. 35(2000):37-43.

[3] J.H. He. Comparison of homotopy perturbtaion method and homotopy analysis method. Appl. Math. Comput.

156(2004):527-539.

[4] J.H. He. Homotopy perturbation method, a new nonlinear analytical technique. Appl. Math. Comput. 135(2003):73-79.

[5] J. Biazar, H. Ghazvini. Exact solution for non-linear Schrodinger equations by Hes Homotopy perturbation method.

Physics Letters A 366(2007):79-84.

[6] Zaid Odibat, Shaher Momani. A reliable treatment of homotopy perturbation method for Klein-Gordon equations.

Physics Letters A 356(2007):351-357.

[7] Figen Kangalgil, Fatma Ayaz. Solitary wave solutions for hirota-satsuma coupled KdV equation and coupled mKdV

equation by differential transfom method. The Arabian Journal for Science and Engineering. 35(2010):203-213.

[8] Mikhail V. Foursov and Mark Moreno Maza. On Computer-assisted Classifications of Coupled Integrable Equations.

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[9] Ahmet Bekir. Applications of the extended tanh method for coupled nonlinear evolution equations. Communications

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[10] Roba Al-Omary. Soliton solutions to systems of nonlinear partial differential equations using trigonometric-functionmethod. Master Thesis. Jordan University of Science and Technology. 2011.

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Math. Comput. 217(2010): 1997-2006.

[12] S. J. Liao. Beyond Perturbation, Interoduction to Homotopy Analysis Method. Chapman and Hall/Crc Press, Boca

Raton. 2003.

[13] N. H. Sweilam, M. M. Khader. Exact solutions of some coupled nonlinear partial differential equations using the

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