exact solutions for nonlinear integro-partial differential ...to solve the nonlinear integro-...

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 10 (2019) pp. 2449-2461

© Research India Publications. http://www.ripublication.com

2449

Exact solutions for nonlinear integro-partial differential equations using the

( , )-expansion method

Ameana Awad Al-Sekhary

Master student, Mathematics Department, Faculty of Science,

Taif University, Taif, Saudi Arabia.

Khaled A. Gepreel

Associate Professor, Mathematics Department, Faculty of Science,

Zagazig University, Zagazig, Egypt.

Abstract

In this paper, we improve the (G'/G, 1/G) - expansion method

to solve the nonlinear integro- partial differential equations.

We use the proposed method to construct the traveling wave

solutions for some nonlinear equations of evolution in

mathematical physics via (1+1)- dimensional Ito nonlinear

integro- partial differential equation, first and second integral

differential KP hierarchy equations. The (G'/G, 1/G) -

expansion method is companied between two different

method namely G'/G expansion method and (1/G) expansion

method. This proposed method will be submitted to literature

to extract exact solutions with arbitrary parameters which

include solitary and periodic wave solutions of nonlinear

integro- differential equations..

Keywords: the nonlinear integro- partial differential

equations, KP hierarchy equations, the (G'/G, 1/G) -

expansion method.

Introduction

Nonlinear partial differential equations are useful in

describing the various phenomena in disciplines and have

many applications not only in physics but also in as medical

sciences, geometry, biology and chemistry. There are many

ways to find accurate solutions for nonlinear partial

differential equations (NPDEs) such as homogenous balance

method [1-2], Darboux transform method [3-4], first integral

method [5-6], tanh function method [7], modified simple

equation method [8-10], auxiliary equations method [11-12],

(G′/G)-expansion method [13-14], F-expansion method [15-

16], Jacobi elliptic function method [17-18] and so on. The

expansion method was proposed by Wang et al. Guo

and Zhou developed extended expansion method [19].

Recently Lü improved generalized expansion method

[20]. In the trial auxiliary equation satisfies the

second order linear equation as .

More recently, Li et al. [21] have presented the two variables

( , )-expansion method. In this paper, we have

applied ( , )-expansion method to find the

traveling wave solutions for some of the following nonlinear

integro- partial differential equations:

(i) the (1+1)- dimensional Ito nonlinear partial

differential equation[22]:

(ii) The first integral differential KP hierarchy

equation[23]:

(iii) The second integral differential KP hierarchy

equation [24-25]:

where

The contants of this chapter have submitted to publication in

advanced difference equations.

The algorithm of (G'/G , 1/G )- expansion method

In this section we will describe the main steps of the (

, )-expansion method to search for the traveling wave

solutions for nonlinear development equations. Suppose that

the equation is given by:

https://www.sciencedirect.com/topics/mathematics/transform-method

https://www.sciencedirect.com/topics/computer-science/integral-method

https://www.sciencedirect.com/topics/computer-science/integral-method

https://www.sciencedirect.com/science/article/pii/S1110256X17300597#bib0007

https://www.sciencedirect.com/topics/mathematics/auxiliary-equation








2450

Q is a polynomial in v(x, t) and its partial derivatives in

which the highest order derivatives and nonlinear terms are

involved. The ( , )-expansion to solve Eq. (2.1) is

summarized in the following steps:

Step 1: Firstly we use the transformation to convert the

nonlinear integro- differential equations to nonlinear partial

differential equations and using the traveling wave

transformation:

, (2.2)

where and are arbitrary constants.

The traveling wave transformation (2.2) permits us to reduce

Eq. (2.1) to the following ODE’s

(2.3)

Step 2: Assume that the solution of ODE (2.3) can be

expressed by a polynomial in φ,

ψ as follows:

(2.4)

where , are

arbitrary constants. In the solution formula (2.4)

are satisfying the following relations

and the auxiliary equation is given by

(2.6)

The Eq.(2.5) and Eq.(2.6) lets to get:

(2.7)

The solution of the Eq.(2.6) in the following three cases:

Case 1. If , the general solution of Eq. (2.6) is given

by

(2.8)

and

where are two arbitrary constants and

Case 2. If , the general solution of Eq. (2.6) is given

by

, (2.10)

and

where

Case 3. If , the general solution of Eq. (2.6) is

Subsequently, the relation between are given by:

Step 3 The balance number n is determined using the

homogeneous balance between the non-linear terms and the

higher order derivatives in the equation (2.3),

Step 4: Substituting (2.4) to (2.3) using (2.7) and (2.9), we

will get many polynomials in and . From equations (2.9),

(2.11) and (2.13) the degree of is not greater than 1.

Setting the coefficient of

to be zero we produces a set of algebraic equations that can be

solved by Maple software package, to obtain the values of

.

Step 5: Substituting the values of in (2.4)

and (2.8), (2.10) and (2.12), we construct many different kind

of the traveling wave solutions to Eq. (2.3).

Application of (G'/G , 1/G)- expansion method

In this section, we apply the proposed method (G'/G, 1/G)-

expansion method to construct the traveling wave solution to

solve the following nonlinear partial differential equation of

Ito (1 + 1) and also to discuss the analytical solution of the

Kadomtsev- Petviashvili hierarchy equations by using (G'/G,

1/G)- expansion method :

the (1+1)- dimensional Ito nonlinear partial differential

equation

In this subsection we will study the exact solutions to the

following (1+1)- dimensional Ito integro differential equation

(3.1.1)

We use the transformations to reduce the integra-

differential equation (3.1.1) to the following nonlinear partial

differential equations

(3.1.2)

The traveling wave transformation

leads to reduce Eq. (3.1.2) to the ODE in the form



2451

Integrating Eq.(3.1.3) twice time, we have:

where is the integration constant further setting ,

we have :

Balancing the highest order derivative with the nonlinear

term , we get n = 2.

We suppose that the Eq. (3.1.6) has the formal solution:

where and are constants.

Case 1. when (hyperbolic function solutions):

Substituting (3.1.7) into (3.1.6) and using (2.7) and (2.9) we

get the polynomials in and , setting the coefficients of

this polynomial to zero to obtain the following set of algebraic

equations:

By solving the previous algebraic equations using the Maple

software packages we get the following results:

Result 1

(3.1.8)

By substituting (3.1.8) into (3.1.7) with (3.1.3) by using (2.5)

and (2.8), we get the

exact solutions of Eq. (3.1.1) as follows:

(3.1.9)

In particular, if we put and in Eq. (3.1.9),

we produce a solitary solution:

(3.1.10)

Setting we illustrate the behavior

of the solitary wave solution (3.1.10) and its projection at

as we shown in the Figure 1.

10 5 5 10x

0.1

0.1

0.2

u11

Fig. 1. The exact solutions of Eq.(3.1.10)

While, if we put and in Eq. (3.1.9), we

produce a solitary solution:

(3.1.11)

Setting we illustrate the behavior of

the solitary wave solution (3.1.11) and its projection at

as we shown in the Figure 2.

Fig. 2. The exact solutions of Eq. (3.1.11)

where and

Result 2

(3.1.12)

10 5 5 10x

1.2

1.0

0.8

0.6

0.4

0.2

u12

https://www.sciencedirect.com/science/article/pii/S1110256X17300597#eqn0003




2452

By substituting (3.1.12) into (3.1.7) with (3.1.3) , (2.5) and

(2.8), we get the


(3.1.13)

In particular, if we put and in

Eq. (3.1.13), We produce a solitary solution:

(3.1.14)

While, if we put and then we


(3.1.15)

The behavior of the exact solution (3.1.13) and and its

projection at when as

we shown in the Figure 3.


where and .

Case 2. when (trigonometric function solutions):

Substituting (3.1.7) into (3.1.6) and using (2.7) and (2.11), we

get the polynomials in and . Setting the coefficients of

this polynomial to be zero to obtain a set of algebraic

(3.1.16)

Solving the system (3.1.16) using the Maple or Mathematica

package we have:

Result 1

(3.1.17)

By substituting (3.1.17) into (3.1.7) with (3.1.3) by using (2.5)


exact solutions of Eq. (3.1.1) has the form:

(3.1.18)

In particular, if we put and in Eq.

(3.1.18), we produce a solitary solution:

(3.1.19)



as shown in the Figure 4.


10 5 5 10x

8

6

4

2

u11

10 5 5 10x

0.1

0.2

0.3

u2





2453

While, if we put and ,then we produce a

solitary solution:

(3.1.20)



as shown in the Figure 5.


where and

Result 2

(3.1.21)

By substituting (3.1.21) into (3.1.7) with (3.1.3) , (2.5) and

(2.10), we get the


(3.1.22)

The behavior of the exact solution (3.1.22) and its projection

at when as we

shown in the Figure 6.

10 5 5 10x

14

12

10

8

6

4

2

u4


In particular, if we put and in

Eq. (3.1.22), we produce the solitary wave solution as

following:

(3.1.23)

While, if we put and , we

produce the solitary solution

(3.1.24)

where and

Case 3. when

By substituting (3.1.7) into (3.1.6) and using (2.7) and (2.13)

we get the polynomials in and By equating coefficients

of equation with zero, we obtain a set of idioms which, by

solving them, we obtain the following results:

(3.1.25)

where and are arbitrary constants . By

substituting (3.1.25) into (3.1.7) with (3.1.3) , (2.5) and (2.12),

we get the rational traveling wave solutions of Eq. (3.1.1)

has the form:

(3.1.26)

Setting we

illustrate the behavior of the rational wave solution (3.1.26)

and its projection at as shown in the Figure 7.


where .

The generalized KP hierarchy equations:

In this subsection we will study the exact solutions to the

following the first and second generalized KP hierarchy

equations integro differential equation

(4.1)

and

where

10 5 5 10x

14

12

10

8

6

4

2

u12

20 10 10 20x

0.4

0.2

0.2

0.4

0.6

u3






2454

The KP equation can be used to model water waves of

long wavelength with weakly non-linear restoring forces

and frequency dispersion. If surface tension is weak compared

to gravitational forces, is used; if surface tension is

strong, then . Because of the asymmetry in the

way x- and y-terms enter the equation, the waves described by

the KP equation behave differently in the direction of

propagation (x-direction) and transverse (y) direction;

oscillations in the y-direction tend to be smoother (be of

small-deviation).

The KP equation can also be used to model waves

in ferromagnetic media, as well as two-dimensional matter–

wave pulses in Bose–Einstein condensates.

The first equation of integral differential KP hierarchy

equations

We use the transformations to reduce the integro-

differential equation (4.1) to the following partial differential

equations

(4.1.1)

The traveling wave transformation

leads to reduce Eq. (4.1.1) to the following ODE in the form

, (4.1.3)

where is the integration constant. When we take the

conversion , Eq. (4.1.3) is written on the formula:

(4.1.4)

Balancing the highest order derivative with the nonlinear

term , we get n = 2.

Consequently the solution of Eq. (4.1.3) has the formal

solution:

(4.1.5)


Case 1. When , we substitute (4.1.5) into (4.1.4) and

using (2.7) and (2.9) and setting the coefficient of and to

be zero , we obtain the following set of algebraic equations:

By solving the previous algebraic equations , we get the

following results:

Result 1

(4.1.7)

Substituting (4.1.7) into (4.1.5), (2.5) ,(2.8), we get the

traveling wave exact solutions of Eq. (4.1) can be represented

in the following form:

(4.1.8)


We produce a solitary solution:

(4.1.9)

Setting we illustrate the

behavior of the solitary wave solution and its projection at

as we shown in Figure 8.


10 5 5 10x

0.40

0.35

0.30

v11

https://en.wikipedia.org/wiki/Water_wave

https://en.wikipedia.org/wiki/Wavelength

https://en.wikipedia.org/wiki/Dispersion_(water_waves)

https://en.wikipedia.org/wiki/Surface_tension

https://en.wikipedia.org/wiki/Earth%27s_gravity

https://en.wikipedia.org/wiki/Ferromagnetism

https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate




2455

While, if we put and in Eq. (4.1.8), We


(4.1.10)

Setting we illustrate the

behavior of the solitary wave solution and its projection at

as we shown in Figure 9.


where and .

Result 2

(4.1.11)

By substituting (4.1.11) into (4.1.5) and using (2.5) and (2.8),

we get the traveling wave solutions of. (4.1.1) take the forms:

(4.1.12)

The behavior of the exact solution (4.1.12) and and its

projection at when as

we shown in the Figure 10.

10 5 5 10x

1.0

0.8

0.6

0.4

v2



(4.1.12), we produce a solitary solution

(4.1.13)

The behavior of the real solitary wave solution (4.1.13) and its

projection at when are



While, if we put and , then ,we

produce the solitary solution:

(4.1.14)

where and

Case 2. when :

Substituting (4.1.5) into (4.1.4) and using (2.7) and (2.11), we

get the polynomials in and setting the coefficients of this

polynomial of and to be zero to obtain a set of algebraic

equations:

10 5 5 10x

0.40

0.39

0.38

0.37

u21

10 5 5 10x

1.0

0.9

0.8

0.7

0.6

0.5

v12






2456

(4.1.15)

Using the Maple software package to solve the system of

algebraic equations (4.1.15) to get the following results:

Result 1

(4.1.16)

Equations (4.1.16), (4.1.5) , (2.5) and (2.10) lead to write the

traveling wave solutions of (4.1.1) as the follows

(4.1.17)


we produce a solitary solution

(4.1.18)

The behavior of the solitary wave solution (4.1.18) and its

projection at , when are



While, if we put and ,then we produce a

solitary solution

(4.1.19)

where and

Result 2

(4.1.20)

Substituting Eq. (4.1.20) into (4.1.5) and using (2.5) , (2.8),

we get the traveling wave solution of Eq. (4.1.1) as follows:

(4.1.21)


(4.1.21), We produce a solitary solution:

(4.1.22)

We illustrate the behavior solution (4.1.22) and its projection

at when as shown in the

Figure 13.


While, if we put and then we

produce a solitary solution

(4.1.23)

where and .

Case 3. when

By substituting (4.1.5) into (4.1.4) and using (2.7) and (2.13),

we get the polynomials in and . Equating coefficients of

this polynomial to be zero, we obtain a set of algebraic

10 5 5 10x

0.9

0.8

0.7

0.6

0.5

0.4

v11

10 5 5 10x

7

6

5

4

3

2

1

v21





2457

equations . Solve this system of algebraic equations to obtain

the following results

(4.1.24)

Equations (4.1.24) , (4.1.5) (2.5) and (2.12), lead to write the

rational traveling wave solutions as the following form:

(4.1.25)

The behavior solution (4.1.25) and its projection when

are shown in the Figure

14.


where .

The Second equation of two members of integral

differential KP hierarchy equations:

In this section, we study the traveling exact solutions of the

integral differential equations (1.3). Notice that the equation

(1.3) contains one, two, and three integral operators. The

traveling wave transformation (4.1.2) transfer the integral

differential equations (1.3) to the following ordinary

differential equation:

(4.2.1)

By using integration , we have

(4.2.2)

where is the integration constant. Balancing the highest

order derivative with the nonlinear term , we get n =

2. Consequently, we suppose the solution of (4.2.2) has the

formal

(4.2.3)


Case 1. When

Substituting (4.2.3) into (4.2.2) and using (2.7) , (2.9), we get

the polynomials in and Setting the coefficients of

and to be zero, we obtain a set of algebraic equations,

which can be solved by using the Maple software packages to

get the following results:

Result 1

(4.2.4)

By substituting (4.2.4) into (4.2.3) with (2.5) and (2.8), we

get the traveling wave solutions of Eq. (4.2.1) as follows:

(4.2.5)



(4.2.6)

The behavior of the solitary wave solution (4.2.6) when

and its projection at are shown in

the Figure 15.


10 5 5 10x

0.3

0.2

0.1

0.1

0.2

0.3

v3

10 5 5 10x

0.50

0.45

0.40

0.35

0.30

0.25

v11





2458

While, if we put and in Eq. (4.2.5),

we produce a solitary solution:

(4.2.7)

The behavior of the solitary wave solution (4.2.7) when

and its projection at are shown in the

Figure 16.


where and .

Result 2

(4.2.8)

In this case the solitary wave solution take the following

(4.2.9)

The behavior of the traveling wave solution (4.2.9) and its

projection at when

are shown in the Figure 17.




(4.2.10)

Also if we put and , then we produce

a solitary solution:

(4.2.11)

where

and .

Case 2. When

Substituting (4.2.3) into (4.2.2) and using (2.7) , (2.11), we

get the polynomials in and Setting the coefficients of

and to be zero, we obtain a set of algebraic equations,

which can be solved by using the Maple software packages to

get the following results:

Result 1

(4.2.12)

By substituting (4.2.12) into (4.2.5) and using (2.5) and

(2.10), we get the exact solutions of Eq. (1.3) as follows:

(4.2.13)



(4.2.14)


projection at when are shown in the

Figure 18.

10 5 5 10x

1.2

1.0

0.8

0.6

v12

10 5 5 10x

0.4

0.2

0.2

v2





2459


While, if we put and , then we


(4.2.15)

where and .

Result 2

(4.2.16)

Substituting (4.2.16) into (4.2.5) using (2.5) and (2.8), we get

the traveling wave exact solutions of Eq. (1.3) as follows:

(4.2.17)




10 5 5 10x

0.4

0.2

0.2

v4




(4.2.18)




20 10 10 20x

0.6

0.4

0.2

0.2

v41


While, if we put and then We


(4.2.19)

where

and .

Case 3. When ,

substituting (4.2.3) into (4.2.2) and using and using (2.7) and

(2.13) we get the polynomials in and . Equating

coefficients of this polynomial to be zero, we obtain a set of

algebraic equations . Solve this system of algebraic equations

to obtain the following results

, (4.2.20)

Consequently by substituting (4.2.20) into (4.2.5) using (2.5)


exact solutions of Eq. (1.3) as follows:

(4.2.21)


projection at when

are shown in the Figure 21.

10 5 5 10x

2.0

1.5

1.0

v11






2460


where

Conclusion

The (G'/G, 1/G)- expansion method is one of the direct

method to solve the nonlinear partial differential equations.

This method is combined between two different method

(G'/G)- expansion method and (1/G)-expansion method . We

improved the (G'/G, 1/G) - expansion method to solve some

of nonlinear problem in mathematical physics namely (1+1)-

dimensional Ito nonlinear integro- partial differential

equation, first and second integral differential KP hierarchy

equations. This proposed method will be submitted to

literature to extract exact solutions with arbitrary parameters

which include solitary and periodic wave solutions of

nonlinear integro- differential equations. This method is

powerful and effective method to solve more complicated

nonlinear evolution equations in mathematical physics.

References

[1] M.Wang, Exact solutions for a compound KdV-

Burgers equation, Phys. Lett. A, 213 (1996) 279-287.

[2] M.Wang, Y. Zhou and Z. Li, Application of a

homogeneous balance method to exact solutions of

nonlinear equations in mathematical physics, Phys.

Lett. A, 216 (1996) 67-75.

[3] S.B. Leble and N.V. Ustinov, Darboux transforms,

deep reductions and solitons, Phys. A, 26 (1993) 5007-

5016.

[4] H.C. Hu, X.Y. Tang, S.Y. Lou and Q.P. Liu, Variable

separation solutions obtained from Darboux

transformations for the asymmetric Nizhnik–Novikov–

Veselov system, Chaos Solit. Fract., 22 (2004) 327-

334.

[5] R.M. El-Shiekh and A.G. Al-Nowehy, Integral

methods to solve the variable coefficient NLSE,

Zeitschrift fur Naturforschung, 68 (2013) 255-260.

[6] Z. Feng and X. Wang, The first integral method to the

two-dimensional Burgers–Korteweg–de Vries equation,

Phys. Lett. A, 308 (2003) 173-178.

[7] H.A. Abdusalam, On an improved complex tanh-

function method, Int. J. Nonlinear Sci. Numer.

Simul., 6 (2005) 99-106.

[8] A.J.M. Jawad, M.D. Petkovic and A. Biswas, Modified

simple equation method for nonlinear evolution

equations, Appl. Math. Comput., 217 (2010) 869-877.

[9] E.M.E. Zayed and S.A.H. Ibrahim, Exact solutions of

nonlinear evolution equations in mathematical physics

using the modified simple equation method, Chin.

Phys. Lett., 29 (2012), Article 060201.

[10] A.V. Porubov, Periodical solution to the nonlinear

dissipative equation for surface waves in a convecting

liquid, Phys. Lett. A, 221 (1996) 391-394.

[11] S. Jiong, Auxiliary equation method for solving

nonlinear partial differential equations, Phys. Lett.

A, 309 (2003) 387-396.

[12] S. Zhang and T. Xia, A generalized new auxiliary

equation method and its applications to nonlinear

partial differential equations, Phys. Lett.

A, 363 (2007) 356-360.

[13] M.E. Islam, K. Khan, M.A. Akbar and R. Islam,

Traveling wave solutions of nonlinear evolution

equations via enhanced (G'/G)-expansion method,

GANIT: J. Bangladesh Math. Soc., 33 (2014) 83-92.

[14] M. Wang, X. Li and J. Zhang, The (G′/G)-expansion

method and travelling wave solutions of nonlinear

evolution equations in mathematical physics, Phys.

Lett. A, 372 (2008) 417-423.

[15] Yun-Mei Zhao, F-Expansion Method and Its

Application for Finding New Exact Solutions to the

Kudryashov-Sinelshchikov Equation, Journal of Appl.

Math. 2013 (2013) 7 pages

[16] M. S. Islam, K. Khan and M. Ali Akber, Application of

the improved F-expansion method with Riccati

equation to find the exact solution of the nonlinear

evolution equations, Journal of the Egyptian

Mathematical Society, 25 (2017) 13-18

[17] M. K. Elboree, The Jacobi elliptic function method and

its application for two component BKP hierarchy

equations, omputers & Mathematics with Applications,

62 (2011) 4402-4414

[18] Zhenya Yan, Jacobi elliptic function solutions of

nonlinear wave equations via the new sinh-Gordon

equation expansion method, MMRC, AMSS,

Academia Sinica,(2003) 363–375

[19] S. Guo and Y. Zhou, The extended G′G-expansion

method and its applications to the Whitham–Broer–

Kaup–Like equations and coupled Hirota–Satsuma

KdV equations, Applied Mathematics and

Computation, 215, (2010) 3214-3221.

[20] H.L. Lü, X.Q. Liu and L. Niu, A generalized (G′/G)-

expansion method and its applications to nonlinear

evolution equations, Applied Mathematics and

Computation, 215, 11, ( 2010) 3811-3816.

[21] L. Li, E. Li and M. Wang, The (G′/G, 1/G)-expansion

method and its application to travelling wave

10 5 5 10x

1.5

1.0

0.5

v3


https://www.hindawi.com/69151638/

https://www.sciencedirect.com/science/journal/1110256X


https://www.sciencedirect.com/science/journal/08981221



https://www.sciencedirect.com/science/article/pii/S0096300309010108







2461

solutions of the Zakharov equations, Applied

Mathematics-A Journal of Chinese Universities, 25,

4, (2010) 454–462.

[22] K. A. Gepreel, T. A. Nofal and A. A. Alasmari, Exact

solutions for nonlinear integro-partial differential

equations using the generalized Kudryashov method,

Journal of the Egyptian Mathematical Society, 25

(2017) 438-444

[23] X.L. Wang, L. Yu, Y.X. Yang and M.R. Chen On

generalized Lax equation of the Lax triple of KP

hierarchy, J. Nonlinear Math. Phys., 22 (2015) 194-203

[24] A.M. Wazwaz Kadomtsev–Petviashvili hierarchy. N-

soliton solutions and distinct dispersion relations, Appl.

Math. Lett., 52 (2016) 74-79

[25] Khaled. A. Gepreel, Exact solutions for nonlinear

integral member of Kadomtsev–Petviashvili hierarchy

differential equations using the modified (w/g)-

expansion method Comput. Math.

Appl., 72 (2016) 2072-2083.

https://link.springer.com/journal/11766

https://link.springer.com/journal/11766


exact solutions for nonlinear integro-partial differential ...to solve the nonlinear integro-...

Documents