Application of the modified exponential function

Application of the modified exponential function method to the Cahn-Allen equation Hasan Bulut

Citation: 1798, 020033 (2017); doi: 10.1063/1.4972625 View online: http://dx.doi.org/10.1063/1.4972625 View Table of Contents: http://aip.scitation.org/toc/apc/1798/1 Published by the American Institute of Physics

Application of the Modified Exponential Function Method to the Cahn-Allen Equation Hasan Bulut Department of Mathematics, Faculty of Science, Firat University, Elazig, Turkey [email protected]

  -expansion function method to the Cahn-Allen equation. We

Abstract. In this study, we have applied the modified exp Ǧȳ Ɍ

have obtained some new analytical solutions such hyperbolic function solutions. Then, we have constructed the two and three dimensional surfaces for all analytical solutions obtained in this paper by using Wolfram Mathematica 9.

INTRODUCTION The Scottish engineer John Russel has firstly investigated a solitary wave [1]. He has followed a water wave travelling through a canal [2]. The investigation of the travelling wave solutions for nonlinear partial differential equation plays an important role in the study of nonlinear physical phenomena. In recent years effective methods have been developed to be using for solving nonlinear differential equation such as (G'/G)-expansion method [3,4, 23-25], the improved (G'/G)-expansion method [5-7], the modified simple equation method [8], the Sumudu transform method [9-12,29], the Bäcklund transform method [13], Natural transform decomposition method [30], the homotopy analysis method [14,15], the exponential function method [16-18,26], the modified exponential function method [19], generalized Bernoulli sub-ODE method [20], homotopy perturbation method [31], extended trial equation method [32], improved Bernoulli sub-ODE method [21,27,28]. In [22], Taşcan and Bekir obtained travelling wave solutions of Cahn-Allen equation by using first integral method. When it comes to this paper, we consider the Cahn-Allen equation [22] defined by; (1) —– ൌ — šš Ǧ — ͵ ൅ —Ǥ





We introduce the general properties of the modified exp Ǧȳ  Ɍ -expansion function method (MEFM) in section

2. We apply MEFM in section 3 for obtaining some new analytical hyperbolic function solutions to the Eq.(1).

FUNDAMENTAL PROPERTIES OF METHOD





The general properties of MEFM have been proposed in this section. MEFM is based on the exp Ǧȳ  Ɍ expansion function method [16-18]. In order to apply this method to the nonlinear partial differential equations, we consider following differential equation: (2)  —ǡ— š ǡ—– ǡ— šš ǡ—–– ǡǥ ൌ Ͳǡ



where — ൌ — šǡ–



is an unknown function,  is a polynomial in — ൌ — šǡ– and its derivative in which highest

order derivatives and nonlinear terms are involved and the subscripts stand for the partial derivatives. The basic phases of method are expressed as follows: Step 1: Let’s consider the following travelling transformation defined by —  šǡ– ൌ  Ɍ ǡ Ɍ ൌ š Ǧ …–Ǥ (3)

ICNPAA 2016 World Congress AIP Conf. Proc. 1798, 020033-1–020033-8; doi: 10.1063/1.4972625 Published by AIP Publishing. 978-0-7354-1464-8/$30.00

020033-1

Using Eq.(3), we can convert Eq.(2) into nonlinear ordinary differential equation (NODE) defined by;

 ǡ cǡ ccǡ cccǡ " ൌ ͲǤ

where U

U [ , U c

(4)

d 2U , ". d[ 2

dU , U cc d[

Step 2: Suppose that the travelling wave solution of Eq.(4) can be rewritten as following manner;





 Ɍ ൌ



‹



Œ

¦ ‹ ª¬‡š’ Ǧȳ Ɍ º¼ ‹ൌͲ



¦  Œ ª¬‡š’ Ǧȳ Ɍ º¼



ŒൌͲ

ൌ

 ǡ Ͳ ൅ ͳ‡š’ Ǧȳ ൅ "൅   ‡š’  Ǧȳ Ͳ ൅ ͳ‡š’ Ǧȳ ൅ "൅  ‡š’  Ǧȳ

(5)

where ‹ ǡ  Œ ǡ Ͳ d ‹ d ǡͲ d Œ d  are constants to be determined later, such that  z Ͳǡ  z Ͳǡ and

ȳ ൌ ȳ Ɍ verifies the following ordinary differential equation;









ȳ c Ɍ ൌ ‡š’ Ǧȳ Ɍ ൅ Ɋ‡š’ ȳ Ɍ ൅ ɉǤ

(6)

Eq. (6) has the following solution families [26]: ʹ

Family-1: When Ɋ z Ͳǡ ɉ Ǧ ͶɊ ൐ Ͳǡ

§ Ǧ ɉ ʹ Ǧ ͶɊ § ɉ ʹ Ǧ ͶɊ · ɉ · ȳ Ɍ ൌ Ž ¨ –ƒŠ ¨ Ɍ ൅ …1 ¸ Ǧ ǡ  ¨ ʹ ¸ ʹɊ ¸¸ ¨ ʹɊ © ¹ © ¹

(7)

ʹ

Family-2: When Ɋ z Ͳǡ ɉ Ǧ ͶɊ ൏ Ͳǡ

§ Ǧɉ ʹ ൅ ͶɊ § Ǧɉ ʹ ൅ ͶɊ · ɉ · –ƒ ¨ Ɍ ൅ …1 ¸ Ǧ ǡ  ¨ ¸ ʹɊ ¸¸ ¨ ʹɊ ʹ © ¹ © ¹

ȳ Ɍ ൌ Ž ¨

(8)

ʹ

Family-3: When Ɋ ൌ Ͳǡ ɉ z Ͳǡ and ɉ Ǧ ͶɊ ൐ Ͳǡ

§

ȳ Ɍ ൌ ǦŽ ¨



ɉ

¨ ‡š’ ɉ Ɍ ൅ … 1 ©



· ¸ǡ Ǧͳ ¸ ¹

(9)

ʹ

Family-4: When Ɋ z Ͳǡ ɉ z Ͳǡ and ɉ Ǧ ͶɊ ൌ Ͳǡ

§ ʹɉ Ɍ ൅ … ൅ Ͷ · 1 ¸ǡ ¨ ɉ ʹ Ɍ ൅ … ¸ 1 ¹ ©

ȳ Ɍ ൌ Ž ¨Ǧ

(10)

ʹ

Family-5: When Ɋ ൌ Ͳǡ ɉ ൌ Ͳǡ and ɉ Ǧ ͶɊ ൌ Ͳǡ





ȳ Ɍ ൌ Ž Ɍ ൅ …1 ǡ

(11)

being Ͳ ǡ ͳ ǡ ʹ ǡ " ǡ  ǡ Ͳ ǡ ͳ ǡ  ʹ ǡ " ǡ   ǡ… ͳ ǡ ɉǡ Ɋ are constants to be determined later. The positive integer N and M can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms occurring in Eq. (5). § Step 3: Setting Eq.(6) and Eq.(7) in Eq. (5), we get a polynomial of ‡š’ ¨©Ǧȳ Ɍ § coefficients of same power of ‡š’ ¨©Ǧȳ Ɍ

·¸¹ .

We equate all the

·¸¹ to zero. This procedure yields a system of equations whichever can be

solved to find Ͳ ǡ ͳ ǡ ʹ ǡ " ǡ  ǡ Ͳ ǡ ͳ ǡ  ʹ ǡ " ǡ   ǡ… ͳ ǡ ɉǡ Ɋ with the aid of commercial software programming Mathematica 9. Substituting the values in Eq. (5), the general solutions of Eq. (5) complete the determination of the solution of Eq. (1).

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APPLICATION In this sub-section of study, we apply above mentioned method to the Cahn-Allen equation [22] for obtaining some new analytical solutions such as new hyperbolic function solution. Example-1 Let's consider the travelling wave transformation as following; —  šǡ– ൌ  Ɍ ǡ Ɍ ൌ š Ǧ …–ǡ (12) where k is the wave number and c is the frequencies of waves . When we substitute Eq.(12) in the Eq.(1), we can obtain following nonlinear differential equation; ʹ

͵

  cc Ǧ  ൅  ൅ … c ൌ Ͳǡ

(13) having …ǡ  are both real constants and non-zero. When we rearrange to Eq.(5), with the help of balance principle between  cc and

U 3 , we obtain the following relationship between

M and N ; N M  1. This relationship gives some new analytical solutions for the Cahn-Allen equation Eq.(1) as follows;

Case 1: If we choose M

ൌ

1 and N

(14)

2 , we can write following equations;



Ͳ ൅ ͳ‡š’ Ǧȳ ൅ ʹ ‡š’ ʹ Ǧȳ Ͳ ൅ ͳ‡š’ Ǧȳ

ൌ b

Ȳ

ǡ (15)

and

b c˕  ˕ cb

8c





(16)

b ccȲ ǦȲ b cȲ c Ǧ Ȳ ccb ൅Ȳ cb c Ȳ ൅ ʹ Ȳ c bȲ ͵

 cc ൌ

˕

ʹ

ʹ

ʹ

Ȳ

Ͷ

ǡ

#, where ʹ

z

(17)

Ͳ and ͳ z Ͳ . When we use Eqs.(15,16,17) into Eq.(13), we get a system of algebraic equations

from the coefficients of polynomial of ‡š’ ¨©Ǧȳ Ɍ §

·¸¹ Ǥ

Solving this system with the help of Wolfram Mathematica 9

yields the following coefficients; Case 1.1: ʹ ൌ Ǧ

Pൌ

ʹ…ͳ ͵

ʹ

ǡͲ ൌ Ǧ

Ǧ͵ͳ ൅ ͻͳ ൅ ʹͶ…Ͳ ͳ Ͷ…



ʹ

ǡɉൌ Ǧ



͵ §¨Ͷ…Ͳ ͳ൅  ͳ Ǧͳ ͵ͳ൅ ͻͳʹ ൅ʹͶ…Ͳ ͳ ·¸ © ¹ ͺ… ʹ ͳʹ

͵ͳ Ǧ ͵ͳ ൅ ͻͳ ൅ ʹͶ…Ͳ ͳ ʹ…ͳ

ǡ ൌ Ǧ

ʹ… ͵

,

(18)

.

Case 1.2: ʹ ൌ Ǧ

Ɋൌ

ʹ…ͳ ͵

ʹ

ǡ Ͳ ൌ Ǧ

͵ͳ ൅ ͻͳ ൅ ʹͶ…Ͳ ͳ Ͷ…

ʹ

ǡɉൌ

ͳʹ…Ͳ ͳ Ǧ͵  ͳ Ǧͳ §¨Ǧ͵ͳ൅ ͻͳʹ ൅ʹͶ…Ͳ ͳ ·¸ © ¹ ͺ… ʹ ͳʹ

Ǧ͵ͳ ൅ ͵ͳ ൅ ͻͳ ൅ ʹͶ…Ͳ ͳ

Ǥ

020033-3

ʹ…ͳ

ǡ ൌ Ǧ

ʹ… ͵

ǡ

(19)

Case 1.3: ʹ ൌ Ǧ

Pൌ

ʹ

ʹ…ͳ

ǡ Ͳ ൌ Ǧ

͵

Ǧ͵ͳ ൅ ͻͳ Ǧ ʹͶ…Ͳ ͳ Ͷ…



ʹ

ǡɉൌ



͵ §¨ǦͶ…Ͳ ͳ൅  ͳ൅ͳ ͵ͳ൅ ͻͳʹ Ǧ ʹͶ…Ͳ ͳ ·¸ © ¹ ͺ… ʹ ͳʹ

Case 1.4: ͳ ൌ

ʹ Ͳ ͳ

൅

ʹ

Ͳ ͳ ǡ ൌ Ͳ

ʹͳ

ǡɉൌ

͵ͳ ൅ ͵ͳ ൅ ͻͳ Ǧ ʹͶ…Ͳ ͳ ʹ…ͳ

ǡ ൌ Ǧ

ʹ… ͵

ǡ

(20)

Ǥ

ʹͲ ൅ Ͳ ͳ ʹ Ͳ

Ͳ  Ͳ ൅ Ͳ ͳ

ʹ

ǡɊ ൌ

ʹ ʹ ʹ Ͳ

ǡ… ൌ

͵ʹ Ǥ ʹͳ

(21) Case 1.5:

Ͳ ൌ

Ͳ Ǧʹ Ͳ ൅ͳͳ ʹ ͳ

ǡ ൌ

ʹ ʹͳ

ǡɉൌ

   Ǧ    ʹ Ͳ൅ͳ Ǧͳ൅ͳ ǡ ʹͳ Ǧ ͳ ʹͲ Ǧ ǡɊ ൌ ʹ Ͳ ͳ ͳ ʹ ʹ ͳ    ʹ

ʹ

ͳ

͵ …ൌǦ ʹ Ǥ ʹͳ

(22) Using coefficients of Eq.(18) along with Eqs.(3-7) in Eq.(15), we find new hyperbolic function solution for Eq.(1) as following; —ͳ  šǡ– ൌ Ǧ

where f ( x, t )

͵ ൅ ͳ

ʹ

ͻͳ ൅ ʹͶ…Ͳ ͳ

͸ͳ ൅ ʹ ͻͳ ൅ ʹͶ…Ͳ ͳ Ǧ ͸ͳ ͳ ൅ ƒŠ >ˆሺšǡ–ሻ @ ʹ

3c1  3ct  2cx 4c

1

2 1

4c 2 B12

0 1

(23)

ǡ

and

3 A 3B  9 A 24cA B P z 0, 1

Ǧͳ ൅ ƒŠ >ˆሺšǡ–ሻ @

2





3§¨ 4cA0 B1  A1  B1 3 A1  9 A12  24cA0 B1 ·¸ © ¹  ! 0. 2c 2 B12

u#x,t'

1.0 0.8 0.6 0.4 0.2

15

10

5

5

10

15

x

FIGURE 1. The 3D and 2D surfaces of the analytical solution Eq.(23) being hyperbolic function solution by considering the values c 0.6, A0 3, B1 5, A1 1, c1 0.1, 15  x  15,  2  t  2, and – ൌ ͲǤͲͳ for 2D.

Using coefficients of Eq.(19) along with Eqs.(3-7) in Eq.(15), we find another hyperbolic function solution for Eq.(1) as following;

020033-4

— 2  šǡ– ൌ Ǧ

where f ( x, t )

Ǧ͵ ൅

ʹ

ͻͳ ൅ ʹͶ…Ͳ ͳ

ͳ

Ǧ͸ͳ ൅ ʹ ͻͳ ൅ ʹͶ…Ͳ ͳ ൅ ͸ͳ ͳ ൅ƒŠ >ˆሺšǡ–ሻ @ ʹ

3c1  3ct  2cx 4c 2 1

1

(24)

ǡ

and

 3 A 3B  9 A 24cA B P z 0, 1

Ǧͳ ൅ƒŠ >ˆሺšǡ–ሻ @

2

0 1

4c 2 B12





3¨§ 4cA0 B1  A1  B1 3 A1  9 A12  24cA0 B1 ·¸ © ¹  ! 0. 2c 2 B12 u#x,t'

2

1

15

10

5

5

10

15

x

1

FIGURE 2. The 3D and 2D surfaces of the analytical solution Eq.(24) being hyperbolic function solution by considering the values c 0.6, A0 3, B1 5, A1 1, c1 0.1, 15  x  15,  2  t  2 and t 0.01 for 2D.

Using coefficients of Eq.(20) along with Eqs.(3-7) in Eq.(15), we find another hyperbolic function solution for Eq.(1) as following; — 3  šǡ– ൌ

͵ ൅ ͳ

ʹ

ͻͳ Ǧ ʹͶ…Ͳ ͳ

͸ͳ ൅ ʹ ͻͳ Ǧ ʹͶ…Ͳ ͳ ൅ ͸ͳ ͳ ൅ƒŠ > ‰ሺšǡ–ሻ @ ʹ

3c1  3ct  2cx

where g ( x, t ) 1

2 1

4c 2 B12

0 1

(25)

ǡ

and

4c

3 A 3B  9 A 24cA B P z 0, 1

Ǧͳ ൅ƒŠ >‰ሺšǡ–ሻ @

2





3¨§ 4cA0 B1  A1  B1 3 A1  9 A12 24cA0 B1 ¸· © ¹  ! 0. 2c 2 B12 u#x,t'

1

15

10

5

5

10

15

x

1

2

FIGURE 3. The 3D and 2D surfaces of the analytical solution Eq.(25) being hyperbolic function solution by considering the values c 0.6, A0 3, B1 5, A1 1, c1 0.1, 15  x  15,  2  t  2, and t 0.01 for 2 D.

020033-5

Using coefficients of Eq.(21) along with Eqs.(3-7) in Eq.(15), we find another new hyperbolic function solution for Eq.(1) as following; Ǧͳ

§ · ¨ ¸ ¨ ¸ ʹ  Ͳ ൅Ͳ —Ͷ šǡ– ൌ Ͳ ¨ Ͳ ൅ ¸ ª º § · ͳ ʹ…  ͳ ͳ ¨ ¸ Ǧͳ൅ƒŠ Ǧ͵–൅ ʹ š൅ « » ¨ ¸ ¨ ʹ ¹ ¼ ¸¹ ¬Ͷ © ©





(26)

ͶͲ  Ͳ ൅Ͳ ͳʹ ʹͲ ൅Ͳ ͳʹ ൅ ൐ ͲǤ ʹʹ Ͳʹ ʹʹ Ͳʹ ʹ

Ɋ z Ͳǡ Ǧ

where

u#x,t'

1

15

10

5

5

10

15

x

1

2

FIGURE 4. The 3D and 2D surfaces of the analytical solution Eq.(26) being hyperbolic function solution by considering the values c 0.6, A0 3, B0 15, A2 12, c1 0.1, 15  x  15,  2  t  2, and t 0.02 for 2D.

Using coefficients of Eq.(22) along with Eqs.(3-7) in Eq.(15), we find another hyperbolic function solution for Eq.(1) as following; §

—ͷ



ªͳ § ʹ… ͳͳ · º · »¸ ¨ ͵–൅ ʹ š൅ Ͷ ʹ ¸¹ »¼ ¸¹ ¬« © © šǡ– ൌ § § ªͳ § ʹ…  · º · · Ǧʹʹ Ͳ ൅ͳ ¨ ʹͳ൅ͳ ¨¨Ǧͳ൅ƒŠ « ¨ ͵–൅ ʹ š൅ ͳ ͳ ¸ » ¸¸ ¸ ¨ Ͷ ʹ ¹ ¼» ¹ ¸¹ ¬« © © ©

where

Ǧʹ Ͳ ൅ ͳͳ ¨¨ͳ൅ƒŠ «



P z 0,

B12 A22

(27)

! 0. u#x,t'

1.0 0.8 0.6 0.4 0.2

15

10

5

5

10

15

x

FIGURE 5. The 3D and 2D surfaces of the analytical solution Eq.(27) being hyperbolic function solution by considering the values c 0.6, A0 3, B1 0.1, B0 5, A1 12, A2 12, c1 5, 15  x  15,  2  t  2, and t 0.02 for 2D.

020033-6

CONCLUSION In this paper, we have applied the MEFM to the Eq.(1). Then, we have obtained some new analytical solutions such as hyperbolic function solutions. This method has given a lot of coefficients such as Eqs.(18-24). Some of them have been considered in this paper for obtaining new analytical solutions. If other coefficients is considered, of course, ones can obtain more different analytical solutions to the Eq.(1). We have observed that all analytical solutions have been verified the Cahn-Allen equation by using Wolfram Mathematica 9. Therefore, it can be said that this method is a powerful tool for obtaining the solutions of such Eq.(1).

ACKNOWLEDGMENTS The authors would like to thank the reviewers for their comments that help improve the manuscript.

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