A note on the modified simple equation method applied to Sharma–Tasso–Olver equation

Applied Mathematics and Computation 218 (2011) 3962–3964

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A note on the modified simple equation method applied to Sharma–Tasso–Olver equation Elsayed M.E. Zayed Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt

a r t i c l e

i n f o

Keywords: Modified simple equation method Sharma–Tasso–Olver equation Nonlinear evolution equations Exact solutions

a b s t r a c t Jawad et al. have applied the modified simple equation method to find the exact solutions of the nonlinear Fitzhugh–Naguma equation and the nonlinear Sharma-Tasso–Olver equation. The analysis of the Sharma–Tasso–Olver equation obtained by Jawad et al. is based on variant of the modified simple equation method. In this paper, we provide its direct application and obtain new 1- soliton solutions. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The authors [1] have outlined the implementation of the modified simple equation method for solving evolution equations which are very important in applied sciences, via the nonlinear Fitzhugh–Naguma equation and the nonlinear Sharma–Tasso–Olver equation. Although the solutions are correct, authors in [1] used a variant the modified simple equation method when dealing with Sharma–Tasso–Olver equation. The objective of this note is to modify the analysis and results of the nonlinear Sharma–Tasso–Olver equation obtained in [1] using the direct application of the modified simple equation method.

2. Description of the proposed method Consider a nonlinear evolution equation:

Fðu; ut ; ux ; utt ; uxx ; uxt ; . . .Þ ¼ 0;

ð2:1Þ

where F is a polynomial in u and its partial derivatives. In order to solve Eq. (2.1) using the proposed method, we give the following main steps [1]: Step 1. Using the wave transformation

u ¼ uðnÞ;

n ¼ x  t;

ð2:2Þ

From (2.1) and (2.2) we have the following ODE:

Pðu; u0 ; u00 ; . . .Þ ¼ 0; d where P is a polynomial in u and its total derivatives and 0 ¼ dn .

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.09.025

ð2:3Þ

E.M.E. Zayed / Applied Mathematics and Computation 218 (2011) 3962–3964

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Step2. We suppose that Eq. (2.3) has the formal solution:

uðnÞ ¼

N X

Ak

k¼0

 0 k w ðnÞ ; wðnÞ

ð2:4Þ

where Ak are arbitrary constants to be determined such that AN – 0, while w(n) is an unknown function to be determined later. Step3. We determine the positive integer N in (2.4) by balancing the highest order derivatives and the nonlinear terms in Eq. (2.3). Step4. We substitute (2.4) into (2.3), we calculate all the necessary derivatives u0 , u00 , . . . and then we account the function 0 ðnÞ w(n). As a result of this substitution, we get a polynomial of wwðnÞ and its derivatives. In this polynomial, we equate with zero all the coefficients of it. This operation yields a system of equations which can be solved to find Ak and w(n). Consequently, we can get the exact solution of Eq. (2.1). 3. Application In this section, we use the direct application of the modified simple equation method, rather than a variant shown in [1], to find the exact solutions of the nonlinear Sharma–Tasso–Olver equation

3 ut þ aðu3 Þx þ aðu2 Þxx þ auxxx ¼ 0; 2

ð3:1Þ

where a is a positive constant. To this end, we use the traveling wave (2.2) to reduce Eq. (3.1) to the following ODE:

u0 þ 3au2 u0 þ 3auu00 þ 3aðu0 Þ2 þ au000 ¼ 0: 000

ð3:2Þ

2 0

Balancing u with u u yields N = 1. Consequently, we have

uðnÞ ¼ A0 þ A1



 w0 ðnÞ ; wðnÞ

ð3:3Þ

where A0 and A1 are constants to be determined such that A1 – 0, while w(n) is an unknown function to be determined. It is easy to see that

"  0 2 # w00 w ; u ¼ A1  w w "  0 3 # w000 3w0 w00 w u00 ¼ A1 þ 2 ;  w w w2 " 0000  00 2  0 4 # w 4w0 w000 w 12ðw0 Þ2 w00 w u000 ¼ A1  3 þ  6  : w w w w2 w3 0

ð3:4Þ ð3:5Þ ð3:6Þ

Substituting (3.3)–(3.6) into (3.2) and equating the coefficients of w1, w2, w3 and w4 to zero, we respectively obtain



0000



aw þ 3aA0 w000 þ 3aA20  1 w00 ¼ 0; 00 2

0

00

ð3:7Þ 0

000

A20 Þðw0 Þ2

3aðA1  1Þðw Þ þ 3aA0 ð2A1  3Þw w þ að3A1  4Þw w þ ð1  3a ðA1  1Þ½ðA1  4Þw00  2A0 w0  ¼ 0

¼ 0;

ð3:8Þ ð3:9Þ

and

ðA1  1ÞðA1  2Þðw0 Þ4 ¼ 0:

ð3:10Þ

Eq. (3.10) directly implies A1 = 1, A1 = 2. Case 1. If A1 = 1. In this case, we deduce from Eq. (3.8) that





aw000 ¼ 3aA0 w00 þ 1  3aA20 w0 :

ð3:11Þ

The general solution of Eq. (3.11) is

wðnÞ ¼ c0 þ c1 em1 n þ c2 em2 n ;

ð3:12Þ

while ci(i = 0, 1, 2) are arbitrary constants, while m1 and m2 are given by

3 1 m1;2 ¼  A0  2 2a

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4a  3a2 A20 :

ð3:13Þ

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E.M.E. Zayed / Applied Mathematics and Computation 218 (2011) 3962–3964

Consequently, the exact solution of Eq. (3.1) has the form

uðx; tÞ ¼ A0 þ

c1 m1 em1 ðxtÞ þ c2 m2 em2 ðxtÞ ; c0 þ c1 em1 ðxtÞ þ c2 em2 ðxtÞ

ð3:14Þ

where A0 is an arbitrary constant. Case 2. If A1 = 2. In this case, we deduce from (3.9) that

w00 þ A0 w0 ¼ 0:

ð3:15Þ

The general solution of Eq. (3.15) has the form

wðnÞ ¼ c1 eA0 n þ c2 ;

ð3:16Þ

where c1 and c2 are arbitrary constants. Substituting (3.16) into (3.8) yields

1 A0 ¼  pffiffiffi ;

a

a > 0:

ð3:17Þ

Consequently, we have n p ffi wðnÞ ¼ c1 e a þ c2 :

ð3:18Þ

Now, the exact solution of Eq. (3.1) has the form

( ) ðxtÞ pffi 1 2c1 e a : uðx; tÞ ¼ pffiffiffi 1  ðxtÞ pffi a c1 e a þ c2

ð3:19Þ

Acknowledgement The author thanks the referees for their suggestions and comments. Reference [1] A.J.M. Jawad, M.D. Petkovic, A. Biswas, Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput. 217 (2010) 869–877.