Improved (G /G)- expansion method for constructing exact traveling wave solutions for a nonlinear PDE of nanobiosciences

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1 Vol 8(5), pp , 5 ugust, 3 DOI 5897/SRE3555 ISSN cademic Journals Scientific Research and Essays Full Length Research Paper Improved (G /G)- expansion method for constructing exact traveling wave s for a nonlinear PDE of nanobiosciences Elsayed M E Zayed*, Yasser mer and Reham M Shohib Mathematics Department, Faculty of Sciences, Zagazig University, Zagazig, Egypt ccepted ugust, 3 In the present paper, the improved (G /G) - expansion method combined with suitable transformations is used to construct many exact s involving parameters of a nonlinear equation describing the nano-ionic currents along microtublues s a result, hyperbolic function s, trigonometric function s and rational function s with parameters are obtained When these parameters are taken as special values, some solitary wave s and the periodic wave are derived from the exact s This method can be employed for many other nonlinear evolution equations in mathematical physics and engineering Comparisons between our results and the wellknown results are given Key words: Improved (G /G) - expansion method, exact traveling wave s, solitary and periodic s, nano-ionic currents along microtublues, nanobiosciences INTRODUCTION Nonlinear phenomena play crucial roles in applied mathematics and physics Exact s for nonlinear partial differential equations (PDEs) play an important role in many phenomena in such as fluid mechanics, hydrodynamics, optics, plasma and physics and so on Many powerful methods have been presented, such as the inverse scattering transform method (blowitz and Clarkson, 99), the bilinear method (Hirota, 97; Ma, ), the Painleve expansion method (Weiss et al, 983; Kudryashov, 988, 99, 999), the Backlund truncated method (Miura, 978), the exp-function method (He and Wu, 6; Yusufoglu, 8; Bekir, 9, ; slan, a; Ma and Zhu, ), the tanh-function method (bdou, 7; Fan, ; Zhang and Xia, 8; Yusufoglu and Bekir, 8), the Jacobi elliptic function method (Chen and Wang, 5; Liu et al, ; Lu, 5), the (G /G)-expansion method (Wang et al, 8; Zhang, 8; Zhang et al, 8; Zayed 9, ; Bekir, 8; yhan and Bekir, ; slan,, b, a,b), the generalized Riccati equation mapping method (Zhu, 8; Zayed and rnous, 3; Ma and Fuchssteiner,996; Ma et al, 7; Ma and Lee, 9), local fractional variation iteration method (Yang and Baleanu, 3), local fractional series expansion method (Yang et al, 3) and so on The objective of this paper is to apply the improved (G /G) - expansion method combined with suitable transformations to construct many exact s involving parameters of the nonlinear PDE of special interest in nanobiosciences namely, the following transmission line model for nano-ionic currents along microtublues (Sekulic et al, ; Sataric et al, ): *Corresponding author emezayed@hotmailcom

2 Zayed et al 54 L Z 3 ZC 3 xxx L t x L t L u ( G C ) uu u u ( RZ G Z ) u () 9 R 34 the resistance of the elementary 9 5 rings (ER) L8 m, C 8 F is the total 3 maximal capacitance of the ER G si is the conductance of pertaining nano-pores (NPS) and Z 556 is the characteristic impedance of the system The parameters and describe the nonlinearity of ER capacitor and conductance of NPS in ER respectively The physical details of the derivation of Equation () can be elaborated in Sataric et al () Recently, Equation () has been discussed by using the modified extended tanh-function method (Sekulic et al, ) and by using the improved Riccati equation mapping method (Zayed et al, 3) its exact s have been found Comparison between our results and the well - known results obtained in Sekulic et al () and Zayed et al (3) will be investigated in the discussions and conclusions part of this work DESCRIPTION OF THE IMPROVED (G /G) - EXPNSION METHOD Suppose that we have the following nonlinear evolution equation: F( u, ut, ux, utt, uxx,) () F is a polynomial in u(x, t) and its partial derivatives, in which the highest order derivatives and the nonlinear terms are involved In the following, we give the main steps of this method (Liu et al, ; Lu, 5) as follows: Step We use the traveling wave transformation u( ), kx t (3) to reduce Equation () to the following ordinary differential equation (ODE): P( u, u, u,) (4) k, are constants Here, P is a polynomial in u( ) and its total derivatives, while the dashes denote the derivatives with respect to Step We assume that Equation 4 has the formal : m G( ) u( ) ai im G( ) i (5) ai ( i m,, m) later and G( ) satisfies the following linear ODE: are constants to be determined G ( ) G( ) G( ), (6) and are constants Step 3 The positive integer m in (5) can be determined by balancing the highest-order derivatives with the nonlinear terms appearing in Equation 4 Step 4 We substitute (5) along with Equation 6 into Equation 4 to G obtain polynomials in,( i,,,) G i Equating all the coefficients of these polynomials to zero, yields a set of algebraic equations which can be solved by using a k the Maple to find the unknowns,, i Step 5 Since the s of Equation 6 are well-known for us, then we have the following ratios: (i)if 4, we have c ( ) 4 ccosh 4 sinh 4 G (7) G( ) csinh 4 ccosh 4 (ii)if 4, we have c G c sin 4 cos 4 G( ) ccos 4 csin 4 (8) ( ) 4

3 54 Sci Res Essays (iii)if 4, we have G( ) c G( ) c c (9) c and c are arbitrary constants Step 6 We substitute the values of ai, k, as well as the ratios (7)-(9) into (5) to obtain the exact s of Equation () MNY FMILIES OF EXCT TRVELING WVE SOLUTIONS FOR EQUTION () Here, we apply the proposed method of description of the improved (G /G) - expansion method part of this work to find many families of exact traveling wave s of Equation () To this end, we use the wave transformation c u( ), x t () L RC and c is the dimensionless 6 6 s, velocity of the wave, to reduce Equation () into the following ODE: u cuu (6 Bc) ucu () G G 5 5 G G G G 4 4 : 4a a c, 3 : 4a a c, : 54a 6 a c(a 3 a a ), 3 G : 54a 6 a c(a 3 aa ), G 3 G : 4a 38a a c(a 3a a a aa ) a (6 Bc), G 3 G G G G G G G G G G : 4a 38a a c(a 3a a a a a ) a (6 Bc), : 5a 8a 7a 8 a c(3a a a a a a a a a ) 3 ( a a )(6 Bc) Ca, : 5a 8a 7a 8 a c(3a a a a a a a a a ) G G 3 ( a a )(6 Bc) Ca, : 6a 8a 4 a a c( a a a a a a a ) 3 ( a a )(6 Bc) Ca, : 6a 8a 4 a a c( a a a a a a a ) 3 ( a a )(6 Bc) Ca, : 6a a a a 6a a c( a a a a a a a a ) ( a a )(6 Bc) Ca By solving the above algebraic equations with the aid of Maple or Mathematica, we have the following cases: Case Where 3 3Z 3ZC ( C G ), B, C 3( RZ GZ ) ( B a ) ( 86),, c, a a, a, B a ( 8 6) ( B a ) a, a, a ( 86) (3) By balancing u with uu, we have m = Hence, the formal of Equation () takes the form G G G G u( ) a a a a a G G G G a, a, a, a, a determined later, such that () are parameters to be a or a Inserting () with the aid of Equation (6) into Equation (), we get the following system of algebraic equations: Case ( B a ) 8 6,, c, a a, a, a, a, a B a (8 6) (4) Case 3 3 B, (a a ( )), c, a a, a, 4a a 9 B a a a ( ), a, a 6a (5)

4 Zayed et al 543 Exact s of Equation () for case Substituting (3) into () and using (7) (9), we have the following exact s for the model (): 4 (Hyperbolic function s),we have (i)if the exact c 3( 4 ) ccosh 4 sinh 4 B a a 86 csinh 4 ccosh 4 3 B a 86 (6) If we set c = and c in (6), we have the solitary 3 B a a sech 4 4 tanh 4 86 (7) while if we set c = and c in (6), we have the solitary 3 B a a cosech 4 4 coth 4 86 (8) c and c c If, then we have the solitary 3(, ) a tanh 4 Where 3( 4 ) B a 3 B a (9) c tanh c then we have the solitary, while if c and c c, 4(, ) tanh 4 a 3( 4 ) B a 3 B a () c coth c 4 (Trigonometric function s), we (ii) If have the exact 3 B a 3(4 ) B a a () csin 4 ccos 4 ccos 4 csin 4 If we set c = and c in (), we have the periodic 3 B a a sec 4 4 tan 4 86 () while if we set c = and c in (), we have the periodic 3 B a a csc 4 4 cot 4 86 (3) c and c c If, then we have the periodic 3 B a 3(4 ) B a (4) 3(, ) a tan 4 while if c tan c c and c c,, then we have the periodic 3 B a 3(4 ) B a (5) 4(, ) cot 4 a (iii) If c cot c 4 (Rational function s), we have 3 (, ) B a a 4 c 8 6 cc 86 t x L B a constants (6) and c, c are arbitrary

5 544 Sci Res Essays Exact s of Equation () for case Substituting (4) into () and using (7) (9), we have the following exact s for the model (): 4 (Hyperbolic function s), we (i) If have the exact u( x, t ) c sinh B a ccosh a 8 6 csinh ccosh (7) If we set c = and c in (7), we have the solitary B a 8 6 (, ) a coth (8) while if we set c = and c in (7), we have the solitary a B a 8 6 (, ) tanh If c oand c c, then we have the solitary B a u3( x, t ) a tanh 8 6 while if c coth c c and c c (9) (3) then we have the solitary B a u 4( x, t ) a tanh 8 6 c tanh c (ii)if have the exact (3) 4 (Trigonometric function s), we c cos B a csin a (3) 8 6 ccos csin If we set c = and c in (3), we have the periodic B a 8 6 (, ) a cot (33) while if c = and c in (3), we have the periodic B a a 8 6 (, ) tan If we set : c and c c (34) then we have the periodic B a u3( x, t ) a tan 8 6 while if c cot c c and c c, B a u4( x, t) a tan 8 6 (iii)if c tan c (35) then we have the periodic (36) 4 (Rational function s), we have B a c a 86 cc 8 6 t x L B a Exact s of Equation () for case 3 (37) Substituting (5) into () and using (7) (9), we have the following exact s for the model (): 4 (Hyperbolic function s), we (i)if have the exact

6 Zayed et al 545 c cosh u( x, t ) a a c sinh c a c c sinh c cosh c sinh c cosh cosh sinh (38) If we set c = and c in (38) we have the solitary u( x, t ) a a tanh coth (39) while if we set c = and c in (38) we have the same solitary (39) 4 (Trigonometric function s), we (ii)if have the exact c sin a a c cos c sin a c cos c cos c sin c cos c sin (4) If we set c = and c in (4) we have the periodic a a tan cot (4) while if we set c = and c in (4) we have the same periodic (4) (iii)if 4 (Rational function s), we have c c u( x, t ) a a a c c c c t x L a DISCUSSION (4) In this article, we have applied the improved (G /G) - expansion method to find many exact s as well as many solitary wave s and periodic s (6) - (4) of the nonlinear model () which are of special interests in nanobiosciences, namely the transmission line models for nano-ionic currents along microtublues On comparing our results obtained in this article with the well-known results obtained in Sekulic et al () and Zayed et al (3) we deduce that some of these results are in agreement together, while the others are new which are not discussed else CKNOWLEDGEMENT The authors wish to thank the referees for their comments on this paper REFERENCES bdou M (7) The extended tanh- method and its applications for solving nonlinear physical models ppl Math Comput 9(): blowitz MJ, Clarkson P (99) Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform Cambridge University Press, New York slan I () note on the (G'/G) -expansion method again ppl Math Comput 7(): slan I (a) Exact and explicit s to the discrete nonlinear Schrödinger equation with a saturable nonlinearity Phys Lett 375(47):44-47 slan I (b) Comment on application of exp-function method (3+) dimensional nonlinear evolution equation Comput Math ppl 56: Comput Math ppl 6(6):7-73 slan I (a) Some exact s for Toda type lattice differential equations using the improved (G'/G) expansion method Math Meth ppl Sci 35(4): slan I (b) The discrete (G'/G) -expansion method applied to the differential-difference Burgers equation and the relativistic Toda lattice system Numer Meth Par Diff Eqs 8():7-37 yhan B, Bekir () The (G'/G) -expansion method for the nonlinear lattice equations Comm Nonlin Sci Numer Simul 7(9): Bekir (8) pplication of the (G'/G) - expansion method for nonlinear evolution equations Phys Lett 37(9): Bekir (9) The exp-function for Ostrovsky equation Int J Nonlinear Sci Numer Simul : Bekir () pplication of exp-function method for nonlinear differential-difference equations ppl Math Comput 5(): Chen Y, Wang Q (5) Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function s to (+)-dimensional dispersive long wave equation Chaos Solit Fract 4(3): Fan E () Extended tanh-function method and its applications to nonlinear equations Phys Lett 77:-8 He JH, Wu XH (6) Exp-function method for nonlinear wave equations Chaos Solit Fract 3(3):7-78 Hirota R (97) Exact s of the KdV equation for multiple collisions of s Phys Rev Lett 7:9-94 Kudryashov N (988) Exact s of a generalized evolution of wave dynamics J ppl Math Mech 5: Kudryashov N (99) Exact s of a generalized Kuramoto- Sivashinsky equation Phys Lett 47(5-6):87-9 Kudryashov N (99) On types of nonlinear nonintegrable equations with exact s Phys Lett 55(4-5):69-75 Liu S, Z Fu Z, Zhao Q () Jacobi elliptic function expansion method and periodic wave s of nonlinear wave equations Phys Lett 89(-):69-74

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