EXACT TRAVELING WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING THE IMPROVED (G /G) EXPANSION METHOD

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1 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 EXACT TRAVELIN WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USIN THE IMPROVED ( /) EXPANSION METHOD Elsayed M. E. Zayed Yasser. A. Amer and Reham M. A. Shohib Mathematics Department Faculty of Sciences Zagazig University Zagazig Egypt of corresponding author e.m.e.zayed@hotmail.com ABSTRACT In this article we apply the improved ( /)-expansion method to construct the exact solutions of the nonlinear fractional partial differential equations (PDEs) in the sense of the modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation certain fractional partial differential equations can be turned into other nonlinear ordinary differential equations (ODEs) of integer orders. For illustrating the validity of this method we apply it to four nonlinear fractional PDEs equations namely the space-time fractional Potential Kadomtsev- Petviashvili (PKP) equation the space-time fractional Symmetric Regularized Long wave (SRLW) equation the space-time fractional Sharma- Tasso-Olver (STO) equation and the space-time fractional Kolmogorov-Petrovskii- Piskunov (KPP) equation. This method can be applied to many other nonlinear fractional PDEs in mathematical physics. Keywords Nonlinear space-time fractional PDES Improved ( / ) - expansion method Nonlinear fractional complex transformation Exact solutions; Modified Riemann- Liouville derivative. INTRODCUTION In recent years nonlinear fractional differential equations have been attracted great interest. It is caused by both the development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics engineering and biology [5 578]. For better understanding the mechanisms of the complicated nonlinear physical phenomena as well as further applying them in practical life the solutions of these equation [696] are much involved. In the past many analytical and numerical method have been proposed to obtain the solutions of nonlinear fractional differential equations such as the finite difference method [7] the finite element method [7] the differential transform method [] the Adomian decomposition method [45] the variational iteration method [49] the homotopy - perturbation method [ 8] the improved ( / ) expansion method [69] the fractional subequation method [4] and so on. - In this article we will apply the improved ( / ) expansion method [6-8] for solving the nonlinear fractional PDEs in the sense of the modified Riemann-Liouville derivative obtained in [48]. The modified Riemann-Liouville derivative of order is defined by the following expression D f () t t d ( ) dt t ( t ) [ f ( ) f ()] d () ( n ) ( n ) f t n n n ( ) We list some important properties for the modified Riemann-Liouville derivative as follows 8

2 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 r ( r) r Dt t t r ( r ) () D f ( t ) g ( t ) f ( t ) D g ( t ) g ( t ) D f ( t ) () t t t D f ( g ( t )) f ( g ( t )) D g ( t ) t g t D f ( g ( t ))[ g ( t )] g (4) The rest of this article is organized as follows In Sec. the improved ( / ) -expansion method for solving nonlinear fractional PDEs is given. In Sec. we apply this method to establish the exact solutions for the space-time nonlinear fractional PKP equation the space-time nonlinear fractional SRLW equation the space-time nonlinear fractional STO equation and the space-time nonlinear fractional KPP equation. In Sec. 4 some conclusions and discussions are obtained. DESCRIPTION PF THE IMPROVED ( /)-EXPANSION METHOD FOR SOLVIN PDES Suppose that we have the following nonlinear fractional partial differential equation F ( u D u D u...) (5) D u D u t t x x are the modified Riemann- Liouville derivatives and F is a polynomial in u(x t) and its partial fractional derivatives in which the highest order derivatives and the nonlinear terms are involved. In the following we give the main steps of this method Step. Using the nonlinear fractional complex transformation [4] are constants with k c to reduce Eq. (5) to the following ordinary differential equation (ODE) with integer order P( U U ' U ''...) (7) P is a polynomial in U ( ) and its total derivatives while the dashes denote the derivatives with respect to. Step. We assume that Eq. (7) has the formal solution m ( ) U( ) ai im ( ) (8) ai ( i m... m) are constants to be determined later and ( ) satisfies the following linear ODE ( ) ( ) ( ) (9) and are constants. Step. The positive integer m in Eq. (8) can be determined by balancing the highest-order derivatives with the nonlinear terms appearing in Eq. (7). Step4. We substitute (8) along with Eq. (9) into Eq. (7) to obtain polynomials in i i ( i...). Equating all the coefficients of these polynomials to zero yields a set of algebraic equations. Step5. We solve the algebraic equations of step 4 using the Maple or Mathematica to find the values of ai k c. Substituting these values into (8) and using the ratios u ( x t ) U ( ) kx ct ( ) ( ) (6) 9

3 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN c cosh c sinh c sinh c cosh c sin c cos ( ) 4 ( ) c cos c sin if c c c () and We obtain the exact 4 4. solutions of Eq. (5) c and c are arbitrary constants. APPLICATIONS In this section we construct the exact solutions of the following four nonlinear fractional PDEs using the proposed method of Sec. as follows Example. The space-time nonlinear fractional PKP equation This equation is well-known [] and has the form 4 D u D ud u D u D ( D u ) () x x x y t x 4 4. Eq. () has been investigated in [] using the fractional sub-equation method. Let us now solve Eq. () using the proposed method of Sec.. To this end we use the nonlinear fractional complex transformation u ( x y t ) U ( ) k x k y ct ( ) ( ) ( ) () U ( ) a a a (4) a a a are constants to be determined later such that a or a. Substituting (4) along with Eq. (9) into Eq. () collecting all the terms of the same orders i i... and setting each coefficient to zero we have the following set of algebraic equations 4 4 a k a k 4 a k a k 4 k (8a 7 a ) k ( a a a a ) a k k c ( 4 ) 4 k (8 a a ) k (a 4 a a ) a k k c ( 4 ) 4 k (a a a a ) k ( a 4a a a a a ) ( a a )(k 4 k c ) 4 k (8 a a ) k (a 4 a a ) a k k c ( 4 ) k k c are constants to reduce Eq. () to the following ODE with integer order k U k U (k 4 ck ) U () 4 By balancing U with U we have m=. Consequently Eq. () has the formal solutions

4 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN k (7a 8 a ) k ( a a a a ) a k k c ( 4 ) 4 a 4 k a k a 4 k a k. On solving these algebraic equations with the aid of Maple or Mathematica we have the following cases Case. k k k k. 4 c (6 k k ) a k a k 4k Case. k k k k 4 c (4 k k ) a k a. 4k Case. k k k k c k k a k a 4k 4 ( ( 4 ) ). Let us now write down the following exact solutions of the space-time fractional PKP equation () for case (Similarly for cases which are omitted here for simplicity) (i) If (Hyperbolic function solutions) In this case we have the exact solution u ( x y t ) c sinh( ) c cosh( ) k c cosh( ) c sinh( ) c cosh c sinh c sinh a k c cosh If we set c = and c in (5) we have the solitary solution u ( x y t ) k coth( ) a k tanh( ) (6) while if we set c andc in (5) we have the solitary wave solution u ( x y t ) u ( x y t ) (7) (ii)if (Trigonometric function solutions) In this case we have the exact solution u ( x y t ) k a c sin( ) c cos( ) c cos( ) c sin( ) c sin( ) c cos( ) k c cos( ) c sin( ) (5) (8) If we set c = and c in (8) we have the periodic solution u ( x y t ) k tan( ) a k cot( ) (9) while if we set c = and c in (8) we have the periodic solution u ( x y t ) u ( x y t ) 4 () k x k y t 4 (6 k k ) ( ) ( ) 4 k ( ) (iii) If (Rational function solutions) c c u ( x y t ) k a k c c c c ()

5 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 k x k y k t ( ) ( ) k ( ) Example. The space-time nonlinear fractional SRLW equation This equation is well-known [] and has the form D u D u ud D u D u D u D D u t x t x t x t x ( ) ( ) (). Eq. () has been investigated in [] using the fractional sub-equation method. Let us solve Eq. () using the proposed method of Sec.. To this end we use the nonlinear fractional complex transformation kx ct u ( x t ) U ( ) () ( ) ( ) k c are constants to reduce Eq. () to the following ODE with integer order k c U ( k c ) U U (4) By balancing U with U we have m=. Consequently Eq. (4) has the formal solutions U ( ) a a a a a (5) a a a a a are constants to be determined later such that a or a. Substituting Eq.(5) along with Eq. (9) into Eq. (4) collecting all the terms of the same orders i i... and setting each coefficient to zero we have the following set of algebraic equations 4 a a k c 6 a a k c (a a ) a ( k c ) ( a a a ) k c (8a a 4 a ) a ( k c ) ( a a a a ) k c ( a 6 a a ) a ( k c ) ( a a a a a ) k c a a a a ( ) a ( k c ) ( a a a a ) k c ( a 6 a a ) a ( k c ) ( a a a ) a a k c (a a ) 4 a a k c k c (8a a 4 a ) 6 On solving the above set of algebraic equations with the aid of Maple or Mathematica we have the following cases Case.

6 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN k c c c k k [ k c ( c k )] a k c c k a a [ ( )] a k c c k a k c c k 4k c [ ( )] [ ( )] Case. c c k k c k a a 576k c [44( ) ] a [48( c k ) a ] a a a a a 48 Let us now write down the following exact solutions of the space-time fractional PKP equation () for case (Similarly for case which is omitted here for simplicity) (i) If 4 (Hyperbolic function solutions) In this case we have the exact solution ( c k )] k c ( k c ) k c u ( x t ) [ k c k c c cosh c sinh k c k c k c c sinh c cosh k c k c c cosh c sinh [ k c ( c k )] k c 4k c k c k c c sinh c cosh (6) If we set c = and c in (6) we have the solitary solution u ( x t ) k c ( k c ) k c k c [ k c ( c k )] tanh [ k c ( c k )] k c k c tanh 4k c (7) while if we set c = and c in (6) we have the solitary wave solution u ( x t ) k c ( k c ) k c k c [ k c ( c k )] coth [ k c ( c k )] k c k c coth 4k c. (8) If c and c c then we have the solitary wave solution u ( x t ) k c ( k c ) k c k c [ k c ( c k )] coth [ k c ( c k )] k c k c coth 4k c (9) c tanh c while if c and c then we have the solitary wave solution 4 u ( x t ) k c ( k c ) k c k c [ k c ( c k )] tanh [ k c ( c k )] k c k c tanh 4k c ()

7 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 c coth and c kx ct ( ) ( ) Exampe. The space-time nonlinear fractional STO equation This equation is well-known [4] and has the form D u D u u D u ud u D u () ( ) t x x x x. Eq. () has been investigated in [4] using the fractional sub-equation method. Let us now solve Eq. () using the proposed method of Sec.. To this end we use the transformation (6) to reduce Eq. () to the following ODE with integer order cu k UU ku k U () By balancingu with U we have m=. Consequently Eq. () has the formal solutions. U ( ) a a a a a () a are constants to be determined later such that a or a. Substituting () along with Eq. (9) into () collecting all the terms of the same order i and setting each coefficient to zero we have the following set algebraic equations i a k a k a k k ( a a a ) a a k a k ca k ( a a a ) k ( a a a a ) k a a ( ) ca k a ( a a ) k ( a 6 a a a ) k ( a a ) ca k ( a a a ) k ( a a a a ) k a a ( ) k ( a a a ) a a k a k a k a k a k By solving these algebraic equations with the aid of Maple or Mathematica we have the following cases Case. k k c k ( 4 ) a k a k a Case. k k c k ( 4 ) a a k a k Case. k k c k ( 4 ) a a k a k Let us now write down the following exact solutions of the space-time fractional STO equation () for 4

8 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 case (Similarly for cases which are omitted here for simplicity) (i) If 4 (Hyperbolic function solutions) In this case we have the exact solution u x t ( ) k 4 c c c c cosh 4 sinh 4 sinh 4 cosh 4 If we set c = and c in (4) we have the solitary solution 4 u ( x t ) k 4 tanh c (5) while if we set c = and in (4) we have the solitary wave solution 4 ( ) 4 coth. u x t k c and c c If wave solution (6) then we have the solitary (4) u ( x t ) k 4 coth 4 (7) c tanh c c while if c c then we have the solitary wave solution ( ) 4 tanh 4 4 and u x t k (8) (ii)if c coth c. 4 (Trigonometric function solutions) In this case we have the exact solution c c c sin 4 cos 4 ( ) k 4 u x t c cos 4 sin 4 If we set c = and c in (9) we have the periodic solution 4 u ( x t ) k 4 cot (4) while if we set c = and c in (9) we have the periodic solution 4 ( ) 4 tan u x t k c and c c If solution (9) (4) then we have the periodic u ( x t ) k 4 cot 4 (4) c tan c c c then we have the periodic solution while if c and u ( x t ) k 4 tan 4 (4) 4 c cot c kx t k ( 4 ). ( ) ( ) and Example4. The space-time nonlinear fractional KPP equation This equation is well-known [6] and has the form Dt u Dx u u u u (44) and are non zero constants. This equation is important in the physical 5

9 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 fields and it includes the Fisher equation. Huxlay equation Burgers equation Chaffee- Infanfe equation and Fitzhugh-Nagumo equation. When Eq. (44) has been discussed in [6] using the ( / ) expansion method. Let us solve Eq. (44) using the proposed method of Sec.. To this end we use the nonlinear fractional complex transformation (6) to reduce Eq. (44) to the following ODE with integer order cu k U U U U (45) By balancing U with U we have m=. Consequently Eq. (45) has the same formal solution (44). Substituting (44) along with Eq. (9) into Eq. (4) collecting all the terms of the same orders i i and setting each coefficient to zero we have the following set algebraic equations a k a a c a k a a a a c k ( a a ) a a a ( a a a a ) c ( a a ) k ( a a ) a ( a a a ) ( a 6 a a a ) a c k ( a a ) a a a ( a a a a ) a c a k a a a a k a By solving the above set of algebraic equations with the aid of Maple or Mathematica we have the following cases Case. k k k ( 4 ) a ( 4 ) a k c k a. k Case. 4 k k ( k ( 4 ) ) a a ( k ) a k c k Let us now write down the following exact solutions of the space-time fractional KPP equation (44) for case (Similarly for case which is omitted here for simplicity) (i) If (Hyperbolic function solutions) In this case we have the exact solution u ( x t ) k ( 4 ) c cosh c sinh c cosh c sinh c sinh c cosh 6k c sinh c cosh (46) If we set c = and c in (46) we have the solitary solution u ( x t ) k tanh ( 4 ) coth 6k (47) while if we set c = and c in (46) we have the solitary wave solution 6

10 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 u ( x t ) k coth ( 4 ) tanh 6k (48) u ( x t ) k tan ( 4 ) cot 6k (5) c and c If c wave solution ( 4 ) then we have the solitary u ( x t ) k tanh 4 (49) coth 6k c coth. c (Trigonometric function solutions) (ii) If In this case we have the exact solution c u ( x t ) k c sin c cos cos c sin c cos ( 4 ) c sin 6k c cos c sin (5) If we set c = and c in (5) we have the periodic solution u ( x t ) k cot ( 4 ) tan 6k (5) while if we set c = and c in (5) we have the periodic solution Figures of some exact solutions c and c c If solution ( 4 ) tan then we have the periodic u ( x t ) k cot (5) 6k c tan c. If then we have the periodic solution ( 4 ) cot c and c c u ( x t ) k tan 4 (54) 6k c cot and c kx k t ( ) ( ). (Rational function solution) (iii) If k c c u ( x t ) c c k c c kx t k ( ) ( ) (55) In this section we give some figures to illustrate the solutions of the equations () (4) () and (46) as follows 7

11 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 Fig. Numerical simulation solution of the spacetime fractional PKP equation. Fig. Numerical simulation solution of the spacetime fractional SRLW equation. 8

12 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 Fig. Numerical simulation solution of the spacetime fractional STO equation. 9

13 Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 differential equations the homogeneous balancing principle is satisfied. References. J. F. Alzaidy The fractional sub-equation method and exact analytical solutions for some fractional PDEs Amer. J. Math.Anal. () 4-9. D. Baleanu K. Diethelm E. Scalas and J. J. Trujillo Fractional Calculus Models and Numerical Methods Vol. of Series on Complexity Nonlinearity and Chaos World Scientific Publishing Boston Mass USA. W. Deng Finite element method for the space and time fractional Fokker-Planck equation SIAM J. Numer. Anal. 47 (8) A. M. A. El-Sayed and M. aber The Adomian decomposition method for solving partial differential equations of fractal order in finite domains Phys. Lett.. A 59(6) 75 8 Fig. 4 Numerical simulation solution of the spacetime fractional KPP equation. Some conclusions and discussions In this article we have extended successfully the improved ( / ) - expansion method to solve four nonlinear fractional partial differential equations. As applications abundant new exact solutions for the space-time nonlinear fractional PKP equation the space-time nonlinear fractional SRLW equation the space-time nonlinear fractional STO equation and the space-time nonlinear fractional KPP equation have been successfully found. As one can see the nonlinear fractional complex transformation for is very important which ensures that certain nonlinear fractional PDEs can be turned into other nonlinear ODEs of integer orders whose solutions can be expressed in the form (8) ( ) satisfies the linear ODE (9). Some numerical examples with diagrams have been given for fractional and non fractional orders to illustrate our results. Besides as this method is based on the homogeneous balancing principle so it can also be applied to other nonlinear fractional partial 5. A. M. A. El-Sayed S. H. Behiry and W. E. Raslan Adomian s decomposition method for solving an intermediate fractional advection-dispersion equation Computers & Math. Appl. 59() J. Feng W. Li Q. Wan Using ( / ) expansion method to seek traveling wave solution of Kadomtsev- Petviashvili- Piskkunov equation Appl. Math. Comput 7 () H. ao Z. Z. Sun and Y. N. Zhang A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions J. Comput. Phys. () K. A. epreel The homotopy perturbation method applied to nonlinear fractional Kadomtsev-Petviashvili- Piskkunov equations Appl. Math. Lett. 4 () K. A. epreel and S. Omran Exact solutions for nonlinear partial fractional differential equations Chin. Phys. B () 4-7. S. uo L. Mei Y. Li and Y. Sun The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics Phys. Lett. A 76() R. Hilfer Applications of Fractional Calculus in Physics World Scientific Publishing River Edge NJ USA. Y. Hu Y. Luo and Z. Lu Analytical solution of the linear fractional differential equation by Adomian

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