Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics Using the Modified Simple Equation Method
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Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics Using the Modified Simple Equation Method The modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)dimensional modified KdV equation, and the (1+1)dimensional reactiondiffusion equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. ^{Nonlinear phenomena play crucial roles in applied mathematics and physics.} Calculating exact and numerical solutions, in particular the traveling wave solutions of nonlinear equations in mathematical physics, plays an important role in soliton theory.^{[1] }Recently, it has become more interesting to obtain exact solutions of nonlinear evolution equations by using symbolical computer programs such as Matlab, Maple, and Mathematica that facilitate complicated and tedious algebrical computations. These equations are mathematical models of complex physical phenomena that arise in engineering, chemistry, biology, mechanics, and applied physics. Several powerful methods for obtaining explicit traveling wave solitary solutions of nonlinear equations have been proposed, such as Hirotaβs bilinear method,^{[2] }Painleve expansions,^{[3] }the inverse scattering transform,^{[4] }the homogeneous balance method,^{[5] }the Fexpansion method,^{[6] }the Jacobielliptic function method,^{[7,8] }the tanhfunction method,^{[9,10] }the Backlund transform method,^{[11] }the (πΊ^{β²}/πΊ)expansion method,^{[12,13] }the expfunction method,^{[14,15] }and the modified simple equation method.^{[16,17] }In this Letter, we apply the modified simple equation method to seek the exact solutions of some nonlinear evolution equations via the (1+1)dimensional modified KdV equation and the (1+1)dimensional reactiondiffusion equation. The proposed method can be described as follows: suppose that we have a nonlinear evolution equation in the form^{[16,17]} πΉ(π’,π’_{π‘},π’_{π₯},π’_{π₯π₯},...) = 0, (1) where πΉ is a polynomial in π’ and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following, we give the main steps of this method.
π’(π₯,π‘) = π’(π), π = π₯ + π‘, (2) to reduce Eq.(1) to the following ODE: π(π’,π’^{β²},π’^{β²β²},...) = 0, (3) where π is a polynomial in π’ and its total derivatives with respect to π.
where π,π,π,π are positive integers which could be freely chosen, π_{π}_{ }and π_{π}_{ }are unknown constants to be determined. To determine the values of π and π, we balance the linear term of the highest order with ^{*}Corresponding author. Email: e.m.e.zayed@hotmail.com; dhoda_isa@yahoo.com Β©2012 Chinese Physical Society and IOP Publishing Ltd 0602011
CHIN.PHYS.LETT. Vol.29, No.6(2012)060201 the highestorder nonlinear term in Eq.(3). Similarly, to determine the values of π and π, we balance the linear term of the lowest order in Eq.(3) with the lowest order nonlinear term. To determine π_{π}_{ }and π_{π}, we substitute Eq.(5) into Eq.(3) and collect all the terms with the same order of exp(ππ)(π = 0,1,2,...) and set them to zero, yielding a set of algebraic equations which can be solved using computer programs such as Maple or Mathematica to get π_{π}_{ }and π_{π}. Consequently, we can obtain the exact solutions to Eq.(1). Further, the wellknown (πΊ^{β²}/πΊ)expansion method^{[12] }can be described as follows: we assume that the solution to Eq.(3) can be expressed in the form π π’(π) = ^{β}^{οΈ}πΌ_{π}_{ }(οΈ^{πΊ}β²^{(}^{π}^{)})οΈπ, πΌ_{π}_{ }= 0, (6) π=0 where πΊ = πΊ(π) satisfies the secondorder linear ODE πΊβ²β² (π) + ππΊ^{β²}(π) + ππΊ(π) = 0, (7) where πΌ_{π}_{ }(π = 0,...,π), π and π are constants to be determined, the primes represent the derivatives with respect to π.
π^{β²}(π) = ππ(π)π(π), π^{β²}(π) = ππ^{2}(π) β ππ(π) + π, (9) where π, π and π are constants, such that π = Β±1, π = 0, and the prime represents the derivative with respect to π.
where π΄_{0 }and π΄_{1 }are constants to be determined such that π΄_{1 }= 0. It is easy to see β² β²β² β²^{2} π’ = π΄_{1}( _{π}_{ }β _{π}_{2 }), (13) β²β² β²β²β² β² β²β² β²^{3} π’ = π΄_{1}( _{π}_{ }β 3 _{π}_{2 }^{ }+ 2 _{π}_{3 }), (14) and so on. Substituting Eqs.(12) and (14) into Eq.(11) and equating all the coeficients of π^{0}, π^{β1}, π^{β2}, π^{β3 }to zero, we obtain π΄_{0 }^{(}^{οΈ}3 β π½π΄_{0}^{)}^{οΈ}^{ }= 0, (15) π΄_{1}π^{β² }1 β π½π΄^{2 }+ π΄_{1}π^{β²β²β² }= 0, (16) π΄_{1}π^{β² }(π½π΄_{0}π΄_{1}π^{β² }+ 3π^{β²β²}) = 0, (17) π΄_{1}π^{β²3 }6 β π½π΄^{2 }= 0. (18) Equations (15) and (18) yield π΄_{0 }= 0, π΄_{0 }= Β±^{β}^{οΈ}3/π½, π΄_{1 }= Β± 6/π½. Let us now discuss the following cases: Case 1. If π΄_{0 }= 0, then we can obtain the trivial solution, which is rejected. Case 2. If π΄_{0 }= Β± 3/π½, then we can deduce that π^{β²β²β²}/π^{β²β² }= β^{β}2. (19) Integrating Eq.(19) yields π^{β²β² }= π_{1 }exp(β^{β}2π), (20) 0602012
CHIN.PHYS.LETT. Vol.29, No.6(2012)060201 and consequently, we obtain β ^{β}^{ }^{ }(οΈ )οΈ β π^{β² }= βπ_{1 }_{2 }exp β 2π , (21) (οΈ )οΈ π = π_{2 }+ _{2}π_{1 }exp β 2π , (22) where π_{1 }and π_{2 }are constants of integration. Now, the exact solution to Eq.(10) has the form βοΈ βοΈ [οΈ ]οΈ π’(π₯,π‘) = Β± _{π½}_{ }β 2π_{1 }_{π½} ^{{}^{οΈ}^{ }exp β 2(π₯ + π‘) ^{}}^{οΈ} _{2}π_{2 }+ π_{1 }exp β 2(π₯ + π‘) _{(23)} Integrating Eq.(32) yields βοΈ π^{β²β² }= π_{1 }exp[β _{πΌ}_{ + 1 }π]. (33) Consequently, we have [οΈ ]οΈ π^{β² }= βπ_{1 }^{πΌ}_{2}_{π½}^{1 }exp β _{πΌ}_{ + 1 }π , (34) π = π_{2 }+ π_{1 }(οΈ^{πΌ}^{ + 1})οΈexp[οΈββοΈ^{πΌ}^{2}_{π½} _{1 }π]οΈ, (35) If we set π_{1 }= Β±1, π_{2 }= 1/2 in Eq.(23) we will have the following solitary wave solutions: β [οΈ ]οΈ π’_{1}(π₯,π‘) = Β± _{π½}_{ }tanh _{2}^{2}(π₯ + π‘) , (24) β βοΈ [οΈ ]οΈ π’_{2}(π₯,π‘) = Β± ^{3 }coth _{2}^{2}(π₯ + π‘) . (25) The second equation is the nonlinear reactiondiffusion equation π’_{π‘π‘}_{ }+ πΌπ’_{π₯π₯}_{ }+ π½π’ β πΎπ’^{3 }= 0, (26)
where π_{1 }and π_{2 }are constants of integration. Now, the exact solution of Eq.(26) has the form βοΈ βοΈ π’(π₯,π‘) = Β± ^{π½}^{ }β 2π_{1 }^{π½} [οΈ βοΈ ]οΈ exp β (π₯ + π‘) Β· 2π½ If we set π_{1 }= Β±1 and π_{2 }= ^{πΌ}^{+1}^{ }in Eq.(36) we will have the following solitary wave solutions: where πΌ, π½ and πΎ are positive constants. The solution to Eq.(26) has been investigated by using different methods namely, the projective Riccati equation method,^{[18] }and the (πΊ^{β²}/πΊ)expansion method.^{[13] }Let us now solve Eq.(26) using the modified simple equation method. To this end, we use the wave transformation (2) to reduce Eq.(26) to the following ODE: βοΈ _{[}_{οΈ}_{ }_{β}_{οΈ}_{ }_{ }_{]}_{οΈ} π’_{1}(π₯,π‘) = Β± _{πΎ}_{ }tanh _{2 }_{πΌ}_{ + 1 }(π₯ + π‘) , (37) βοΈ _{[}_{οΈ}_{ }_{β}_{οΈ}_{ }_{ }_{]}_{οΈ} π’_{2}(π₯,π‘) = Β± _{πΎ}_{ }coth _{2 }_{πΌ}_{ + 1 }(π₯ + π‘) . (38) (πΌ + 1)π’^{β²β² }+ π½π’ β πΎπ’^{3 }= 0. (27) ]οΈ Balancing π’^{β²β² }with π’^{3 }yields π = 1. Consequently, Eq.(27) has the same formal solution (12). Substituting Eqs.(12) and (14) into Eq.(27) and equating all the coeficients of π^{0},π^{β1},π^{β2},π^{β3 }to zero, we obtain π΄_{0 }^{(}^{οΈ}π½ β πΎπ΄_{0}^{)}^{οΈ}^{ }= 0, (28) π΄_{1 }(πΌ + 1)π^{β²β²β² }+ π½ β 3πΎπ΄^{2 }π^{β² }= 0, (29) 3π΄_{1}π^{β² }[(πΌ + 1)π^{β²β² }+ πΎπ΄_{0}π΄_{1}π^{β²}] = 0, (30) π^{β²3 }2(πΌ + 1) β πΎπ΄_{1 }= 0. _{ }(31) Equations (28) and (31) yield π΄_{0 }= 0, π΄_{0 }= Β±^{β}^{οΈ}π½/πΎ, π΄_{1 }= Β± 2(πΌ + 1)/πΎ. Now, let us now discuss the following cases: Case 1. If π΄_{0 }= 0, then we will obtain the trivial solution which is rejected. Case 2. If π΄_{0 }= Β± π½/πΎ, then we can deduce that βοΈ π^{β²β²β²}/π^{β²β² }= β _{πΌ}_{ + 1}. (32) In summary, the modified simple equation method has been proposed to find exact solutions and solitary wave solutions of the (1+1)dimensional nonlinear modified KdV equation and the (1+1)dimensional nonlinear reactiondiffusion equation. Comparing the presently proposed method with other methods, namely, the expfunction method, the (πΊ^{β²}/πΊ)expansion method and the projective Riccati equation method, we can conclude: (i) the exact solutions to Eqs.(10) and (26) can be investigated using these methods with the aid of the computer programs, such as Matlab, Maple, and Mathematica to facilitate the complicated algebraic computations. (ii) The exact solutions to these equations have been obtained here without using the computer programs since the computations are simple. From (i) and (ii) we can conclude that the modified simple equation method is much more simple than the other methods. Also, we can see that the proposed method is direct, effective, and can be applied to many other nonlinear evolution equations. 0602013 CHIN.PHYS.LETT. Vol.29, No.6(2012)060201 References [1] Jawad A J M Petkovic M D and Biswas A 2009 Appl. Math. Comput. 216 2649 [2] Hirota R1971 Phys. Rev. Lett. 27 1192 [3] Weiss J, Tabor M and Carnevalle G 1983 J. Math. Phys. 24 1405 [4] Ablowitz M J and Clarkson P A 1991 Solitons, Nonlinear Evolution Equation and Inverse Scattering (New York: Cambridge University) [5] Fan E 2000 Phys. Lett. A 265 353 [6] Zhou Y, Wang M and Wang Y 2003 Phys. Lett. A 308 31 [7] Inc M and Ergut M 2005 Appl. Math. E 5 89 [8] Lu D and Shi Q 2010 Int. J. Nonlinear Sci. 10 320 [9] Jawad A J M Petkovic M D and Biswas A 2009 Appl. Math. Comput. 216 3370 [10] Parkes E J and Duffy B R 1996 Comput. Phys. Commu. 98 288 [11] Miura M R 1978 Backlund Transformation (Berlin: Springer) [12] Wang M, Li X and Zhang J 2008 Phys. Lett. A 372 417 [13] Zayed E M E and Gepreel K A 2009 J. Math. Phys. 50 013502 [14] ELWakil S A, Madkour M A and Abdou M A 2007 Phys. Lett. A 369 62 [15] He J H and Wu X H 2006 Chaos Solitons Fractals 30 700 [16] Jawad A J M Petkovic M D and Biswas A 2010 Appl. Math. Comput. 217 869 [17] Zayed E M E 2011 Appl. Math. Comput. 218 3962 [18] Mei J, Zhang H and Jiang D 2004 Appl. Math. E 4 85 0602014 Download 134.48 Kb. Do'stlaringiz bilan baham: 
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