Article Explicit and Exact Traveling Wave Solutions of the Cahn Allen Equation Using the MSE Method
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1 Preprints (wwwpreprintsorg) NOT PEER-REVIEWED Posted: January 07 doi:009/preprints070008v Article Explicit and Exact Traveling Wave Solutions of the Cahn Allen Equation Using the MSE Method Harun-Or-Roshid * M Zulfikar Ali and Md Rafiqul Islam * Department of Mathematics Pabna University of Science and Technology Pabna-6600 Bangladesh Department of Mathematics Rajshahi University Rajshahi-06 Bangladesh; zulfi06@gmailcom * Correspondence: harunorroshidmd@gmailcom (H-O-R);harun_math@pustacbd (MRI); Tel: Abstract: By using the modified simple equation method we study the Cahn Allen equation which arises in many scientific applications such as mathematical biology quantum mechanics and plasma physics As a result the existence of solitary wave solutions of the Cahn Allen equation is obtained Exact explicit solutions interms of hyperbolic solutions of the associated Cahn Allen equation are characterized with some free parameters Finally the variety of structures and graphical representations make the dynamics of the equations visible and provide the mathematical foundation in mathematical biology quantum mechanics and plasma physics Keywords: the modified simple equation method; Cahn Allen equation; soliton solution; kink type solutions Introduction The mathematical modeling of events in nature can be explained by differential equationsit is well-known that various types of the physical phenomena in the fieldsof fluid mechanics quantummechanics electricity plasma physics chemical kinematicspropagation of shallow water waves and optical fibers aremodeled by nonlinear evolution equations and the appearanceof solitary wave solutions in nature is somewhat frequenthowever the nonlinear process is one of the major challengesand not easy to control because the nonlinear characteristicsof the system abruptly change due to some small changes in parameters including time Thus this issue becomes moredifficult and hence a crucial solution is needed The solutions of these equations have crucial impact in mathematical physics and engineering The variety of solutions of NLEEs which are mutual different operating mathematical techniques is very important in many fields of science such as fluid mechanics optical fibers technology of space control engineering problems hydrodynamics meteorology plasma physics applied mathematics Advanced nonlinear techniques are importantfor solving inherentnonlinear problems particularly those involving dynamicalsystems and allied areas In recent years there have beenbigimprovements in finding the exact solutions of NLEEsMany powerful methods have been establishedand enhanced such as the modified extended Fan sub-equation method [] the homogeneous balance method [] the Jacobi elliptic function expansion [] the Backlund transformation method [56] the Darboux transformation method [7]the Adomian decomposition method [8 9] the auxiliary equation method[0] the ( G / G) -expansion method [ 8] the Exp- ( φ( ξ)) -expansion method [9] the sine-cosine method [0 ] the tanh method []the F-expansion method [5] the exp-function method [67]the modified simple equation method [8 0]the first integral method [] the simple equation method [] the bilinear method[] the transformed rational function method[]and so on Most of the above methods are dependent on computational software except the MSE method The objective of this paper is to look for new exact traveling wave solutions includingtopological soliton single soliton solutions of the well-recognized Cahn Allen equation [0] viathe MSE method
2 Preprints (wwwpreprintsorg) NOT PEER-REVIEWED Posted: January 07 doi:009/preprints070008v of 9 Description of the MSE Method Consider a general form of a nonlinear evolution equation Huu ( t ux uxt u xx ) = 0 () u( ξ ) = u( x t) is an unknown function H is a polynomial of u( x t ) and its partial derivatives in which the highest order derivatives and nonlinear terms are involved In the followingwe present the main steps of the method: Step :Combine the real variablesxand t by acompound variableξ : urt ( ) = u( ξ ) ξ= P r± wt () Here P = liˆ+ mj ˆ+ nkˆ and P = xiˆ+ yj ˆ+ zkˆ axes of x yzrespectively μis wave number and w is the speed of the traveling wave lmnare the constant magnitudes along the This travelling wave transformation permits us to reduce Equation () to the following ordinary differential equation (ODE): Guu ( u ) = 0 () G is a polynomial in u( ξ) and its derivatives in Step : We also consider that the Equation () has the formal solution: n S ( ξ) u( ξ ) = Ai i= 0 S ( ξ) A(0 i n) are constants to be determined and () i i du u () ξ = u () ξ = dξ dξ d u and so on () S ξ is also unknown function to be evaluated Step : The value of positive integer n in Equation () can be determined by taking into account the homogeneous balance between the highest order nonlinear terms and the derivatives of highest order occurring in Equation () If the degree of u( ξ) is D( u( ξ )) = n then the degree of the other expression will be u( ξ) as follows: s p d u( ξ) q p du( ξ) D = n+ p p D u = np+ s( n+ q) q dξ dξ Step :Inserting Equation () into Equation () we get a polynomial of ( S ( ξ)/ S( ξ )) and its i derivatives and ( S( ξ )) ( i = 0 n) In the resultant polynomial we equate all the i coefficients of ( S( ξ )) ( i = 0 n) to zero This technique produces a system of algebraic and differential equations that can be solved receiving Ai ( i = 0 n) S() ξ and the value of the other needful parametersthis completes the determination of the solution to the Equation () RemarkIn comparisonin the modified simple equation method with the simple equation method [] it is seen that the simple equation method gets help froman auxiliary equation (the Riccati equation) but the modified simple equation method can perform directly without help from an auxiliary equation On the other hand the simple equation method yields results that are a special case from the modified equation method Traveling Wave Solution of the Cahn Allen Equation Let us consider the nonlinear parabolic partial differential equation given by:
3 Preprints (wwwpreprintsorg) NOT PEER-REVIEWED Posted: January 07 doi:009/preprints070008v of 9 m ut uxx u u = + ; (5) for m = Equation (5) becomes the Cahn Allen equation [90] This equation arises in many scientific applications such as mathematical biology quantum mechanics and plasma physics To solvethis example we can use transformation ξ = kx + wt ( k and w are the wave number andthe wave speed respectively) and then Equation (5) becomes ordinary differential equation: Balancing u with u then gives n = : + = 0 (6) wu k u u u S () ξ u() ξ= A0+ A (7) S () ξ S ( ξ) S ( ξ) ( ξ ) = u A A (8) S( ξ) S( ξ) S ( ξ) S ( ξ) S ( ξ) S ( ξ) ( ξ ) = + u A A A (9) S( ξ) S ( ξ) S( ξ) get: Putting Equations (6) (9) in Equation (6) and equating coefficients of like powers of Coefficient of ( S( ξ )) 0 : S () ξ S() ξ we A 0 A0 0 = ; (0) Coefficient of ( ) S ( ξ ) : ( ξ ) Coefficient of kas () ξ + AAS () ξ + was () ξ AS () ξ = 0 () ; S ( ) : 0 ( ) ( ) 0 wa S ( ξ ) + k A S ( ξ) S ( ξ ) + A A S ( ξ ) = 0 () S ( ) : ( ξ ) and Coefficient of ( ) A( A k ) S ( ξ ) = 0 () From Equation(0) we achieve A 0 = 0 and from Equation() A 0 thus A =± k Integrating we have S k (A0 ) + w( w A0A) = S k ( w A0A) ()
4 Preprints (wwwpreprintsorg) NOT PEER-REVIEWED Posted: January 07 doi:009/preprints070008v of 9 From Equation () k (A0 ) + w( w A0A) S = c exp( ξ) k ( w A0A) (5) S ck k (A ) + w( w A A) exp( ) 0 0 = ξ w A0A k ( w A0A) (6) and ck k (A0 ) + w( w A0A) S = exp( ξ ) + c k (A0 ) + ww ( AA 0 ) k ( w AA 0 ) (7) Using Equations (6) and (7) we attain k (A ) + w( w A A) 0 0 exp( ξ) cak k ( w A0A) 0 w A0A ck k A0 + w w A0A k (A0 ) + ww ( AA 0 ) k ( w AA 0 ) u = A + with w Case-I:For set 0 k A = 0 A =± k we get ( ) ( ) exp( ξ ) + c =± Here c and c are arbitrary constants (8) If with c kc w k = then w k ) exp( ξ) ck kw) u =± w ck w k ) exp( ξ ) + c w k k w) w =± k (9) w k w k u =± + tanh ξ wk (0) wk If c kc = w k with then w=± k w k w k u =± + coth ξ wk () wk
5 Preprints (wwwpreprintsorg) NOT PEER-REVIEWED Posted: January 07 doi:009/preprints070008v 5 of 9 Since c and c are arbitrary constants for other choices of c and c it might yield many new and more general exact solutions of the nonlinear Cahn Allen equation withoutany aid of symbolic computation softwarethe solutions u(xt) obtained in Equations (0) and () arepresented in the following figures: (see Figures and ) Figure Kink wave of Equation (0) with k = Case-II:For set 0 Figure Single soliton solution of Equation() with k = A =± A =± k we get If 6 k + w( w k)) exp( ξ) ck k ( w k) u =± ± w k ck 6 k + w( w k)) exp( ξ ) + c 6 k + w( w k) k ( w k) with w k kc c = 6 k + w( w k) then =± Here c and c are arbitrary constants ()
6 Preprints (wwwpreprintsorg) NOT PEER-REVIEWED Posted: January 07 doi:009/preprints070008v 6 of 9 If {6 k + w( w k)} 6 k + w( w k)) u =± ± + tanh ξ kw ( k) k( w k) () c w=± with kc = 6 k + w( w k) then k {6 k + w( w k)} 6 k + w( w k)) u =± ± + tanh ξ kw ( k) k( w k) () w=± k with Since c and c are arbitrary constants for other choices of c and c it might yield many new and more general exact solutions of the nonlinear Cahn Allen equation withoutany aid of symbolic computation software The solutions u( x t ) obtained in Equation()are similar to Figure and Equation () is similar to Figure and omitted for convenience Again with commercial software we can find some solutions of the Cahn Allen equation (solving fromequations () and ()) For A0 = 0 A =± k we get ( ) exp( / ) And thus b uxt ( ) =± with ξ= k( x± t / ) ξ ξ acosh sinh + b k k (5) If we consider a / b = exp( c) then Equation(5) reduces to well known solution: For 0 and thus uxt ( ) tanh x t c =± + ± + + (6) A = A =± k we get S () ξ= a + b exp( ±ξ / ) k uxt ( ) = acosh b ξ ξ ± sinh + b k k with ξ= k( x± t / ) If we consider a / b = exp( c) then Equation(6) reduces to well known solution: A = A =± k For 0 (7) uxt ( ) tanh x t c = + ± + + (8) we get S () ξ= a+ b exp( ξ / ) k
7 Preprints (wwwpreprintsorg) NOT PEER-REVIEWED Posted: January 07 doi:009/preprints070008v 7 of 9 and thus b uxt ( ) = ξ ξ acosh sinh + b k k with ξ= k( x t / ) If we consider a / b = exp( c) then Equation(6) reduces to well known solution: (9) uxt ( ) tanh x t c = + ± + + (0) Since a and b are arbitrary constants for other choices of a and b it might yield many new and more general exact solutions of the nonlinear Cahn Allen equation When we choose a / b = exp( c) we get special type solutions like Equations (8) and (0) but if we choose a and b in different ways we can get different types of solutions ThusEquations (8) and (0) are special types of our solutions Graphs of Equations (5) (7) and (9) represent kink type like Figure both for positive/negative values of the arbitrary constants c and c like single solitons such as Figure fortheir opposite values Comparison In this section we compare our solution with some well-known methods namely the exp-function method and the first integral method as follows: (a) Comparison with Exp-method Reference [7]:Ugurlu [7] obtained some solutions of the Cahn Allen equation via the exp-function methodin which solutions u 8 u 9 are identical with Equation (5) when b = a = b and the other solutions are different from their solutions(for 0 more see Ref [7]) (b)comparison with First Integral method Reference []:Tascan and Bekir [] obtained some solutions of the Cahn Allen equation via the first integral method in which Equation(0) is identical with Equation (5) (when in our study a = b = k = / and in their study c 0 = 0) u8 u9is identical with Equation (5) when b = a = b0 and the other solutions are different from their solutions On the contrary by using themse method in this article we obtained four solutions withless calculations 5 Conclusions In this article we have successfully implemented the MSE method to find the exact traveling wave solutions of the Cahn Allen equation Comparing the MSE method to other methods we claim that the MSE method is straight forward efficient and can be used in many other nonlinear evolution equations In the existing methods such as the (G /G)-expansion method the exp-function method and the tanh-function method it is requires making use of symbolic computation software such as Mathematica or Maple- to facilitate complex algebraic computations To solve non-linear evolution equations via the MSE method no auxiliary equations are needed On the other hand via the MSE method the exact and solitary wave solutions to these equations have been achieved without using any symbolic computation software because the method is very simple and has easy computations Acknowledgements: The authors are very grateful to the referees whose valuable suggestions tremendously improved the quality of the manuscript
8 Preprints (wwwpreprintsorg) NOT PEER-REVIEWED Posted: January 07 doi:009/preprints070008v 8 of 9 Author Contributions: HOR proposed the algorithm MZA developed analyzed and implement the methods MRI contributed to the implementation of the methods All authors approved the final manuscript Conflicts of Interest: The authors declare that they have no competing interests References Yomba EThe modified extended Fan sub-equation method and its application to the (+)-Dimensional Broer-Kaup-Kuperschmidt equations Chaos Solitons Fractals Zayed EME; Zedan HA; Gepreel KA On the solitary wave solutions for nonlinear Hirota-Sasuma coupled KDV equations Chaos Solitons Fractals Wang ML Exact solutions for a compound KdV-Burgers equationphys LettA Yan Z Abundant families of Jacobi elliptic function solutions of the (G /G)-dimensional integrable Davey-Stewartson- type equation via a new methodchaos Solitons Fractals Miura MRBacklund Transformation; Springer: Berlin Germany978 6 Ma WX;Fuchssteiner B Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equationint J Non-Linear Mech Matveev VB; Salle MADarboux Transformation and Solitons; Springer: Berlin Germany 99 8 Wu TY; Zhang JEOn Modeling Nonlinear Long WavesIn Mathematics Is for Solving Problems;Cook P Roytburd VTulin M Eds; SIAM: Philadelphia PA USA 996; pp 9 9 Adomain GSolving Frontier Problems of Physics: The Decomposition Method; Kluwer Academic Publishers: Boston MA USA 99 0 Sirendaoreji; Sun J Auxiliary equation method for solving nonlinear partial differential equations Phys LettA Sirendaoreji Auxiliary equation method and new solutions of Klein-Gordon equations Chaos Solitions Fractals Wang ML; Li XZ; Zhang J The ( G / G) -expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics Phys Lett A Bekir A Application of the ( G / G) -expansion method for nonlinear evolution equationsphys Lett A Roshid HO; Hoque MF; Akbar MA New extended ( G / G) -expansion method for traveling wave solutions of nonlinear partial differential equations (NPDEs) in mathematical physics Ital J PureAppl Math Roshid HO; Alam MN; Hoque MF; Akbar MA A new extended ( G / G) -expansion method to find exact traveling wave solutions of nonlinear evolution equations Math Stat Neyrame A; Roozi A; Hosseini SS; Shafiof SM Exact travelling wave solutions for some nonlinear partial differential equationsj King Saud Univ-Sci Roshid HO; Rahman N; Akbar MA Traveling wave solutions of nonlinear Klein-Gordon equation by extended ( G / G) -expansion methodannpure Appl Math Salehpour E; Jafari H; Kadkhoda N Application of (G /G)-expansion Method to Nonlinear Lienard Equation Indian J Sci Tech Roshid HO; Rahman MA The exp( Φ( η)) -expansion method with application in the (+)-dimensional classical Boussinesq equations Results Phys
9 Preprints (wwwpreprintsorg) NOT PEER-REVIEWED Posted: January 07 doi:009/preprints070008v 9 of 9 0 Wazwaz AMExact solutions of compact and noncompact structures for the KP BBM equationappl Math Comput Wazwaz AMCompact and noncompact physical structures forthe ZK-BBM equationappl MathCompact Wazwaz AMThe sine-cosine method for obtaining solutions with compact and noncompact structures Appl Math Comput Wazwaz AM The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equationschaos Solitons Fractals Abdou MAThe extended F-expansion method and its application for a class of nonlinearevolution equations Chaos Solitons Fractals Abdou MAExact periodicwave solutions to some nonlinear evolution equations Int JNonlinear Sci Mahmoudi J; Tolou N; Khatami I; Barari A; Ganji DDExplicit solution of nonlinear ZK-BBM wave equation using Exp-function method J Appl Sci Ugurlu Y Exp-function method for the some nonlinear partial differential equations Math Aeterna Jawad AJM; Petkovic MD; Biswas A Modified simple equation method for nonlinear evolution equations Appl Math Comput Zayed EME A note on the modified simple equation method applied to Sharma Tasso Olver equation Appl Math Comput Zayed EME; Ibrahim SAH Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method Chin Phys Lett Tascan F; Bekir A traveling wave solutions of the Cahn-Allen equation by using first integral method Appl Math Comput Jafari H; Kadkhoda N Application of simplest equation method to the (+ )-dimensional nonlinear evolution equationsnew Trends Math Sci MaWX Generalized Bilinear Differential Equations Studies Nonlinear Sci0 0 Ma WX;Lee JH A transformed rational function method and exact solutions to the + dimensional Jimbo Miwa equationchaos Solitons Fractals by the authors; licensee Preprints Basel Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (
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