Application of Exp-Function Method to. a Huxley Equation with Variable Coefficient *

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1 Interntionl Mthemticl Forum, 4, 9, no., 7-3 Appliction of Exp-Function Method to Huxley Eqution with Vrible Coefficient * Li Yo, Lin Wng nd Xin-Wei Zhou. Deprtment of Mthemtics, Kunming College Kunming,Yunnn, 63, P. R. Chin kmliyo6@6.com, km_xwzhou@63.com. College of Sttistics nd Mthemtics Yunnn University of Finnce nd Economics Kunming, 6, P. R. Chin WL64mil@yhoo.com.cn Abstrct This pper suggests method clled the Exp-function method to find the solutions of the Huxley eqution with vrible coefficient. The obtined result includes ll solutions in open literture s specil cses, nd the generlized solution with some free prmeters might imply some fscinting menings hidden in the eqution. Keywords: Exp-function method, Huxley eqution, generlized solution. Introduction The core mthemticl frmework for modern biophysiclly bsed neurl modeling ws developed hlf century go by Hodgkin nd Huxley []. In series of ppers published in 9, they presented the results of n elegnt series of electrophysiologicl experiments in which they investigted the flow of electric current through the surfce membrne of the gint nerve fiber of squid. The Huxley eqution is nonliner prtil differentil eqution of second order of the form: The work ws supported by Scientific Reserch Foundtion of Yunnn Province Eductionl Bureu. * Corresponding uthor. Emil:WL64mil@yhoo.com.cn.

2 8 Li Yo, Lin Wng nd Xin-Wei Zhou ( ) u = u + u( k u) u. (.) t xx Recently, Huxley eqution with vrible coefficient is discussed in [], which red, ut α() t uxx + u ( u ) =. (.) In this pper, we will pply the exp-function method to the discussed problem.. Appliction to the Huxley eqution with vrible coefficients The bsic ide of the Exp-function ws proposed in J.H. He s monogrph[3]. Some illustrtive exmples in Refs.[4-7] showed tht this method is very effective to serch for vrious solitry nd periodic solutions of nonliner equtions. Zhng pplied the method to some differentil equtions with vrible coefficient [8]. Using the trnsformtion u = U, = kx + τ() t dt, (.) ( ) where k is constnt, τ () t is n integrble function of t to be determined lter, Eq.(.) becomes ( ) τ() tu α() t ku + U U = (.) where prime denotes the differentil with respect to. According to the Exp-function method[4], we ssume tht the solution of Eq.(.) cn be expressed in the following form: d n exp( n ) n= c cexp( c ) + L+ d exp( d) U ( ) = = (.3) q b exp( ) bpexp( p ) qexp( q ) n p m m + L+ = where c, d, p nd q re positive integers which re unknown to be further determined, n nd b m re unknown constnts. To determine the vlues of c nd p, we blnce the liner term of highest order in Eq.(.) with the highest order nonliner term. Similrly to determine the vlues of d nd q. By simple clcultion, we hve, c exp[(3 p+ c) ] + L U = c exp[4 p ] + L, 3 c3exp[3 c ] + L c3exp[(3 c+ p) ] + L U = = c4exp[3 p] + L c4exp[4 p] + L. (.4) Blncing the highest order of Exp-function in (.4), we hve 3p + c= 3c+ p, nd this gives p = c. Using the sme method, we cn lso obtin tht q= d. We cn freely choice the prmeters c nd d. He nd Wu[] systemticlly studied the choice of the vlues of the prmeters, reveling the solution very wekly depends upon the vlues of the prmeters. An illustrting exmple of Dodd Bullough Mikhilov eqution ws given, He nd Wu considered three cses: Cse : p = c =, q= d = ; Cse :

3 Appliction of Exp-function method 9 p= c=, q= d = ; Cse 3: p= c=, q= d =, nd ll cses led to the equivlent result. Bekir nd Boz[9] pointed out p = c = nd q= d = re vlid for most nonliner prtil differentil equtions. For simplicity, we set p = c = nd q= d =, so Eq. (.3) reduce to exp( ) + + exp( ) U ( ) =, (.) exp( ) + b + b exp( ) Substituting Eq. (.) into Eq.(.), nd by the help of symbolic computtion system, we hve, 3 3 Ce 3 + Ce + Ce + C + C e + C e + C 3e = A, (.6) where A ( e b b e ) 3 = + +. Equting to zero the coefficients of ll powers of n equtions s following: 3 α() t + α() t = e yields set of lgebric τ() t k α() t α() t + 3 α() t + τ() t b + k α() t b α() t b = τ() t 4 kα() t α() t α() t + 3 α() t+ 3 α() t + τ() tb + 4 k α() t b α() t b τ() t b + k α() t b α() t b + τ() t b k α() t b = 3 α() t + α() t + 6 α() t + 6 k α() t b α() t b 3 τ() t b 3 kα() t b α() tb α() t b+ 3 τ() tbb 3 kα() tbb = α() t + 3 α() t + 3 α() t τ() t b + 4 k α() t b α() t b α() t b + () τ tb 4 kα() tb α() t b τ() tbb+ kα() tbb τ() t b kα() t b = 3 α() t α() t b + τ() t b k α() t b α() t b + k α() t b b τ() t b b = α() t α() t b = (.7) 3 Solving the system (.7) simultneously using symbolic computtion system, we obtin the following results. Cse =, =, =, b =, b =, k =±, τ () t = α() t, α() t. (.8) Cse =, =, =, b =, b + =, k =±, τ () t = α() t, (.9) Cse3 =, =, =, b =, b = b, k =±, τ () t = α() t, α() t b. (.)

4 3 Li Yo, Lin Wng nd Xin-Wei Zhou Cse4 b =,, =, b, b + =, k =±, τ () t = α() t, (.) Inserting Eqs.(.8)-(.) into (.), respectively, yields the following generlized solitry solutions(see Figs. -4): u( x, t) =, (.) + e + e u( x, t) =, (.3) e e where = = x α() or = = x α(). e u3( x, t) =, (.4) b + e + e u4( x, t) = (.) b e b e where = 3 = x + α() or = 4 = x + α(). To compre our results with those obtined in Ref.[], we set = in Eq.(.), Eq. (.) becomes u () ( x, t) = = tnh + e if we set b = in Eq.(.4), Eq. (.4) becomes e u3() ( x, t) = = + tnh + e These re the solutions obtined in [].,, 3 = or = (.6) = or = 4 (.7)

5 Appliction of Exp-function method Fig. Solution u ( x, t ) is shown t = Fig. Solution u ( x, t ) is shown t = = = nd α () t = t. = nd α () t = cost Fig.3 Solution is shown t = = nd α () t =. + t 3 Fig.4 Solution is shown t b =, b = = 4 nd () t t α =.

6 3 Li Yo, Lin Wng nd Xin-Wei Zhou 3. Discussions nd Conclusions The Exp-function method leds to generlized solitry solutions with some free prmeters involving the known solutions in open literture. The free prmeters might imply some physiclly meningful results in biologicl process. References [] A. L. Hodgkin nd A. F. Huxley, A quntittive description of membrne current nd its ppliction to conduction nd excittion in nerve, J Physiol (London) 7 (9). [] B.A. Li nd X.Y. Li, A Bcklund Trnsformtion nd exct solutions to Huxley Eqution with vrible coefficient. Journl of Henn University of Science nd Technology:Nturl Science, 8(7)78-8. [3] J.H. He, Non-Perturbtive Methods for Strongly Nonliner Problems, Disserttion. de-verlg im Internet GmbH, Berlin, Germny, 6. [4] J.H. He nd X.H. Wu, Exp-function method for nonliner wve equtions, Chos, Solitons & Frctls, 3(6), [] X.H.(Benn)Wu nd J.H. He, Solitry solutions, periodic solutions nd compcton-like solutions using the Exp-function method, Computers & Mthemtics with Applictions, 4(7), [6] X.W.Zhou, Exp-function method for solving Huxley eqution, Mthemticl Problems in Engineering, Volume8(8), Article ID: [7] X.W. Zhou, Exp-function method for solving Fisher s Eqution, Journl. of Physics: Conference Series, 96(8), 63. [8] S.Zhng, Appliction of Exp-function method to KdV eqution with vrible coefficients, Physics Letters A, 36(7) [9] A. Bekir, A.Boz Exct solutions for clss of nonliner prtil differentil equtions using exp-function method, Interntionl Journl of Nonliner Sciences nd Numericl Simultion, 8(7)-. Received: September 4, 8