Exact Traveling Wave Solutions for Nano-Solitons of Ionic Waves Propagation along Microtubules in Living Cells and Nano-Ionic Currents of MTs

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1 World Journal of Nano Science and Engineering, 5, 5, Published Online September 5 in SciRes. Exact Traveling Wave Solutions for Nano-Solitons of Ionic Waves Propagation along Microtubules in Living Cells and Nano-Ionic Currents of MTs Emad H. M. Zahran Department of Mathematical and Physical Engineering, College of Engineering, University of Benha, Shubra, Egypt e_h_zahran@hotmail.com Received April 5; accepted August 5; published 4 August 5 Copyright 5 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). Abstract In this work, the extended Jacobian elliptic function expansion method is used as the first time to evaluate the exact traveling wave solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to nano-solitons of ionic waves propagation along microtubules in living cells and nano-ionic currents of MTs which play an important role in biology. Keywords Extended Jacobian Elliptic Function Expansion Method, Nano-Solitons of Ionic Waves Propagation along Microtubules in Living Cells, Nano-Ionic Currents of MTs, Traveling Wave Solutions. Introduction The nonlinear partial differential equations of mathematical physics are major subjects in physical science []. Exact solutions for these equations play an important role in many phenomena in physics such as fluid mechanics, hydrodynamics, optics, and plasma physics. Recently many new approaches for finding these solutions have been proposed, for example, tanh-sech method []-[4], extended tanh-method [5]-[7], sine-cosine method [8]- [], homogeneous balance method [] [], F-expansion method []-[5], exp-function method [6] [7], trigonometric function series method [8], ( G G) expansion method [9]-[], Jacobi elliptic function method []-[6] and so on. How to cite this paper: Zahran, E.H.M. (5) Exact Traveling Wave Solutions for Nano-Solitons of Ionic Waves Propagation along Microtubules in Living Cells and Nano-Ionic Currents of MTs. World Journal of Nano Science and Engineering, 5,

2 The objective of this article is to apply the extended Jacobian elliptic function expansion method for finding the exact traveling wave solution of nano-solitons of ionic waves propagate on along microtubules in living cells and nano-ionic currents of MTs which play an important role in biology and mathematical physics. The rest of this paper is organized as follows: In Section, we give the description of the extended Jacobi elliptic function expansion method. In Section, we use this method to find the exact solutions of the nonlinear evolution equations pointed out above. In Section 4, conclusions are given.. Description of Method Consider the following nonlinear evolution equation F uu,, u, u, u, (.) t x tt xx where F is polynomial in u(x, t) 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 []-[6]. Step. Using the transformation, c, u = u ξ ξ = x t (.) where k and c are the wave number and wave speed, to reduce Equation (.) to the following ODE: where P is a polynomial in P uu,, u, u, (.) u ξ and its total derivatives, while u = du d ξ. Step. Making good use of ten Jacobian elliptic functions, we assume that (.) have the solutions in these forms: with N j ( ε) = ( ε) ( ε) + ( ε) u a + f,,,, j i aj fi bg j i i = = (.4) f ξ = snξ, g ξ = cnξ, f ξ = snξ, g ξ = dnξ, f ξ = nsξ, g ξ = csξ, f ξ = nsξ, g ξ = dsξ, 4 4 f ξ = scξ, g ξ = ncξ, 5 5 f ξ = sdξ, g ξ = ndξ, (.5) 6 6 where snξ, cnξ, dnξ, are the Jacobian elliptic sine function. The jacobian elliptic cosinefunction and the Jacobian elliptic function of the third kind and other Jacobian functions which is denoted by Glaisher s symbols and are generated by these three kinds of functions, namely that has the relations cnξ nsξ =, ncξ =, ndξ =, scξ =, snξ cnξ dnξ snξ snξ dnξ snξ csξ =, dsξ =, sdξ = cnξ snξ dnξ sn ξ + cn ξ =, dn ξ + m sn ξ =, ns ξ = + cs ξ, ns ξ = m + ds ξ, sc ξ + = nc ξ, m sd + = nd ξ with the modulus m ( < m < ): In addition we know that d d d d sn ξ cn ξ dn ξ d cn ξ sn ξ dn ξ d dn ξ m sn ξ cn ξ ξ ξ ξ (.6) (.7) =, =, =. (.8) 79

3 The derivatives of other Jacobian elliptic functions are obtained by using Equation (.8). To balance the highest order linear term with nonlinear term we define the degree of u as D[u] = n which gives rise to the degrees of other expressions as q q s d u p d u D n q, D u np s( n q). q = + q = + + dξ dξ According the rules, we can balance the highest order linear term and nonlinear term in Equation (.) so that n in Equation (.4) can be determined. In addition we see that when m, snξ, cnξ, dnξ degenerate as tanhξ, sechξ, sechξ, respectively, while when therefore Equation (.5) degenerate as the following forms N j ( ξ) = + tanh ( ξ) tanh ( ξ) + sech ( ξ) (.9) j= i j j, (.) u a a b N j ( ξ) = + coth ( ξ) coth ( ξ) + coth ( ξ) j= i j j, (.) u a a b N j ( ξ) = + tan ( ξ) tan ( ξ) + sec( ξ) j= i j j, (.) u a a b N j ( ξ) = + cot ( ξ) cot ( ξ) + csc( ξ) j= i j j. (.) u a a b Therefore the extended Jacobian elliptic function expansion method is more general than sine-cosine method, the tan-function method and Jacobian elliptic function expansion method.. Application.. Example : Nano-Solitons of Ionic Waves Propagation along Microtubules in Living Cells [7] We first consider an inviscid, incompressible and non-rotating flow of fluid of constant depth (h). We take the direction of flow as x-axis and z-axis positively upward the free surface ingravitational field. The free surface elevation above the undisturbed depth h is ƞ(x; t), so that the wave surface at height z = h + ƞ(x; t), while z = is horizontal rigid bottom. Let ϕ(x; z; t) be the scalar velocity potential of the fluidlying between the bottom (z = ) and free space ƞ(x; t), then we could write the Laplace and Euler equation with the boundary conditions at the surface and the bottom, respectively, as follows: ϕ ϕ + = ; < z < h+ η; < x< + x z (.) ϕ ϕ ϕ + i+ k + gη z = h+ η (.) t z z η η ϕ ϕ + t x x z ϕ = ; z = z It is useful to introduce two following fundamental dimensionless parameters: σ η h = < δ = < h ;, l where η is the wave amplitude, and l is the characteristic length-like wavelength. Accordingly, we also take a complete set of new suitable non-dimensional variables: (.) (.4) (.5) x ; z ; ct η x = z = ; ; h, l h = l ψ = = η η lc ϕ (.6) 8

4 where c = gh is the shallow-water wave speed, with g being gravitational acceleration. In term of (.5) and (.6) the initial system of Equation (.)-(.4) now reads ϕ δ x ϕ z + = ; (.7) Expanding ( xt ; ) in terms of δ σ σ ψ = ; Z = + σψ, x z ψ ψ + σ = ; Z = + σψ, x x δ z ϕ = ; z = z (.8) (.9) (.) = + δ + δ, (.) and using the dimensionless wave particles velocity in x-direction, by definition u (.) into (.7)-(.9), with retaining terms up to linear order of small parameters order in (.9), we get ϕ δ x u + ψ + σ = ψ u u u ψ δ + σ + + = x δ x 6 x u, ( σψ ). Making the differentiation of (.) with respect to x, and rearranging (.), we get u u u ψ u σ δ x x x + + ψ u + u ( + σψ ) δ =. x 6 x Returning back to dimensional variables ƞ(x; t) and = dϕ dx, (.4) now reads v v u v g η + + = h. t x x x t =, then substituting of x σδ, in (.8), and second We could define the new function V(x, t) unifying the velocity and displacement of water particles as follows: implying that (.6) becomes V V v = ; η = h t x V V 4 V gh + h V = t x h x t x t. (.) (.) (.4) (.5) (.6) (.7) (.8) We seek for traveling wave solutions with moving coordinate of the form ξ = x vt and with wave speed v, which reduces Equation (.8) into ordinary nonlinear differential equation as follows: 4 V v V V + = 4 v gh h v ζ h ζ ζ V Integrating Equation (.9) once, and setting = W, we get ξ ζ (.9) 8

5 Balancing w and in the form w = α + β + ξ w w c (.) w yields, N + = N N =. Therefore, we can write the solution of Equation (.) W ξ = a + a sn + b cn + a sn + b sncn (.) ( ξ ) ( ) 4 ( ξ ) 6 4 ( ) + 6b sm sn cn b ( 4 ) sncn W = a cn b sn + a sncn + b b sn dn (.) W = a m + sn + a m sn a m + a m + + m sn cn b + m b cn + a sn (.) Substituting (.) into (.), setting the coefficients of (sn 4, sn, sn cn, sn, sn cn, sncn, sn, cn, sn ) to zero, we obtain the following underdetermined system of algebraic equations for (a, a, a, b, b ): ( ) 6am α a b (.4) am α aa bb (.5) α( ) α 4a m + a b + b + a a βa (.6) a m + a a + bb βa (.7) 6bm α ab (.8) bm α ab + ba (.9) α b m + 4 ab + ab βb (.) b αab βb (.) a αa βa c αb =. (.) Solving the bove system with the aid of Mathematica or Maple, we have the following solution: ( am + a + am ) 6m 4 α =, β =, a = a, a a a ( a + am + aam + aa ) a = a, b b c = a Sothat the solution of Equation (.) will be in the form: if m, we have the hyperbolic solution:.. Example. Nano-Ionic Currents of MTs The nano ionic currents are elaborated in [7] take the form l W ( ξ) sn ( ξ) W (.) β m m = + 6, (.4) α α α α β 6 tanh. α α α ( ξ) = ( ξ) (.5) z zc u + ( xc ss) uu + u + u + ( Rz GZ ) u (.6) l l l xxx t t 8

6 9 9 5 where R =.4 Ω is the resistance of the ER with length, l = 8 m, c =.8 F is the maximal capacitance of the ER, G =. si is conductance of pertaining NPs and z = 5.56 Ω is the characteristic impedance of our system parameters δ and x describe nonlinearity of ER capacitor and conductance of NPs in ER, respectively. In order to solve Equation (.6) we use the travelling wave transforma- c 6 tions u( xt, ) = u( ξ), ξ = x t with = Rc =.6 s, to reduce Equation (.6) to the following nonl linear ordinary differential equation: cz zcc u ( xc sc) uu + u ( Rz GZ ) u, + = (.7) which can be written in the form (.8) u H uu H u H u c ce = = = z H B, H, B z xc ss, E = c, D = H. (.9) Thus Equation (.8) takes the form c ce u + Buu + u + Du =. Balancing u and uu yields, N + = N + N + N =. Consequently, we get (.4) u = a + a sn + b cn + a sn + b sncn (.4), where a, a, a, b, b are arbitrary constants such that a or b. From Equation (.4), it is easy to see that u = dna cn dnb cn + dna sn cn dnb sn + b dn, (.4) u = a m sn + a sn m + m sn b cn 4a m sn + 6a sn m 4 + 6m sn cnb m sncnb a sn b cn + a 4a sn 4 b sncn. (.4) Substituting Equations (.4)-(.4) into Equation (.4) and equating the coefficients of sn dn, sn cndn, sn dn, sncndn, sndn, cndn and dn to zero, we obtain 4 sn dn, sn cndn, cbab 8bm 4 (.44) ( ) cb a b + = (.45) 8am, ( ) cb ab ab bm (.46) ( ) cb a a b b + = (.47) am, 4 cb ce b( 5m + ) + ( ab ab + ab) + ( b) + Da (.48) 8

7 8 cb ce a( m + ) + ( aa a b + b ) + a + Db cb ce b( 4m + ) + ( ab+ ab + ab) + b+ Da cb ce a( m + ) + ( ab+ bb ) + a+ Db cb ce b( m + 4) + ( ab + ba ) + b=. Solving the above system with the aid of Mathematica or Maple, we have the following solution: Case. Case. 4m + ce 4m D m= ma, =, a a =, b b = cb cb ce + ce D m=, a a a =, b b = 4 cb So that the solution of Equation (.4) will be in the form: Case. (.49) (.5) (.5) (.5) Case. u ( ξ ) If m, we have the hyperbolic solution: Case. Case. 4. Conclusion 4m + ce 4m cb cb = + sn (.5) u u + ce = (.54) cb ( ξ ) sn + ce 4 = + (.55) cb cb ( ξ ) sn u + ce = (.56) cb ( ξ ) sn The nano waves propagating along microtubules in living cells play an important role in nano biosciences and cellular signaling where the propagation along microtubules shaped as nanotubes is essential for cell motility, cell division, intracellular trafficking and information processing within neuronal processes. Ionic waves propagating along microtubules in living cells have been also implicated in higher neuronal functions, including memory and the emergence of consciousness and we presented an in viscid, incompressible and non-rotating fluid of constant depth (h). The extended Jacobian elliptic function expansion method has been successfully used to find the exact traveling wave solutions of some nonlinear evolution equations. According to the suggested method we obtained a new and more accurate traveling wave solution of nano ionic-solitons waves propagation along microtubules in living cells and nano-ionic currents of MTs. Let us compare between our results obtained in the present article with the well-known results obtained by other authors using different methods as follows: Our results of nano-solitons of ionic waves propagation along microtubules in living cells and nano-ionic currents of MTs are new and different from those obtained in [7]. Figures - show solitary wave 84

8 Figure. Plot of solution of Equation (.5). Figure. Plot of solution of Equation (.56). Figure. Plot of solution of Equation (.55). 85

9 solution. It can be concluded that this method is reliable and proposes a variety of exact solutions NPDEs. The performance of this method is effective and can be applied to many other nonlinear evolution equations. References [] Ablowitz, M.J. and Segur, H. (98) Solitions and Inverse Scattering Transform. SIAM, Philadelphia. [] Maliet, W. (99) Solitary Wave Solutions of Nonlinear Wave Equation. American Journal of Physics, 6, [] Maliet, W. and Hereman, W. (996) The Tanh Method: Exact Solutions of Nonlinear Evolution and Wave Equations. Physica Scripta, 54, [4] Wazwaz, A.M. (4) The Tanh Method for Travelling Wave Solutions of Nonlinear Equations. Applied Mathematics and Computation, 54, [5] El-Wakil, S.A. and Abdou, M.A. (7) New Exact Travelling Wave Solutions Using Modified Extended Tanh-Function Method. Chaos Solitons Fractals,, [6] Fan, E. () Extended Tanh-Function Method and Its Applications to Nonlinear Equations. Physics Letters A, 77, [7] Wazwaz, A.M. (7) The Extended Tanh Method for Abundant Solitary Wave Solutions of Nonlinear Wave Equations. Applied Mathematics and Computation, 87, [8] Wazwaz, A.M. (5) Exact Solutions to the Double Sinh-Gordon Equation by the Tanh Method and a Variable Separated ODE Method. Computers & Mathematics with Applications, 5, [9] Wazwaz, A.M. (4) A Sine-Cosine Method for Handling Nonlinear Wave Equations. Mathematical and Computer Modelling, 4, [] Yan, C. (996) A Simple Transformation for Nonlinear Waves. Physics Letters A, 4, [] Fan, E.G. and Zhang, H.Q. (998) A Note on the Homogeneous Balance Method. Physics Letters A, 46, [] Wang, M.L. (996) Exact Solutions for a Compound KdV-Burgers Equation. Physics Letters A,, [] Abdou, M.A. (7) The Extended F-Expansion Method and Its Application for a Class of Nonlinear Evolution Equations. Chaos, Solitons & Fractals,, [4] Ren, Y.J. and Zhang, H.Q. (6) A Generalized F-Expansion Method to Find Abundant Families of Jacobi Elliptic Function Solutions of the (+)-Dimensional Nizhnik-Novikov-Veselov Equation. Chaos, Solitons & Fractals, 7, [5] Zhang, J.L., Wang, M.L., Wang, Y.M. and Fang, Z.D. (6) The Improved F-Expansion Method and Its Applications. Physics Letters A, 5, [6] He, J.H. and Wu, X.H. (6) Exp-Function Method for Nonlinear Wave Equations. Chaos, Solitons & Fractals,, [7] Aminikhad, H., Moosaei, H. and Hajipour, M. (9) Exact Solutions for Nonlinear Partial Differential Equations via Exp-Function Method. Numerical Methods for Partial Differential Equations, 6, [8] Zhang, Z.Y. (8) New Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation. Turkish Journal of Physics,, 5-4. [9] Wang, M.L., Zhang, J.L. and Li, X.Z. (8) The (G /G)-Expansion Method and Travelling Wave Solutions of Nonlinear Evolutions Equations in Mathematical Physics. Physics Letters A, 7, [] Zhang, S., Tong, J.L. and Wang, W. (8) A Generalized (G /G)-Expansion Method for the mkdv Equation with Variable Coefficients. Physics Letters A, 7, [] Zayed, E.M.E. and Gepreel, K.A. (9) The (G /G)-Expansion Method for Finding Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics. Journal of Mathematical Physics, 5, Article ID: 5. [] Zayed, E.M.E. (9) The (G /G)-Expansion Method and Its Applications to Some Nonlinear Evolution Equations in Mathematical Physics. Journal of Applied Mathematics and Computing,,

10 [] Dai, C.Q. and Zhang, J.F. (6) Jacobian Elliptic Function Method for Nonlinear Differential-Difference Equations. Chaos, Solitons & Fractals, 7, [4] Fan, E. and Zhang, J. () Applications of the Jacobi Elliptic Function Method to Special-Type Nonlinear Equations. Physics Letters A, 5, [5] Liu, S., Fu, Z., Liu, S. and Zhao, Q. () Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 89, [6] Zhao, X.Q., Zhi, H.Y. and Zhang, H.Q. (6) Improved Jacobi-Function Method with Symbolic Computation to Construct New Double-Periodic Solutions for the Generalized Ito System. Chaos, Solitons & Fractals, 8, [7] Sataric, M., Dragic, M. and Sejulic, D. () From Giant Ocean Solitons to Cellular Ionic Nano-Solitons. Romanian Reports in Physics, 6,