_{1}

In this work, the exp(- φ (ξ )) -expansion method is used for the first time to investigate the exact traveling wave solutions involving parameters of nonlinear evolution equations. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. The validity and reliability of the method are tested by its applications to Nano-ionic solitons wave’s propagation along microtubules in living cells and Nano-ionic currents of MTs which play an important role in biology.

The nonlinear partial differential equations of mathematical physics are major subjects in physical science [

[

The objective of this article is to investigate more applications than obtained in [

onstrate the advantages of the exp

waves’s propagation along microtubules in living cells and Nano-ionic currents of MTs.

Consider the following nonlinear evolution equation

where F is a polynomial in

Step 1. We use the wave transformation

where c is a positive constant, to reduce Equation (2.1) to the following ODE:

where P is a polynomial in u(ξ) and its total derivatives.

Step 2. Suppose that the solution of ODE (2.3) can be expressed by a polynomial in

where

The solutions of ODE (2.5) are

When

When

When

When

When

where

Step 3. Substitute Equation (2.4) along Equation (2.5) into Equation (2.3) and collecting all the terms of the same power

Step 4. substituting these values and the solutions of Equation (2.5) into Equation (2.3) we obtain the exact solutions of Equation (2.1).

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 in gravitational field. The free surface elevation above the undisturbed depth h is

Let

It is useful to introduce two following fundamental dimensionaless parameters:

where

where

Expanding

and using the dimensionless wave particles velocity in x-direction, by definition

Making the differentiation of (3.12) with respect to

Returning back to dimensional variables

We could define the new function

implying that (3.16) becomes

We seek for traveling wave solutions with moving coordinate of the form

Integrating Equation (3.19) once, and setting

Balancing

Substituting (3.21) along (3.23) into (3.20), setting the coefficients of_{0}, a_{1}, a_{2}):

Solving the above system with the aid of Mathematica or Maple, we have the following solution:

Sothat the solution of Equation (3.20) will be in the form:

Consequently, the solution takes the forms:

When

When

When

When

When

The Nano-ionic currents are elaborated in [

where R = 0.34 × 10^{9} Ω is the resistance of the ER with length, l = 8 × 19^{−9} m, c_{0} = 1.8× 10^{−15} F is the maximal capacitance of the ER, G_{0} = 1.1 × 10^{−13} si is conductance of pertaining NPs and z = 5.56 ×10^{10} Ω is the characteristic impedance of our system parameters δ and

Which can be written in the form

where

Thus Equation (3.37) take the form

Balancing

Where a_{0}, a_{1}, a_{2} are arbitrary constants such that a_{2} ≠ 0. From Equation (3.40), it is easy to see that

Substituting Equations (3.40)-(3.42) into Equation (3.39) and equating the coefficients of

Solving above system with the aid of Mathematica or Maple, we have the following solution:

a_{1} = a_{1}, a_{2} = a_{2}.

So that the solution of Equation (3.39) will be in the form:

Consequently, the solution take the forms:

When

When

When

When

When

In nanobiosciences the transmission line models for ionic waves propagating along microtubules in living cells play an important role in cellular signaling where ionic wave’s propagating along microtubules in living cells shaped as nanotubes that are 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 inviscid, incompressible and non-rotating flow of fluid of constant depth (h). The

the traveling wave solutions for As an application, the traveling wave solutions for Nano-ionic solitons wave’s propagation along microtubules in living cells and Nano-ionic currents of MTs, which have been constructed using the