The improved generalized Riccati equation mapping method and its application for solving a nonlinear partial differential equation ( PDE ) describing the dynamics of ionic currents along microtubules

In this paper we apply the improved Riccati equation mapping method to construct many families of exact solutions of a nonlinear partial differential equation involving parameters of a special interest in nanobiosciences and biophysics which describe a model of microtubules as nonlinear RLC transmission lines. As results, we can successfully recover the previously known results that have been found using other methods. This method is straightforward and concise, and it can be applied to other nonlinear PDEs in mathematical physics. Comparison between our new results and the well-known results are given. Some comments on the well-known results are also presented at the end of this article.

In the present paper, we shall use the improved Riccati equation mapping method to find the exact solutions of a nonlinear PDE of nanobiosciences.The main idea of this method is that the traveling wave solutions of nonlinear equations can be expressed by polynomials in Q , where , where , , , r p k  and q are constants.The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in the given nonlinear equation, the coefficients of this polynomial can obtained by solving a set of algebraic equations resulted from the process of using the proposed method.
The objective of this paper is to apply the improved Riccati equation mapping method for finding many families of exact traveling wave solutions of the following nonlinear PDE of special interest in nanobiosciences, namely, the transmission line models for microtubules as nonlinear RLC transmission line (Sekulic et al., 2011a, Sataric et al., 2010):  1) can be elaborated in Sataric et al. (2010).For further references about electrical models of microtubules, see for example Ilic et al. (2009), Sekulic et al. (2011bSekulic et al. ( , 2012)), Sataric et al. (2009); Freedman et al. (2010), andSekulic andSataric (2012).Recently, Equation (1) has been discussed in (Sekulic et al. 2011a) by using the modified extended tanh-function method, where its exact solutions have been found.The rest of this paper can be organized as follows: First is description of the improved generalized Riccati equation method.Many families of exact traveling wave solutions for Equation (1) are next obtained.This is followed by illustrations on physical explanations for some obtained results.Thereafter, conclusions and comments on Sekulic et al. (2011a) as well as comparison between our new results and the well-known results obtained in Sekulic et al. (2011a) are investigated.

Description of the improved generalized Riccati equation mapping method
We suppose that a nonlinear PDE is in the following from: ( , , , , ,...) and its partial derivatives in which the highest order derivatives and nonlinear terms are involved.Let us now give the main steps for solving Equation (2.1) using the improved Riccati equation mapping method (Zhu 2008;Zayed and Arnous, 2013;Zayed et al, 2013): Step 1: We look for its traveling wave solution in the form where , k  are constants.Substituting (3) into Equation (2) gives the nonlinear ODE for ( ) u  as follows: ( , , ,...) 0 where H is a polynomial in ( ) u  and its total derivatives , , ,... .
Step 3: We determine the positive integer m in (5) by balancing the nonlinear terms and the highest order derivatives of ( ) u  in Equation (4).
Step 4: Substituting (5) and along with Equation (6) into Equation ( 4) and then equating all the coefficients of ( 0, 1, 2,..., ) to zero yield a system of algebraic equations which can be solved by using the Maple or Mathematica to find the values of the constants ( ,..., ) i a m m  and , k  .
Step 5: It is well-known (Zhu 2008;Zayed and Arnous, 2013;Zayed et al., 2013) that Equation ( 6) has many families of solutions as follows: where A and B are two non-zero real constants satisfying where A and B are two non-zero real constants satisfying where d is an arbitrary constant.
Type 4: When 0 0 r p and q    we have , where c 1 is an arbitrary constant.
Step 6: Substituting the well known solutions of Equation ( 6) listed above in Step 5 into (5) we have many families of exact solutions of Equation ( 2).

MANY FAMILIES OF EXACT TRAVELING WAVE SOLUTIONS FOR EQUATION (1)
Here we apply the proposed improved generalized Riccati equation mapping method to find many families of exact traveling wave solutions of Equation (1).To the end we use the wave transformation where is the dimensionless velocity of the wave, to reduce Equation (1) into the following ODE: where .

By balancing
2 u with u  , we have m=2.Hence the formal solution of Equation ( 8) takes the form: where 2 1 0 1 2 , , , , a a a a a   are constants to be determined, such that Inserting (9) with the aid of Equation ( 6) into Equation (8) we get the following system of algebraic equations: By solving these algebraic equations with the aid of Maple or Mathematica we have the following cases: , ( 2), , 24 2 3 12 6 , , 0 5 q p p c c p p q q r p a p a q q a c a a

Exact traveling wave solutions of Equation (1) for Case 1
By using the case 1 and according to the values of solutions of type 1 in the proposed method, we obtain the following exact traveling wave solutions for Equation (1): where A and B are two non-zero real constants satisfying where 1 6 .5 According to the values of the solutions of type 3 in Sec.2, we obtain the following exact traveling wave solutions for Equation (1): Exact traveling wave solutions of Equation (1) for case 2.
By using the case 2 and according to the values of solutions of type 1 in the proposed method, we obtain the following exact traveling wave solutions for Equation (1): where A and B are two real constants satisfying B 2 -A 2 >0,

Exact traveling wave solutions of Equation (1) for Case 3
By using the case 3 and according to the values of solutions of type 1 in the proposed method, we obtain the following exact traveling wave solutions for Equation (1): 5 24 24 20 24 32 24 where A and B are two non-zero real constants satisfying 5 24 24 20 24 24 192 24 5 24 24 20 24 24 192 24 where, We have noted in the two cases 1, 2 that 0 6     while in case 3 we have 0 24     .This yields that all the solutions of type 2 in the proposed method cannot be considered in this paper.Therefore, Equation (1) has no trigonometric function solutions.

PHYSICAL EXPLANATIONS FOR SOME OBTAINED RESULTS
Here, we will present some graphs for the obtained solutions of Equation ( 1) by selecting some special values of the parameters in the exact solutions using the mathematical software Maple, which can be shown below in Figures 1 to 6. From these explicit solutions, we see that 1 ( , ) u x t in both cases 1, 2 are kink shaped soliton solutions while 2 ( , ) u x t in these two cases are singular kink shaped soliton solutions.The two solutions 1 ( , ) u x t and 2 ( , ) u x t in case 3 are kink-singular shaped soliton solutions.The graphical representations of these solutions are shown in the following figures:

CONCLUSIONS AND COMMENTS
Here we give some comments on the solutions ( 22) to ( 27) of the second model of microtubules ( 16) in Sekulic et al. (2011a).We have found that some of these solutions are incorrect.Thus, we will correct these solutions and then compare between some of our results in the present article and the corrected solutions as follows: 1) There is a minor error in Equation ( 18) of Sekulic et al. (2011a).The correction is to replace  by 2  in Equation ( 18).
2) The results of Case I in Sekulic et al. (2011a) do not satisfy the algebraic Equation ( 21).After a careful revision, we have shown that the correction of this case should be in the form: It is easy to see that the corrected values (10) satisfy the algebraic Equation ( 21) of Sekulic et al. (2011a).
Hence with replacing  by 2  in (10), the result (22) of Sekulic et al. (2011a) should be rewritten in the corrected form: tanh tanh 2 5 24 24 5 6 2 24 5 6 On comparing the corrected result (11) with our result 1 ( , ) u x t of "Exact traveling wave solutions of Equation (1) for case 1", we conclude that they are equivalent if 0 p  and 1 q  .
3) The results of Case II in Sekulic et al. (2011a) do not satisfy the algebraic Equation ( 21) too.After a careful revision, we have shown that the correction of this case should be in the form: It is easy to see that the corrected values of (13) satisfy the algebraic Equation ( 21) of Sekulic et al. (2011a).Hence with replacing  by 2  in (13), the result (24) of (Sekulic et al, 2011a) should be rewritten in the corrected form: On comparing the corrected result ( 14) with our result 1 ( , ) u x t of 'Exact traveling wave solutions of Equation (1) for case 2', we conclude that they are equivalent if 0 p  and 1 q  .5) The results of Case IV of Sekulic et al. (2011a) still do not satisfy the algebraic Equation ( 21).After a careful revision, we have shown that the correction of this case should be in the form: It is easy to see that the corrected values of (15) satisfy the algebraic Equation ( 21) of Sekulic et al. (2011a).
From the values of Equation ( 15) we deduce that On comparing the corrected result (17) with our result 1 ( , ) u x t of 'Exact traveling wave solutions of Equation (1) for Case 3', we conclude that they are equivalent if 0 p  and 1 q  .7) The results of case VI in Sekulic et al. (2011a) do not also satisfy Equation ( 21).After a careful revision, we have shown that the correction of this case should be in the form: It is easy to see that the corrected values of (18) satisfy the algebraic Equation ( 21) of Sekulic et al. (2011a).
From the values of Equation ( 18) we deduce that From these discussions we deduce that our results in the present article are new and recover the well-known results obtained in Sekulic et al. (2011a) after its corrections obtained above.
maximal capacitance of the ER.The physical details of the derivation of Equation (

Figure 3 .
Figure 3.The plot of 1 ( , ) u x t of case 2, where

Figure 5 .
Figure 5.The plot of 1 ( , ) u x t of case 3, where case IV inSekulic et al. (2011a) should be rejected.6)The results of Case V inSekulic et al. (2011a) also do not satisfy the algebraic Equation (21).After a careful revision, we have shown that the correction of this case should be to see that the corrected values of (16) satisfy the algebraic Equation (21) ofSekulic et al. (2011a).Hence with replacing  by 2  in Equation (13), the result (26) ofSekulic et al. (2011a) should be rewritten in the corrected form: case VI in Sekulic et al. (2011) should be rejected.