Controlled Variational Iteration Method for Bratu Equation Arising in Electro-Spun Organic Nanofibers Elaboration

In this paper, an effective formulation of the variational iteration method is suggested for solving Bratu equation arising in electro-spinning. The suggested formulation depends on embedding a nonzero auxiliary parameter that controls the solution convergence region. An alternative formulation of the Bratu equation is suggested as well. The proposed formula eliminates the complexity that appears when solving using the standard variational iteration algorithms illustrated in [1,2] without approximating the exponential term. A suitable choice of the auxiliary parameter results in an accurate approximation compared with the approximation of the standard variational iteration method.


Introduction
Electro-spinning is a process for elaborating Nano fibers by driving a fluidified polymer through a spinneret with the aid of an electric field. In literature, there are many useful models that describe electro-spinning process e.g., [3][4][5]. Among which Wan-Guo-Pan model is the most famous model driven in [1,6], which reads 2 2 0. This famous equation is called by Bratu equation [7]. Here, where u is the jet velocity in the spinning process.
The authors in [1,2] suggested the following variational iteration formula by the so-called enhanced variational iteration method in [1] which is the same that suggested in [2] using the standard variational iteration method (VIM) firstly proposed in [8].
A suitable choice of the initial guess 0 ( ) v z that based on the initial conditions results in an accurate solution as n increases. The standard variational iteration formula shown in Eq. (2) that used by other authors [9,10]  v v v e = + + + + . This approximation is used in [1] for solving the considered example but not mentioned by the authors. The second drawback is related to the VIM itself which is summarized in the lack the method of a way to improve and control the solution accuracy. It's worth to note that the model of Eq. (1) can be effectively solved by many other methods. Recall, for example, the homotopy analysis method as in [11].
In this paper, we suggest a new formulation of Bratu equation to eliminate the first draw back. A controlled variational iteration method (CVIM) will be considered to eliminate the second draw back as well.

New Formulation of Bratu Equation
Multiplying Eq. (1) by dv dz and integrating from a to z, it can be re-formulated as

Controlled Variational Iteration Method
In order to increase and control the convergence region of the VIM series solution, we propose embedding a nonzero auxiliary parameter χ in the standard variational iteration algorithm which will play an important role in improving and control the solution convergence region and accuracy. So, the controlled variational iteration algorithm can be read as: The suitable initial guess 0 From Eq. (4), it can be noticed that the CVIM will provide a family of solutions, dependent upon the convergence control parameter χ . If 1 χ = , the CVIM is exactly the standard VIM. To obtain an accurate approximation to the considered problem, an optimal value of χ must be found. Firstly, the valid region of χ that achieves the solution convergence can be obtained via the χ curve as follows. n v a χ + versus χ contains a horizontal line segment which corresponds to the valid region of χ . This is due to that all convergent series given by different values of the auxiliary parameter χ converge to its exact value. So, if the solution is unique, then all of these series solutions converge to the same value and therefore there exists a horizontal line segment in the curve, all of these possible values of χ construct a set R χ for the convergence-control parameter. Secondly, a more accurate approximation can be obtained by assigning χ an optimal value. The optimal value χ * of the control parameter can be estimated by minimizing the averaged absolute residual error ( ) n χ ∆ defined as: where z ∆ is the resolution. The low value of z ∆ leads to obtain a more accurate value of χ * and hence the solution accuracy will be increased.
It's worth to note that the considered formulation is firstly proposed by Turkyilmazoglu in [12].

Numerical Examples
To demonstrate the reliability and efficiency of the proposed CVIM, two initial/boundary value problem of Bratu equation are considered. The results of CVIM and standard VIM are compared with the exact solutions at the same number of iterations to demonstrate the accuracy of the considered formulation over the classical VIM. .

Consider the BratuInitial Value Problem [1]
It's noticeably that the new formulation of the Bratu equation, contributed a lot in making the integration and calculations easy.
To find the valid region R χ for the control parameter χ , the χ curves at 0.4, 0.6, 0.8 z = are drawn in Fig. 1 whose exact solution is ( ) v z χ and, can be easily obtained as: In order to obtain more accurate solution, more iteration must be calculated. We used the six iterations approximate solution 7 ( , , ) v z χ α to find the values of the unknown parameter α and the optimal value of the control parameter χ . Firstly, we find a relation between α and χ using the condition (1) 0 v = and then plot this relation implicitly i.e., plotting the relation 7 (1, , ) 0 v χ α = implicitly. The relation between α and χ is plotted in Fig. 3.

Conclusion
In this paper, new formulation of Bratu equation that appears in electro-spinning process is presented. The proposed presentation eliminates the complexity that may appear when solving using the standard VIM. Moreover, the controlled VIM is proposed as well in order to increase the accuracy and control the convergence of the solution. The auxiliary parameter χ plays an important role in increasing the overall solution accuracy. From the obtained numerical results that illustrated graphically, one can conclude that the proposed CVIM display a high accuracy and reliability when compared with the standard VIM.