Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Boundary Value Problems

In this article, the variational iteration algorithm-I with an auxiliary parameter (VIA-I with AP) is elaborated to initial and boundary value problems. The effectiveness, absence of difficulty and accuracy of the proposed method is remarkable and its tractability is well suitable for the use of these type of problems. Some examples have been given to show the effectiveness and utilization of this technique. A comparison of variational iteration algorithm-I (VIA-I) along VIA-I with AP has been carried out. It can be seen that this technique is more appropriate than as VIA-I.


Introduction
The study of nonlinear and physical sciences presents various numerical and analytical techniques. Most of these techniques have deficiencies in connection with where ( ) η λ is a general Lagrange multiplier [19].
Applying the variation δ on the one side as well as the other side of Equ.
(2) with respect to ( ), where ( ) can be identified using optimality conditions. Let k approach to infinity. An exact solution may be written as In short, the variational iteration algorithm-I for (1) is ... , 3 , 2 , 1 , 0 , , guess e appropriat an is This algorithm is called variational iteration algorithm-I (VIA-I ) and can be used for finding the accurate solution of various problems. The modification of this method has been presented in a lot of problems, arising in numerous fields of applied sciences and engineering [20][21][22][23][24][25][26][27], to get the approximate and exact solution of nonlinear problems.

VIA-I with an Auxiliary Parameter
In this section, an auxiliary parameter, say h, is suggested to use with VIM-I, which will improve the accuracy and efficiency of algorithm. So equation (5) involving h may be written as , guess initial an being This algorithm is said to be VIA-I with an auxiliary parameter. An important characteristic of this method is that, it has a lesser size of calculation, nor tough to investigate but have a capability to approximate the solution precisely in domain of wide range. Interested readers are encouraged to study the convergence of this technique in [28].

Applications
In order to show the applicability of the VIA-I with AP, we solve nonlinear and linear partial differential equations. These equations are discussed very much in literature, that is why we are choosing to solve these examples.

Burgers' Equation
Consider a homogeneous nonlinear PDE [29,30]: with conditions ( ) Taking the variation δ w.r.t ( ) t x u k , on both sides of equation (9) Ignoring the restricted terms The stationary conditions are: we obtain the value of ( ) η λ which is ( ) Putting this value of ( ) η λ in equation (9), other iterations can be find by using (10), The absolute error of ( ) is shown in Figure 1. In Figure 1, it is obvious that variational iteration algorithm-I diverges for big values of t.
the following iterations are listed using relation (11) , , 6 h t x u Let the residual function may be defined as So, the square of residual function for 10th order approximation w.r.t h for The least result will occur at .
Hence, the absolute error between approximate and exact solutions is shown in Figure 2. Comparing Figure 1 and Figure 2, it is clear that VIA-I with auxiliary parameter gives better results as compared to VIA-I.
Numerical comparison in both the approximate and exact solutions of both algorithms is given in Table 1. Table 1. Comparison of absolute errors for 6th order approximation by VIA-I and VIA-I WITH AP.

A Nonlinear PDE
Consider the homogeneous nonlinear PDE [29,30]: with conditions First, we solve this nonlinear PDE by VIA-I.
So, the correction functional for (14) becomes Taking the variation δ w.r.t ( ) t x u k , on both sides of equation (16) Ignoring the restricted terms we get the value of ( ) η λ which is ( ) Putting this value of ( ) η λ in equation (16), , 120 120 120 60 20 5 ,  The absolute error between approximate and exact solutions of is shown in Figure 3. In Figure 3, it is obvious that variational iteration algorithm-I diverges for big values of t.
Next, the recurrence relation for VIA-I involving an auxiliary parameter is ).
, , 10 h t x u Let, the residual function may be defined as The square of residual function for 10th order approximation w.r.t h for The least result will occur at . 1542483790 the absolute error between approximate and exact solution is shown in Figure 4. Comparing Figure 3 and Figure 4, it is clear that VIA-I involving auxiliary parameter reveals better performance rather than VIA-I . So, a numerical comparison of absolute errors for both methods are shown in Table  2. Table 2. A comparison of absolute errors for 10th order approximation by VIM-AI and VIA-I with auxiliary parameter.

A Linear PDE
Consider the homogeneous linear PDE [29,30]: First, we solve this linear PDE by variational iteration algorithm-I.
So the correction functional for equation (21) is Taking the variation δ w.r.t ( ) t x u k , on both sides of the equation (23), we get Ignoring the restricted terms, Hence, the iterative scheme after using the value of ( ), η λ may be written as The least result will occur at . 0 . 1 = h Consider this value in equation (25), we acquired

Conclusions
In this paper, we have studied and analyzed two iterative methods, VIA-I with an auxiliary parameter and VIA-I. The presence of the elements of flexibility and capability in solutions of nonlinear equations is the very basic characteristic of VIA-I. One of the key characteristics of this method is that, exact solution or approximate solution of better accuracy can be gained by only a few iterations. Its other properties include the simple procedure of solution, acceptable results and above all, it can be practically utilized to a great number of nonlinear problems. The use of an auxiliary parameter h in the VIA-I, the most favorable choice of it can remarkably improve the accuracy. The numerical and graphical results show that VIA-I with AP is a perfect and dependable technique for the solution of nonlinear and linear problems. The comparison shows that VIM-I including an auxiliary parameter presents efficient results in large domains as compared to VIA-I.