# Preface

This section presents derivation of simple pendulum equation.

# Variational Iteration Method

The variational iteration method (VIM) was developed by Ji-Huan He in 2000--2007. This method is preferable over numerical methods as it is free from rounding off errors as it does not involve discretization and does not require large computer power or memory. The VIM gives rapidly convergent successive approximations of the exact solution if such a solution exists; otherwise, a few approximations can be used for numerical purposes. Many researches in variety of scientific fields applied this method and showed the VIM has many merits and to be reliable for a variety of scientific application.

To illustrate the basic concept of the He’s VIM, consider the following general nonlinear equation:

$L\left[ u(t) \right] = N\left[ u(t) \right] + g(t) ,$
where L is the linear differential operator of order m, N is the nonlinear operator, and g is the input (known) function. He has modified the general Lagrange multiplier method to an iteration method called as correction functional. The basic character of the method is to construct a correction functional for the aforementioned equation, which reads as
$u_{n+1} (t) = u_n (t) + \int_0^t \lambda (s) \left\{ L\left[ u_n (s) \right] - N\left[ \tilde{u}_n (s) \right] - g(s) \right\} {\text d}s ,$
where λ(s,t) is so called a general Lagrange multiplier, which can be identified optimally via variational theory. The subscript n denotes the n-th approximation, and $$\tilde{u}_n$$ is a restricted variation., i.e., $$\delta\tilde{u}_n = 0 .$$ Having λ obtained, an iteration formula should be used for the determinantion of the successive approximations un+1(t) of the solution u(t). The initial approximation u0(t) can be arbitrary function; however, it is usually selected based on the auxiliary conditions (such as the initial or boundary conditions). Consequently, the solution is given by
$u(t) = \lim_{n\to\infty}\,u_n (t) .$

The main problem with VIM is the determination of the Lagrange multiplier, and there are known many forms of them. It could be achieved by making the correction functional stationary. Taking variation with respect to the variable un, noticing δun(0)=0, we get

\begin{eqnarray*} \delta u_{n+1} (t) &=& \delta u_n (t) + \delta \int_0^t \lambda \left\{ L\left[ u_n (s) \right] - N\left[ \tilde{u}_n (s) \right] - g(s) \right\} {\text d}s \\ &=& \delta u_n (t) + \left. \lambda\,\delta u_n^{(m-1)} \right\vert_{s=t} - \left. \lambda ' \,\delta u_n^{(m-2)} \right\vert_{s=t} + \cdots + \left. (-1)^{m-1} \lambda^{(m-1)} \delta u_n \right\vert_{s=t} + \int_0^t L^{\ast} \left[ \lambda \right] \delta u_n \,{\text d}s =0 , \end{eqnarray*}
for all variations δun. This yields the following equation
$L^{\ast} \left[ \lambda \right] =0 ,$
subject to some auxiliary conditions on the diagonal s = t. Here L* is adjoint differential operator for L. In other words, if L is a m-th order linear differential operator, we need to find an extremum form the following functional
$\int_0^t \lambda \,L\left[ u_n (s) \right] {\text d}s = \int_0^t \lambda \,f \left( s, u_n (s) , u'_n (s) , \ldots , u_n^{(m)} \right) {\text d}s .$
So we consider the general functional
$I = \int_0^t F \left( s, y(s) , y' (s) , \ldots , y^{(m)} \right) {\text d}s$
that attains an extremum on solutions of the generalized Euler--Lagrange equation (or simply the Euler equation):
$\frac{\partial F}{\partial y} - \frac{\text d}{{\text d}s} \left( \frac{\partial F}{\partial y'} \right) + \frac{{\text d}^2}{{\text d}s^2} \left( \frac{\partial F}{\partial y''} \right) - \cdots + (-1)^m \frac{{\text d}^m}{{\text d}s^m} \left( \frac{\partial F}{\partial y^{(m)}} \right) =0 .$
Using the above equation, we see that extremum of the variational functional is given as follows
$\frac{\partial \left( \lambda \,f \right)}{\partial y} - \frac{\text d}{{\text d}s} \left( \frac{\partial \left( \lambda \,f \right)}{\partial y'} \right) + \frac{{\text d}^2}{{\text d}s^2} \left( \frac{\partial \left( \lambda \,f \right)}{\partial y''} \right) - \cdots + (-1)^m \frac{{\text d}^m}{{\text d}s^m} \left( \frac{\partial \left( \lambda \,f \right)}{\partial y^{(m)}} \right) =0 .$
Solving these equations is sufficient to determine the Lagrange multipliers. We show how it works in a series on examples.

Example: Consider the IVP for the following Riccati equation

$y' = 3\,y - y^2 + 2\,x, \qquad y(0) =0.$
First, we write the variational iteration sequence:
$y_{n+1} (x) = y_n (x) + \int_0^x \lambda (x,s) \left\{ y'_n (s) - 3\,y_n (s) + \tilde{y}_n^2 - 2\,s \right\} {\text d}s , \qquad n=0,1,2,\ldots .$
Taking variation, we obtain
\begin{eqnarray*} \delta y_{n+1} (x) &=& \delta y_n (x) + \delta \int_0^x \lambda \left\{ y'_n (s) - 3\,y_n (s) + \tilde{y}_n^2 - 2\,s \right\} {\text d}s \\ &=& \delta y_n (x) + \delta \int_0^x \lambda \left\{ y'_n (s) - 3\,y_n (s) \right\} {\text d}s \\ &=& \delta y_n (x) + \left. \lambda\,\delta y_n (s) \right\vert_{s=0}^{s=t} - \int_0^t \left[ \lambda ' + 3\,\lambda \right] \delta y_n (s) \, {\text d}s . \end{eqnarray*}
This gives us the equations for determination of the Lagrange multiplier:
$\begin{split} \lambda ' (s) + 3\,\lambda (s) &=0 , \\ 1 + \left. \lambda (s) \right\vert_{s=t} &= 0. \end{split}$
So the multiplier can be identified as
$\lambda (x,s) = - e^{3(x-s)} .$
Then the iteration formula becomes
$y_{n+1} = y_n - \int_0^x e^{3(x-s)} \left\{ y'_n (s) - 3\,y_n (s) + y_n^2 - 2\,s \right\} {\text d}s , \qquad n=0,1,2,\ldots .$
Using previously defined subroutine seriesDSolve, we find series solution of the given problem:
seriesDSolve[ y'[x] == 2*x + 3*y[x] - (y[x])^2, y, {x, 0, 10}, {y[0] -> 0}]
$y = x^2 + x^3 + \frac{3}{4}\,x^4 + \frac{x^5}{4} - \frac{5}{24}\, x^6 - \frac{25}{56}\,x^7 - \frac{187}{448}\, x^8 - \frac{2551}{12096}\, x^9 + \frac{1217}{40320}\, x^{10} + \cdots .$
\begin{eqnarray*} y_1 (x) &=& x^2 - \int_0^x e^{3(x-s)} \left\{ 2s - 3\,s^2 + s^4 - 2\,s \right\} {\text d}s = x^2 + \frac{1}{81} \left( 27\,x^4 -45\,x^2 -30\,x -10 \right) + \frac{10}{81}\, e^{3x} , \\ y_2 &=& y_1 - \int_0^x e^{3(x-s)} \left\{ y'_1 -3\,y_1 + y_1^2 -2\,s \right\} {\text d}s \approx x^2 - \frac{x^3}{3} - \frac{x^4}{4} - \frac{3}{20}\, x^5 - \frac{3}{40}\, x^6 + \frac{253}{840}\, x^7 + \cdots . \end{eqnarray*}
So we see that calculations become messy very shortly. We ask Mathematica for help.
y1[t_] = t^2 - Integrate[(s^4 - 3*s^2)*Exp[3*t - 3*s], {s, 0, t}];
y2[t_] = t^2 - Integrate[(D[y1[s], s] - 3*y1[s] + (y1[s])^2 - 2*s)* Exp[3*t - 3*s], {s, 0, t}];
y3[t_] = t^2 - Integrate[(D[y2[s], s] - 3*y2[s] + (y2[s])^2 - 2*s)* Exp[3*t - 3*s], {s, 0, t}];
Expand[y3[t]]
Series[%, {t, 0, 10}]
$y_3 (x) \approx x^2 + x^3 + \frac{3}{4}\,x^4 + \frac{x^5}{4} + \frac{31}{56}\, x^7 + \frac{205}{448}\, x^8 + \frac{4505}{12096} \, x^9 + \cdots ,$
which gives correct answer up to the power 6. Then we plot the third approximation y3 and the true solution.
sol = (y[x] /.
DSolve[{y'[x] == 2*x + 3*y[x] - (y[x])^2, y[0] == 0}, y[x], x])[[1]];
Plot[{Callout[sol, "exact"], Callout[y3[x], "three-term VIM approximation"]}, {x,0,1.2}, PlotStyle->Thick]

We demonstrate a modified variational iteration method that leads in our case to the recurrence:

$y_{n+1} (t) = y_n (t) + 3 \int_0^t \left( y_n (s) - y_{n-1} (s) \right) {\text d}s - \int_0^t \left( y_n^2 (s) - y_{n-1}^2 (s) \right) {\text d}s , \qquad n=1,2,\ldots ;$
where $$y_{-1} = 0, \ y_0 = 0$$ (from the initial condition) and
$y_{1} (t) = y_0 + \int_0^t \left[ 3 \left( y_0 - y_{-1} \right) - \left( y_0^2 - y_{-1}^2 \right) + 2s \right] {\text d}s = t^2 .$
Therefore, we can obtain the following successive approximations:
\begin{eqnarray*} y_2 (t) &=& t^2 + \int_0^t \left[ 3 \left( y_1 - y_{0} \right) - \left( y_1^2 - y_{0}^2 \right)\right] {\text d}s = t^2 + t^3 - \frac{t^5}{5} , \\ y_3 (t) &=& t^2 + t^3 + \int_0^t \left[ 3 \left( y_2 - y_{1} \right) - \left( y_2^2 - y_{1}^2 \right)\right] {\text d}s = t^2 + t^3 + \frac{3}{4}\,t^4 - \frac{t^5}{5} - \frac{13}{30}\, t^6 + \frac{t^8}{20} + \frac{2}{45}\,t^9 - \frac{t^{10}}{275} , \\ y_4 (t) &=& t^2 + t^3 + \int_0^t \left[ 3 \left( y_3 - y_{2} \right) - \left( y_3^2 - y_{2}^2 \right)\right] {\text d}s = t^2 + t^3 + \frac{3}{4}\,t^4 + \frac{t^5}{4} - \frac{13}{30}\, t^6 - \frac{19}{35}\, t^7 - \frac{107\,t^8}{560} + \frac{41}{432}\,t^9 + \frac{111}{700} \, t^{10} + \cdots . \end{eqnarray*}
y2[t_] = t^2 + t^3 - t^5 /5
y3[t_] = y2[t] + Integrate[3*y2[s] - 3*s^2 - (y2[s])^2 + s^4, {s, 0, t}]
y4[t_] = y3[t] + Integrate[3*y3[s] - 3*y2[s] - (y3[s])^2 + (y2[s])^2, {s, 0, t}]

Example: Consider the one dimensional steady-state heat conduction problem

$\frac{{\text d}^2 T}{{\text d} x^2} + T(x) + 2x =0, \qquad (0< x < \ell , \qquad T(0) = 4, \quad T(\ell ) =3 .$
Here T is the temperature within the rod of length ℓ, and we set for simplicity ℓ=2. Its correction functional can we written as
$T_{n+1} (x) = T_n (x) + \int_0^x \lambda \left\{ T''_n (s) + T_n (s) + 2s \right\} {\text d}s , \qquad n=0,1,2,\ldots .$
Taking variation with respect to the dependent avriable Tn, and noticing that &deltaTn(0)=0, we have
\begin{eqnarray*} \delta T_{n+1} (x) &=& \delta T_n (x) + \delta \int_0^x \lambda \left\{ T''_n (s) + T_n (s) + 2s \right\} {\text d}s \\ &=& \delta T_n (x) + \left. \lambda \,\delta T'_n (s) \right\vert_{s=x} - \left. \lambda ' \,\delta T_n (s) \right\vert_{s=x} + \int_0^x \left[ \lambda '' + \lambda \right] \delta T_n (s) {\text d}s \end{eqnarray*}
for all variations δTn and δT'n. This yields the following equations:
$\begin{split} \lambda '' + \lambda &= 0, \\ \left. \lambda (s) \right\vert_{s=x} =0 , \\ \left. 1-\lambda ' (s) \right\vert_{s=x} =0. \end{split}$
Therefore, the lagrange multiplier becomes
$\lambda (x,s) = \sin \left( s-x \right) ,$
which gives us the following iteration formula
$T_{n+1} (x) = T_n (x) + \int_0^x \sin \left( s-x \right) \left\{ T''_n (s) + T_n (s) + 2s \right\} {\text d}s , \qquad n=0,1,2,\ldots .$
We choose the initial approximation as a function that satisfies the boundary conditions:
$T_{0} (x) = T(0) \,\frac{\ell -x}{\ell} + \frac{x}{\ell} \, T(\ell ) = 2 \left( 2-x \right) + \frac{3}{2}\, x = 4 - \frac{x}{2} .$
Now we use iteration
\begin{eqnarray*} T_1 (x) &=& T_0 (x) + \int_0^x \sin \left( s-x \right) \left\{ T''_0 (s) + T_0 (s) + 2s \right\} {\text d}s = -2x + 4\,\cos x + \frac{3}{2}\, \sin x , \\ T_2 (x) &=& T_1 (x) + \int_0^x \sin \left( s-x \right) \left\{ T''_1 (s) + T_1 (s) + 2s \right\} {\text d}s = T_1 (x) . \end{eqnarray*}
So the iteration stops after first step. We compare it with the exact solution.
DSolve[{y''[t] + y[t] + 2*t == 0, y[0] == 4, y[2] == 3}, y[t], t] // Flatten
exact[t_] = y[t] /. %
$T(x) = -2x + 4\,\cos x - 4\,\cos 2\,\sin x + 7\,\sec 2\,\sin x .$
T0[x_] = 4 - x/2;
T1[x_] = T0[x] + Integrate[Sin[s - x]*(D[T0[s], s, s] + T0[s] + 2*s), {s, 0, x}]
Plot[{Callout[exact[x], "exact"], Callout[T1[x], "VIM approximation"]}, {x,0,2}, PlotStyle->Thick]

So we see that the linear function as the initial approximation does not work. Instead, we use its complementary function $$T_0 (x) = A\,\cos x + B\,\sin x .$$ Then

$T_{1} (x) = A\,\cos x + B\,\sin x + \int_0^x \sin \left( s-x \right) \left\{ T''_0 (s) + T_0 (s) + 2s \right\} {\text d}s = A\,\cos x + \left( B+2 \right) \sin x -2x .$
By imposing the boundary conditions, we get A = 4 and $$B= 7\,\csc 2 - 4\,\cot 2 -2 .$$ So
$T_{1} (x) = 4\,\cos x + \left( 7\,\csc 2 -4\,\cot 2 \right) -2\,x ,$
Next iteration provides a function that is far away from the true solution. ■

Example: Consider the IVP for harmonic oscillator

$\ddot{y} + \omega^2 y = f(t) \qquad\mbox{with}\quad f(t) = A\,\sin \omega t +B\,\cos t ,$
subject to the initial conditions
$y(0) =1, \qquad \dot{y} (0) =-1.$
Its correction functional can be written as follows
$y_{n+1} (t) = y_n (t) + \int_0^t \lambda \left\{ y''_n (s) + \omega^2 \,y_n (s) - f(s) \right\} {\text d}s , \qquad n=0,1,2,\ldots .$
Making the above correction functional stationary, and noticing that δy(0) = 0, we get
\begin{eqnarray*} \delta y_{n+1} (t) &=& \delta y_n (t) + \delta \int_0^t \lambda \left\{ y''_n (s) + \omega^2 \,y_n (s) - f(s) \right\} {\text d}s \\ &=& \delta y_n (t) + \left. \lambda (s)\,\delta y'_n (s) \right\vert_{s=t} - \left. \lambda ' (s)\, \delta y_n (s) \right\vert_{s=t} + \int_0^t \left[ \lambda '' + \omega^2 \lambda \right] \delta y_n (s) \,{\text d}s . \end{eqnarray*}
From the right hand side, it follows
$\begin{split} \lambda '' + \omega^2 \lambda &= 0 , \\ \left. \lambda (s) \right\vert_{s=t} &= 0 , \\ \left. 1 - \lambda ' (s)\right\vert_{s=t} &= 0 . \end{split}$
The Lagrange multiplier can be readily identified as
$\lambda (t, s) = \frac{1}{\omega}\,\sin \omega (s-t) .$
With this in hands, we obtain the following iteration formula
$y_{n+1} (t) = y_n (t) + \frac{1}{\omega} \int_0^t \sin \omega (s-t) \left\{ y''_n (s) + \omega^2 \,y_n (s) - f(s) \right\} {\text d}s , \qquad n=0,1,2,\ldots .$
Now we consider the homogeneous equation corresponding to the given driven equation $$\ddot{y} + \omega^2 y =0$$ subject to the given initial conditions; its solution, called the complementary function, can be chosen as the initial approximation
$y_{0} (t) = C_1 \cos \omega t + C_2 \sin \omega t = \cos \omega t - \frac{\sin \omega t}{\omega} ,$
where C1 and C2 are such constants to satisfy the given initial conditions, y(0) = 1 and y'(0) = -1. Then the iteration formula gives
\begin{eqnarray*} y_1 (t) &=& y_0 (t) - \frac{1}{\omega} \int_0^t \sin \omega (s-t) \left[ A\,\sin \omega s + B\,\sin \omega s \right] {\text d}s \\ &=& C_1 \cos \omega t + C_2 \sin \omega t - \frac{A}{2} \left[ t\,\cos \omega t - \frac{\sin \omega t}{\omega} \right] + \frac{B\,\omega}{\omega^2 -1} \left[ \cos \omega t - \cos t \right] , \end{eqnarray*}
which is the general solution of the given differential equation (harmonic oscillator).

However, is we apply restricted variations to the correction function, then its exact solution can be arrived at only by successive iterations. Considering the corresponding homogeneous differential equation $$\ddot{y} + \omega^2 y =0 ,$$ we rewrite the correction functional as follows:

$y_{n+1} (t) = y_n (t) + \int_0^t \lambda \left\{ y''_n (s) + \omega^2 \,\tilde{y}_n (s) \right\} {\text d}s , \qquad n=0,1,2,\ldots .$
Here $$\tilde{y}_n (s)$$ is considered a restricted variation, then the stationary conditions ($$\delta\tilde{y}_n =0$$ ) of the above correction functional can be expressed as follows
$\begin{split} \lambda '' (s) &=0 , \\ \left. \lambda (s) \right\vert_{s=t} &= 0, \\ \left. 1 - \lambda ' (s) \right\vert_{s=t} &= 0 . \end{split}$
Therefore, the Lagrange multiplier becomes $$\lambda = s-t .$$ This leads to the following iteration formula
$y_{n+1} (t) = y_n (t) + \int_0^t (s-t) \left\{ y''_n (s) + \omega^2 \,y_n (s) - f(s) \right\} {\text d}s , \qquad n=0,1,2,\ldots .$
It we choose the initial iteration as $$y_0 = 1 -t$$ in order to satisfy the initial conditions, we have
\begin{eqnarray*} y_1 (t) &=& y_0 (t) + \omega^2 \int_0^t \left( s-t \right) y_0 (s)\, {\text d}s = 1-t + \omega^2 \left( -\frac{t^2}{2} + \frac{t^3}{6} \right) , \\ y_2 (t) &=& y_1 (t) + \int_0^t \left( s-t \right) \left( y''_1 (s) + \omega^2 y_1 (s) \right) {\text d}s = 1-t + \omega^2 \left( -\frac{t^2}{2} + \frac{t^3}{6} \right) + \omega^4 \left( \frac{t^4}{24} - \frac{t^5}{120} \right) , \\ y_3 (t) &=& y_2 (t) + \int_0^t \left( s-t \right) \left( y''_2 (s) + \omega^2 y_2 (s) \right) {\text d}s = 1-t + \omega^2 \left( - \frac{t^2}{2!} + \frac{t^3}{3!} \right) + \omega^4 \left( \frac{t^4}{4!} - \frac{t^5}{5!} \right) + \omega^6 \left( - \frac{t^6}{6!} + \frac{t^7}{7!} \right) , \\ y_n (t) &=& \sum_{k=0}^n \frac{(-1)^n}{(2k)!}\, \omega^{2k} t^{2k} - \frac{1}{\omega} \,\sum_{k=0}^n (-1)^k \frac{\omega^{2k+1} t^{2k+1}}{(2k+1)!} . \end{eqnarray*}
Thus, we obtaine
$\lim_{n\to\infty}\, y_n (t) = \cos\omega t - \frac{1}{\omega}\,\sin \omega t ,$
which is the exact solution of the corresponding homogeneous equation. ■

Example: Consider the IVP for the pendulum equation

$\ddot{\theta} + \omega^2 \sin \theta =0 , \qquad \theta (0) = \theta_0 , \quad \dot{\theta} (0) =0.$

Example: Consider the IVP for harmonic oscillator ■

Example: Consider the third order differential equation

$u''' + 4\,u' = f(t) , \qquad u(0) =$
1. Abbasbandy S (2007), A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, Journal of Applied Mathematics and Computing, 207:59–63
2. Tamer A. Abassy, New Applications of the Modified Variational Iteration Method, Studies in Nonlinear Sciences, 2012, Vol. 3, No. 2, pp. 49--61, doi: 10.5829/idosi.sns.2012.3.2.246
3. Tamer A. Abassy, Magdy A. El-Tawil, and H. El-Zoheiry, Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and the Padé technique, Computers and Mathematics with Applications, 2007, Vol. 54, pp. 940--954.
4. Alshibly, N. and Alomari, A.K., Explicate series solution for prey-preditor problem, 2010,
5. Chen, Y.M., Meng, G., Liu, J.K., Variational iteration method for conservative oscillators with complicated nonlinearities, Mathematical and Computational Applications, 2010, Vol. 15, No. 5, pp. 802--809.
6. M.A. Fariborzi Araghi, S. Gholizadeh Dogaheh, and Z. Sayadi, The modified variational iteration method for solving linear and nonlinear ordinary differential equations, Australian Journal of Basic and Applied Science, 2011, Vol. 5, No. 10, pp. 406--416.
7. Inan Ates and Ahmet Yildirim, "Comparison between variational iteration method and homotopy perturbation method for linear and nonlinear partial differential equations with the nonhomogeneous initial conditions," Numerical Methods for Partial Differential Equations, 2010, Vol 26, Issue 6, pp. 1581--1593, doi: https://doi.org/10.1002/num.20511
8. Ganji DD, Sadighi A (2006) Application of He’s methods to nonlinear coupled systems of reactions. Int J Nonlinear Sci Numer Simul 7(4):411–418
9. Ganji DD, Jannatabadi M, Mohseni E (2006) Application of He’s variational iteration method to nonlinear Jaulent–Miodek equa- tions and comparing it with ADM. Journal of Applied Mathematics and Computing, 207:35–45
10. Ganji DD, Tari H, Babazadeh H (2007) The application of He’s variational iteration method to nonlinear equations arising in heat transfer. Physics Letters A, Vol. 363, No. , pp. 213--217
11. Ganji DD, Tari H, Jooybari MB (2007) Variational iteration method and homotopy perturbation method for evolution equations. Comput Math Appl 54:1018–1027
12. Ji-Huan He, "Variational iteration method---a kind of non-linear analytical technique: some examples," International Journal of Non-Linear Mechanics, 1999, Vol 34, 699--708.
13. Ji-Huan He, "Variational iteration method for autonomous ordinary differential systems," Applied Mathematics and Computation, 2000, Vol. 114, 115--123.
14. Ji-Huan He, Variational iteration method — Some recent resultsand new interpretations, Journal of Computational and Applied Mathematics, 2007, Vol. 207, pp. 3--17.
15. Ameina S. Nuseir and Mohammad Al-Towaiq, The modified variational iteration method for solving the Impenetrable Agar model problem, International Journal of Pure and Applied Mathematics, 2014, Vol. 96, No 4, pp. 445--456, doi: http://dx.doi.org/10.12732/ijpam.v96i4.3
16. Rafei M, Daniali H, Ganji D,D., Variational iteration method for solving the epidemic model and the prey and predator problem, Applied Mathematics and Computation, 2007, Vol. 186, No. 2, pp. 1701--1709.
17. S. S. Samaee, O. Yazdanpanah, and D. D. Ganji, "New approaches to identification of the Lagrange multiplier in the variational iteration method," Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2914, doi: 0.1007/s40430-014-0214-3
18. Tari H, Ganji D.D., Rostamian M. (2007) Approximate solution of K (2, 2), KdV and modified KdV equations by variational itera- tion method, homotopy perturbation method and homotopy ana- lysis method. Int J Nonlinear Sci Numer Simul 8:203–210
19. Tatari M, Dehghan M (2007), "He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation," Chaos Solitons Fractals, 33(2):671--677.
20. A.-M. Wazwaz (2006), "A study on linear and nonlinear Schrodinger equations by the variational iteration method," Chaos Solitons Fractals, doi: 10.1016/j.chaos.10.009
21. A.-M. Wazwaz (2007), "The variational iteration method for solving tow forms of Blasius equation on a half-infinite domain," Applied Mathemathics and Computation, 2007, Vol. 188, Issue 1, pp. 485--491.
22. A.-M. Wazwaz (2007), The variational iteration method for the exact solution of Laplace equation, Physics Letters A, Vol. 363, No 4, pp. 260--262, doi: 10.1016/j.physleta.2006.11.014
23. A.-M. Wazwaz (2001) The numerical solution of fifth-order boundary value problems by the decomposition method. Journal of Applied Mathematics and Computing, 136:259–270
24. Wazwaz AM (2007) The variational iteration method for solving a system of PDES. Comput Math Appl 54:895–902
25. Wu, B. S., C. W. Lim, and L. H. He, “A new methodfor approximate analytical solutions to nonlinear oscillations ofnonnatural systems,”Nonlinear Dynamics, Vol. 32, 1–13, 2003.