The Semianalytical Solutions for Stiff Systems of Ordinary Differential Equations by Using Variational Iteration Method and Modified Variational Iteration Method with Comparison to Exact Solutions

The Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) are used to find solutions of systems of stiff ordinary differential equations for both linear and nonlinear problems. Some examples are given to illustrate the accuracy and effectiveness of these methods. We compare our results with exact results. In some studies related to stiff ordinary differential equations, problemswere solved byAdomianDecompositionMethod andVIMandHomotopyPerturbationMethod.Comparisons with exact solutions reveal that the Variational Iteration Method (VIM) and the Modified Variational Iteration Method (MVIM) are easier to implement. In fact, these methods are promising methods for various systems of linear and nonlinear stiff ordinary differential equations. Furthermore, VIM, or in some cases MVIM, is giving exact solutions in linear cases and very satisfactory solutions when compared to exact solutions for nonlinear cases depending on the stiffness ratio of the stiff system to be solved.

In this paper, Variation Iteration Method (VIM) and Modified Variation Iteration Method (MVIM) are proposed to obtain exact solutions of a stiff system of ODEs.This numerical scheme is based upon application of Variational Theory and is capable of finding the exact solutions of many linear differential equations.
The purpose of this paper is to show the merits of VIM and MVIM in solving some stiff systems of ordinary differential equations.The Variation Iteration Method is useful for obtaining exact and approximate solutions of linear and nonlinear differential equations.There is no requirement for linearization or discretization and, hence, large computational work and round-off errors are avoided.The availability of computer symbolic packages gives a mathematical tool to perform some complicated manipulations and to carry out some modifications on a method for a specific problem easily.The results of the Variation Iteration Method (VIM) and Modified Variation Iteration Method (MVIM) are compared with the exact solutions of the problems to show the efficiency of these methods.

Variation Iteration Method
In 1978, Inokuti et al. [26] introduced a general Lagrange multiplier method to solve nonlinear problems.In this method the solution of a mathematical problem with a linearization assumption is used as an initial approximation or trial function.To illustrate this method, consider the following form of a differential equation: where  is a linear operator,  is a nonlinear operator, and () is an inhomogeneous term.
Also as it was indicated by Ramos [27], Inokuti et al. 's Variational Iteration and also He's Variation Iteration Method can be derived by means of Adjoint Operators, Green's Function, Integration by Parts, and the method of Weighted Residuals.
Based on this basic VIM and MVIM formulation knowledge, there are three different VIM algorithms given in [28]: Algorithm 2.
Further details of these iteration procedures with examples can be found in [27].

Stiffness of a System of Ordinary Differential Equations
For a given system of ordinary differential equations, stiffness means a big difference in the time scales of the components in the vector solution.Some of well-known numerical methods and their procedures which are quite satisfactory in general can work unsatisfactorily on stiff problems.For a general formulation for stiff problems, where f(, y) is defined and continuous in a region  ⊂ [, ], that is either singular or stiff or both.The definition of stiffness is given in [15] and in [16] as follows.
The Initial Value Problem of ( 6) is said to be stiff over the finite interval for every  ∈ [, ], the eigenvalues {  (),  = 1, 2, 3, . . ., } of the Jacobian matrix satisfy the following conditions: where   are the eigenvalues of the Jacobian of the system.

Applications
In this section, Variation Iteration Method (VIM) and Modified Variational Method (MVIM) are applied to various stiff systems of ordinary differential equations.

Problem 1.
The stiff differential equation of second order can be considered as a first example of this study elaborated in [29].The problem is as follows: with initial conditions Its exact analytical solution is () =  −19 +  − .Below, this second order ordinary differential equation can be written as a system of first order ordinary differential equations and the stiffness ratio for this system can be calculated as  = 19 by using (7) and one can obtain the following system.
4.1.1.VIM Iteration Formulation.VIM iteration formulation for this system can be written as follows: Their Lagrange multipliers can be calculated as  1 () = −1 and  2 () = − 20(−) .As an exemplary purpose, first and second iterations are given as follows for VIM formulation.
For  = 0, For  = 1, By using the above written formulations, we can get the following approximations as 4.1.2.MVIM Iteration Formulation.MVIM iteration formulation for this system can be written as follows: For  = 0, For  = 1, or as computed iterations as follows: In the interval of 0 ≤  ≤ 1 VIM and MVIM results are obtained for the 15th iteration.These results are compared with the exact result of the stiff ordinary differential equation for error analysis purpose in Tables 1 and 2.

Problem 2.
This stiff ordinary differential equations system example is presented in [9] and has the following form: with initial conditions ) .
The stiffness ratio  of the system is  = 5.393540395 and can be easily found by using (7).The exact solutions of this stiff ordinary differential equations system can be obtained by Laplace transform method as follows:

VIM Iteration Formulation.
We can construct the following correction functionals for vector ( () V() ) as follows: where  1 and  2 are general Lagrange multipliers and ũ and Ṽ denote restricted variations.In this respect, we have ũ  = 0 and δV  = 0.In order to construct correction functional, ( 24) should be elaborated as follows: From here, by using stationary conditions obtained from (25), following iteration formulas, and general Lagrange multipliers can be obtained as follows: for this problem.As an example, first and second VIM iterations of the problem given in ( 21)-( 22) can easily be found by using ( 26)-( 27) together.
For  = 0, or, as a computed result for this first iteration, For  = 1, or, as a computed result for this second iteration, VIM iteration of this problem is conducted up to 15 iterations and results and related errors are given in Tables 3, 4, 5, and 6 for comparison purposes.
and general Lagrange multipliers can be obtained as follows: for this problem.As an example, first and second MVIM iterations of the problem given in ( 21)-( 22) can easily be found by using (32)-(33) together.
For  = 0, or, as a computed result for this first iteration, For  = 1, or, as a computed result for this second iteration, MVIM iteration of this problem is conducted up to 15 iterations and results and related errors are given in Tables 3-6 for comparison purposes.

Problem 3.
Let us consider the following stiff ordinary differential equations system with stiffness ratio as  = 1000 as an our next example which is given in [30] (   () with initial conditions as follows: with exact solutions: (40)

VIM Iteration Formulation.
VIM iteration formulation for this system can be written as follows: Their Lagrange multipliers can be calculated as  1 () = − −998(−) and  2 () = − 1999(−) .As an exemplary purpose, first and second iterations are given as follows for VIM formulation.
The stiffness ratio of the system () is equal to 1000.
For  = 0, For  = 1, By using the above written formulations, we can get the following approximations as

MVIM Iteration Formulation.
MVIM iteration formulation for this system can be written as follows: (47b) or as computed iterations as follows: The results and related error analysis in the interval of 0 ≤  ≤ 0.001 are obtained by using VIM and MVIM.The results are given in Tables 7, 8, 9, and 10.

Problem 4.
Current stiff ordinary differential equations system example is elaborated in [8] and has the following form: () = 0.01 − (0.01 +  () + V ()) × ( 2 () + 1001 () + 1001) , with initial conditions are obtained with general Lagrange multipliers of this problem Furthermore, by using (7), it can be calculated that the Stiffness Ratio  of this current nonlinear ordinary differential equations system is 102312, so  = 102312 for this system.It should be noted that this current  value is much higher than the previous example's  value.For  = 0, the first iteration will be as follows: or, as a computed result for this first iteration, For  = 1, the second iteration will be as follows: × [  1 () − 0.01 or, as a computed result for this second iteration, (58) Here again the same general Lagrange multipliers are used for calculations, for example, For  = 0, or, as a computed result for this first iteration,

Discussion and Conclusions
In this research, we obtain numerical solutions for four examples of stiff systems of ordinary differential equations.In order to analyze the efficiency of the two methods, namely, VIM and MVIM, in comparison with exact solutions, the absolute errors are calculated for different  values.
It can be seen that there are very good agreement between the results of VIM, results of MVIM, and exact solutions for these examples, which confirms the validity of the two methods for these problems.However, when solutions of VIM and MVIM are compared, it can be said that MVIM is providing better results in general.
These methods are very simple and do not require very small time steps for mildly stiff linear ordinary differential equations to obtain results in which error accumulation gets bigger as the time gets bigger for finding the solution.
and MVIM iterations of this problem are conducted up to 2 iterations.The results and related errors are given in Tables11, 12, 13, and 14 for comparison purposes.

Table 1 :
Values for () for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) compared with exact values of ().

Table 2 :
Absolute error values for () for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM).

Table 3 :
Values for () for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) compared with exact values of ().

Table 4 :
Values for V() for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) compared with exact values of V().

Table 5 :
Absolute Error Values for () for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM).

Table 6 :
Absolute Error Values for V() for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM).

Table 7 :
Values for () for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) compared with exact values of ().

Table 8 :
Values for V() for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) compared with exact values of V().

Table 9 :
Absolute Error Values for () for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM).

Table 10 :
Absolute Error Values for V() for the fifteenth iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM).

Table 11 :
Values for () for the second iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) compared with exact values of () in 0.0001 ≤  ≤ 0.1.

Table 12 :
Values for V() for the second iteration by Variational Iteration Method (VIM) and Modified Variational Iteration Method (MVIM) compared with exact values of V() in 0.0001 ≤  ≤ 0.1.