On the Numerical Solution of Differential-Algebraic Equations with Hessenberg Index-3

Many important mathematical models can be expressed in terms of differential-algebraic equations DAEs . Many physical problems are most easily initially modeled as a system of differential-algebraic equations DAEs 1 . Some numerical methods have been developed, using both BDF 1–3 and implicit Runge-Kutta methods 1 , Padé and Chebysev approximations method 4–6 . These methods are only directly suitable for low-index problems and often require that the problem, have special structure. Although many important applications can be solved by these methods, there is a need for more general approaches. There are many new publication in the field of analytical sueveys such as 7–10 . The variational iteration method VIM was developed by He in 11 . The method is used by many researchers in a variety of scientific fields. The method has been proved by many authors 12–16 to be reliable and efficient for a variety of scientific applications, linear and nonlinear as well. The most general form of a DAE is given by


Introduction
Many important mathematical models can be expressed in terms of differential-algebraic equations DAEs .Many physical problems are most easily initially modeled as a system of differential-algebraic equations DAEs 1 .Some numerical methods have been developed, using both BDF 1-3 and implicit Runge-Kutta methods 1 , Padé and Chebysev approximations method 4-6 .These methods are only directly suitable for low-index problems and often require that the problem, have special structure.Although many important applications can be solved by these methods, there is a need for more general approaches.There are many new publication in the field of analytical sueveys such as 7-10 .The variational iteration method VIM was developed by He in 11 .The method is used by many researchers in a variety of scientific fields.The method has been proved by many authors 12-16 to be reliable and efficient for a variety of scientific applications, linear and nonlinear as well.
The most general form of a DAE is given by F t, x, x 0, 1.1 where ∂F/∂x may be singular.The rank and structure of this Jacobian matrix may depend, in general, on the solution x t , and for simplicity we will always assume that it is independent of t.The important special case is of a semiexplicit DAE or an ODE with constraints: This is a special case of 1.1 .The index is 1 if ∂g/∂z is nonsingular, because then one differentiation of 1.2b yields z in principle.For the semi-explicit index-1 DAE we can distinguish between differential variables x t and algebraic variables z t 1 .The algebraic variables may be less smooth than the differential variables by one derivative.In the general case, each component of x may contain a mix of differential and algebraic components, which makes the numerical solution of such high-index problems much harder and riskier.

Special Differential-Algebraic Equations (DAEs) Forms
Most of the higher-index problems encountered in practice can be expressed as a combination of more restrictive structures of ODEs coupled with constraints.In such systems the algebraic and differential variables are explicitly identified for higher-index DAEs as well, and the algebraic variables may all be eliminated using the same number of differentiations.These are called Hessenberg forms of the DAE and are given below.In this paper, the variational iteration method has been proposed for solving differential-algebraic equations with Hessenberg index-3.

Hessenberg Index-1
One has x f t, x, z , 0 g t, x, z .

2.1
Here the Jacobian matrix function g z is assumed to be nonsingular for all t.This is also often referred to as a semi-explicit index-1 system.Semi-explicit index-1 DAEs are very closely related to implicit ODEs.

Hessenberg Index-2
One has x f t, x, z , 2.2a 0 g t, x .

2.2b
Here the product of Jacobians g x f z is nonsingular for all t.Note the absence of the algebraic variables z from the constraints 2.2b .This is a pure index-2 DAE, and all algebraic variables play the role of index-2 variables.

Hessenberg Index-3
One has x f t, x, y, z , y g t, x, y , 0 h t, y .

2.3
Here the product of three matrix functions h y g x f z is nonsingular.
The index of a Hessenberg DAE is found, as in the general case, by differentiation.However, here only algebraic constraints must be differentiated.

He's Variational Iteration Method (VIM)
Consider the differential equation where L and N are linear and nonlinear operators, respectively, and g x is the source inhomogeneous term.In 11 , He proposed the variational iteration method where a correction functional for 3.1 can be written as where λ is a general Lagrange's multiplier, which can be identified optimally via the variational theory and u n as a restricted variation which means δ u n 0. It is to be noted that the Lagrange multiplier λ can be a constant or a function.
The variational iteration method should be employed by following two essential steps.It is required first to determine the Lagrange multiplier λ that can be identified optimally via integration by parts and by using a restricted variation.Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations u n 1 x , n ≥ 0, of the solution u x .The zeroth approximation u 0 can be any selective function.However, using the initial values u 0 , u 0 , and u 0 are preferably used for the selective zeroth approximation u 0 as will be seen later.Consequently, the solution is given by

First-Order ODEs
We first start our analysis by studying the first-order linear ODE of a standard form u p x u q x , u 0 α.

3.4
The VIM admits the use of the correction functional for this equation by where λ is the lagrange multiplier, that in this method may be a constant or a function, and u n is a restricted value where δ u n 0.
Taking the variation of both sides of 3.5 with respect to the independent variable u n we have λ u n dt.

3.9
The extremum condition of u n 1 requires that u n 1 0. This means that the left hand side of 3.9 is 0, and as a result the right hand side should be 0 as well.This yields the stationary conditions 1 λ t x 0, λ t x 0.

3.11
Substituting this value of the lagrange multiplier into the functional 3.5 gives the iteration formula u n t p t u n t − q t dt, 3.12 obtained upon deleting the restriction on u n that was used for the determination of λ.
Considering the given condition u 0 α, we can select the zeroth approximation u o α.Using the selection into 3.12 we obtain the following successive approximations: u n t p t u n t − q t dt.

3.13
Recall that u n 1 x , 3.14 that may give the exact solution if a closed form solution exists, or we can use the n 1 th approximation for numerical purposes.

Applications
Example 4.1.We first considered the following differential-algebraic equations with Hessenberg index-3 form: The exact solutions are where x 2 , x 3 represent the differential variables and x 1 represents the algebraic variables.
After three times of differentiation of 4.1 we have the following ODE system:

4.4
Differential-algebraic equation DAE is a Hessenberg index-3 form.
To solve system 4.4 , we can construct following correction functionals: x δx n 1 x δ x 0 t dt 0.

4.6
Its stationary conditions can be obtained as follows:

4.7
The Lagrange multipliers can be identified as follows: and the following multipliers can be obtained as

4.10
Example 4.2.One has  The exact solutions are v 1 x e −x , v 2 x x, v 3 x 1:  where v 1 , v 2 represent the differential variables and v 3 represents the algebraic variables.
After three times of differentiation of 4.11 we have the following ODE system:

4.14
Differential-algebraic equation DAE is a Hessenberg index-3 form.To solve system 4.14 , we can construct the following correction functionals: t dt.

Conclusion
The method has been proposed for solving differential-algebraic equations with Hessenberg index-3.Results show the advantages of the method.Tables 1-4 and Figures 1-4 show that the numerical solution approximates the exact solution very well in accordance with the above method.

where λ 1 t
, λ 2 t , and, λ 3 t are general Lagrange multipliers and x n

Figure 1 :
Figure 1: Values of x 1 x and its x * 1 x variational iteration.

Figure 2 :
Figure 2: Values of x 2 x and its x * 2 x variational iteration.

Figure 3 :
Figure 3: Values of x 3 x and its x * 3 x variational iteration.

Figure 4 :
Figure 4: Values of v 1 x and its v * 1 x variational iteration.

Table 1 :
Numerical solution of x 1 x .

Table 2 :
Numerical solution of x 2 x .

Table 3 :
Numerical solution of x 3 x .

Table 4 :
Numerical solution of v 1 x .