# Effects of Nonlinearity on the Variational Iteration Solutions of Nonlinear Two-Point Boundary Value Problems with Comparison with Respect to Finite Element Analysis

Authors: MehmetTarık Atay; SafaBozkurt Coşkun
Journal: Mathematical Problems in Engineering
 (2008-01-27)

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## Abstract

Solution of a nonlinear two-point boundary value problem is studied using variational iteration method (VIM) considering its convergence behavior due to the changing nonlinearity effects in the equation. To achieve this, steady Burger equation is first solved by using finite element method (FEM) with a very fine mesh for the comparison of results obtained from VIM. Effect of the nonlinear term in the equation that is multiplied by a constant is taken into account for five different cases by changing the corresponding constant. Results have shown that VIM is a flexible, easy to apply, and promising method for the analysis of nonlinear two-point boundary value problems with the fact that the larger the effect of the nonlinear term of the equation, the slower the convergence rate when compared to FEM solutions.


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## Body

## 1. Introduction

In the scientific world, numerical
solutions of two-point boundary value problems (BVP's) have great importance
because of their wide range of applications. These problems have been studied
immensely using different numerical approaches such as finite difference
method, finite element method, B-spline methods, and so forth.Recently, the variational iteration
method introduced by He [1] has been the subject of numerous studies. This paper
applies the variational iteration method to two-point boundary value problems.
The method was first proposed in 1998 to solve nonlinear equations with
fractional derivatives [2]. Then, the method has been applied to delay
differential equations [3], Duffing equation with nonlinearity of fifth-order
and mathematical pendulum [1], generalized KdV and coupled Schrodinger-KdV [4],
and some other problems including solitary and compacton-like solutions of
nonlinear wave equations, construction of some compact and noncompact
structures of Klein-Gordon equations, and different types of nonlinear
differential equations [5–19].The numerical solution of Burger
equation is very important because of the equations’ application in the
approximate theory of flow through a shock wave traveling in a viscous fluid
and in the Burger’s turbulence model [20]. Some specific one-dimensional Burger
equations as an initial value problem rather than a boundary value problem have
been solved using VIM [20, 21] and modified VIM [22, 23]. However, these studies have not focused on the effect of nonlinearity in the solution process of VIM.In this study, VIM solution of nonlinear two-point boundary value problems is considered  to focus on  the effect of nonlinearity in the
convergence behavior of VIM. As a nonlinear two-point boundary value problem,
steady Burger equation is considered with changing weights of nonlinearity for
the purpose of generalizing and predicting the behavior of nonlinear term in
the equation. Five different cases have been solved by using both VIM and FEM,
and comparisons are made with respect to the number of iterations and to the
error norms based on FEM solutions of the equations with very fine meshes.


## 2. Variational Iteration Method

According to the variational
iteration method, the following differential equation may be considered:(2.1)Lu+Nu=g(x),
where L is a linear operator, N is
a nonlinear operator, and g(x) is an inhomogeneous term.
Based on the variational iteration
method, a correct functional can be constructed as follows:
(2.2)un+1(x)=un(x)+∫0xλ{Lun(τ)+Nu˜n(τ)−g(τ)}dτ,
where λ is a general Lagrangian multiplier,  which can
be identified optimally via the variational theory, the subscript n denotes the
nth-order approximation;  the second term on the right is called the correction
in which u˜ is considered as a restricted variation,  that
is,   δu˜=0.
Equation (2.2) is called correction functional. For further details He’s studies [1, 2, 17, 18]
can be visited.


## 3. Finite Element Method

The equation to be solved in this
study is a nonlinear two-point boundary value problem namely steady Burger
equation given as follows:(3.1)u″−αuu′=0,u(0)=0,u(1)=1,x∈[0,1].
This equation can be linearized
using classical linearization technique. To this aim, value of u at any node in
an element will be assumed to be known. Modified equation to be solved becomes:
(3.2)u″−αu¯u′=0,
where u¯ represents the previously known values of u.Weak form of (3.2) is
(3.3)(∇u,∇ψ)−α(u¯∇u,ψ)=0,
where u¯ is assumed as changing linearly in a two-noded
linear element shown in Figure 1. For such an element, corresponding shape
functions are defined by the following equations:
(3.4)ψ1(x)=1−xLe,ψ2(x)=xLe,
where Le is the element
length. Weak formulation of the equation gives the following coefficient
matrices for each element:
(3.5)[K]=1Le[1−1−11],(3.6)[L*]=α6[-2u¯1−u¯22u¯1+u¯2-u¯1−2u¯2u¯1+2u¯2].
In (3.6), u¯1 and u¯2 are assumed values of u at the nodes 1 and 2
shown in Figure 1. Final form of the finite element formulation resulting from
classical linearization technique is
(3.7)([K]+[L*]){u}={F}.
The analysis procedure is
iterative. First a starting value for u is assumed at the nodes of FEM mesh for
the whole domain, and the matrix [L*] is determined using (3.6). Then (3.7) is solved
for u. The last step in a typical iteration is the comparison of new values
with the previous ones. This comparison is based on an error criterion for the
whole domain. The error criterion used in this study is
(3.8)(∑i=1N|uir+1−uir|2∑i=1N|uir+1|2)<ε,
where N is the total number of unknowns
in the FEM mesh, r  is the iteration number, and  ε is the preselected error tolerance.Figure 1
Linear element used in finite element formulation.If error tolerance obtained from (3.8) is less than preselected error, iteration stops and the results produced by
final iteration become the solution from FEM analysis. However, if error
tolerance is greater than preselected error, then the assumed value of u(i.e.,u¯)) is changed with the new values of current
iteration, and the process goes in the same manner until preselected error tolerance
is reached.100 linear elements are used in the finite element analysis of the problem, and error tolerance is chosen as10−5,
that is, ε=10−5,
for the iteration process. The results obtained from FEM analysis can be seen
between Figures 2–6 with the VIM solutions of the same equation.Figure 2
Consecutive VIM approximations compared to FEM solution forα=0.1.Figure 3
Consecutive VIM approximations compared to FEM solution forα=1.Figure 4
Consecutive VIM approximations compared to FEM solution forα=5.Figure 5
Consecutive VIM approximations compared to FEM solution forα=10.Figure 6
Consecutive VIM approximations compared to FEM solution forα=100.


## 4. VIM Applications

We recall steady Burger equation in
(3.1) to be solved by using VIM:
(4.1)u″−αuu′=0,u(0)=0,u(1)=1,x∈[0,1].
A correct functional can be derived
using VIM as follows:
(4.2)un+1(x)=un(x)+∫0xλ{un″(ξ)−αu˜n(ξ)u˜′n(ξ)}dξ,
where λ is the Lagrange multiplier and can be easily
found as
(4.3)λ=ξ−x.
Hence, the following iteration
formula would be obtained:
(4.4)un+1(x)=un(x)+∫0x(ξ−x){un″(ξ)−αun(ξ)un′(ξ)}dξ.
Iteration process can be started
using initial guess as
(4.5)u0(x)=Ax+B,
where A and B are constants to be determined.For the analysis, differentα  values will be considered based on this
formulation. α takes values between 0.1 and 100 to represent
the increasing power of nonlinearity in the equation.In each case, an error norm is
calculated for each iteration to represent the convergence behavior of the
solution. The error norm is defined as follows:(4.6)Error=(∑i=1N|uFEM−uVIMr|2∑i=1N|uFEM|2),
where N is the number of nodes in
the FEM mesh, r is the iteration number of the VIM, uFEM is the FEM solution, and uVIMr is the VIM solution at the rth iteration.Case 1 (α = 0.1).
The iteration formula for this case is(4.7)un+1(x)=un(x)+∫0x(ξ−x){un″(ξ)−(0.1)un(ξ)un′(ξ)}dξ.
The first iteration gives
(4.8)u1(x)=0.983867x+0.0161332x3.This approximation is very close to
the FEM solution of the equation as shown in Figure2. Error norm defined in (4.6)
is calculated as 0.000202871. VIM solution is in perfect agreement with FEM
solution even in the first iteration due to the fact that there is considerably
weak nonlinearity in the equation.Case 2 (α = 1).
The iteration formula for this case
is(4.9)un+1(x)=un(x)+∫0x(ξ−x){un″(ξ)−un(ξ)un′(ξ)}dξ.
Iterations are
(4.10)u1(x)=0.872983x+0.127017x3,u2(x)=0.85593x+0.122103x3+0.0209023x5+0.00106493x7.
In this case, the number of
iterations increased with the increasing nonlinearity effect. Second
approximation is in perfect agreement with the FEM solution as shown in Figure 3, and error norms for this case are given in Table 1.Table 1
Error norms for differentαvalues.
αErr(U1)Err(U2)Err(U3)Err(U4)Err(U5)Err(U6)0.10.00020287110.0146060.00210350.1563510.0633450.0234280.007741100.3403290.1728090.0846900.0391740.0168891001.8041819581.2297201680.8676959470.6233107930.4517441310.328771988Case 3 (α = 5).
The iteration formula for this case
is(4.11)un+1(x)=un(x)+∫0x(ξ−x){un″(ξ)−(5)un(ξ)un′(ξ)}dξ.
Iterations are
(4.12)u1(x)=0.649x+0.351x3,u2(x)=0.562735x+0.263893x3+0.148501x5+0.0248711x7,u3(x)=0.534962x+0.238487x3+0.127581x5+0.0690635x7+0.0229405x9+0.00590127x11+0.000996742x13+0.0000687683x15,u4(x)=0.525756x+0.230349x3+0.121108x5+⋯+2.88895*10−10x31.
As shown in Figure 4, fourth
approximation is in very good agreement with the FEM solution. It can be
expected that fifth or sixth iteration will be in excellent agreement with the
FEM solution. Table 1 shows the decreasing error norms with the increasing iterations
for this case.Case 4 (α = 10).
The iteration formula for this case
is(4.13)un+1(x)=un(x)+∫0x(ξ−x){un″(ξ)−(10)un(ξ)un′(ξ)}dξ.
Iterations are
(4.14)u1(x)=0.530662x+0.469338x3,u2(x)=0.415184x+287297x3+0.238562x5+0.0589567x7,u3(x)=0.373597x+0.232624x3+0.173815x5+0.13142x7+0.0609713x9+0.0219068x11+0.00516804x13+0.000498014x15,u4(x)=0.356649x+0.211997x3+0.151217x5+⋯+1.90326*10−8x31,u5(x)=0.349601x+0.203702x3+0.142429x5+⋯+1.51797*10−17x63.Figure5 shows that fifth
approximation is in good agreement with the FEM solution. It can be expected
that two or more iterations are required for being in excellent agreement with
the FEM solution. Decreasing error norms with the increasing iterations for
this case can be seen in Table 1.Case 5 (α = 100).
The iteration formula for this case
is(4.15)un+1(x)=un(x)+∫0x(ξ−x){un″(ξ)−(100)un(ξ)un′(ξ)}dξ.
Iterations are
(4.16)u1(x)=0.216779x+0.783221x3,u2(x)=0.108295x+0.195462x3+0.423349x5+0.272895x7,u3(x)=0.0759922x+0.096247x3+0.14628x5+0.22497x7+0.212303x9+0.155159x11+0.0744542x13+0.0145939x15,u4(x)=0.0625732x+0.0652569x3+0.0816667x5+⋯+0.0000153413x31,u5(x)=0.0559616x+0.052195x3+0.0584182x5+⋯+5.24076*10−12x63,u6(x)=0.0523792x+0.0457264x3+0.0479023x5+⋯+1.56741*10−25x127.
In this final case, the equation is
highly nonlinear. As it can be seen from Figure 6, sixth approximation is still not enough for being in good agreement with the FEM solution, and it can be expected
that many more iterations are required to reach the close proximity of the FEM
solution. For the final case, decreasing error norms with the increasing
iterations are also included in Table 1.


## 5. Discussion of Results

When all the cases are considered
consecutively, a pattern emerges from the number of iterations related with the
weight of nonlinearity in the equation. It is obvious that there is a strong
relation between the increasing number of iterations required to reach an
acceptable solution and the  effect of the increasing weight of the nonlinearity in
the equation. This phenomenon can also be seen in Figure7. Based on this figure, it can be inferred that
αvalues, higher than the values considered in
this study, will produce very large errors even for considerably many
iterations.Figure 7
Error estimations of VIM approximations for differentα values.


## 6. Conclusion

In this paper, VIM solution of nonlinear two-point boundary value problems with changing nonlinearity effects
in the equation is considered. Steady Burger equation is chosen because of its
wide range of applications in different disciplines. Five different cases are
considered in detail with respect to their nonlinear terms. The results
obtained from these cases are compared with the corresponding FEM solutions.Although FEM is a powerful
numerical technique for the solution of differential equations, a fixed mesh
with fixed points within the boundary is used for FEM analysis. These fixed
points are called “node" in FEM, and the solutions are obtained at the nodes. Important
merits of VIM are that it is a kind of analytical solution which does not need
a discretized solution domain, and the solution procedure is very simple. The
advantage of VIM at this point is that VIM solution is obtained for any
arbitrary point instead of fixed points in the domain. Another advantage of VIM
is the method which is easy to apply. In addition, application of FEM is much
more difficult when compared to VIM in the analysis of nonlinear boundary value
problems.However, as shown from the case
studies, when the equation becomes highly nonlinear, the advantages of VIM
disappear dramatically compared to FEM. In such a case, many more iterations
are required in VIM for an acceptable solution.As a result, it can be concluded
that VIM can be preferred for the analysis of nonlinear two-point boundary
value problems when the equation is not dominated by its nonlinear part. For
such cases, FEM is much more suitable for the analysis of the problem.



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