nueva página del texto (beta)
Inglés (pdf)
Artículo en XML
Referencias del artículo
Traducción automática
Enviar artículo por email
Citado por SciELO 
Similares en
SciELO 
Permalink
1. Introduction
Troesch equation is relevant in physics because it models the confinement of a plasma column by radiation pressure. Therefore, it is important to search for accurate solutions for this equation. Unfortunately, it is difficult to solve nonlinear differential equations, like many others that appear in the physical sciences.
The perturbation method (PM) is a well established method; it is among the pioneer techniques to approach various kinds of nonlinear problems. This procedure was originated by S. D. Poisson and extended by J. H. Poincare. Although the method appeared in the early 19th century, the application of a perturbation procedure to solve nonlinear differential equations was performed later on that century. The most significant efforts were focused on celestial mechanics, fluid mechanics, and aerodynamics [1, 2, 54, 55].
In general, it is assumed that the differential equation to be solved can be expressed as the sum of two parts, one linear and the other nonlinear. The nonlinear part is considered as a small perturbation through a small parameter (the perturbation parameter). The assumption that the nonlinear part is small compared to the linear is considered as a disadvantage of the method. There are other modern alternatives to find approximate solutions to the differential equations that describe some nonlinear problems such as those based on: Variational approaches [5-7, 29], Tanh method [8], Exp-function [9, 10], Adomian's decomposition method [11-16,40], Parameter expansion [17], Homotopy perturbation method [3,4,18-28,31-36,39,45,46,48-53,56,57] and Homotopy analysis method [30,47], among many others.
Although the PM method provides in general, better results for small perturbation parameters ε<<1 , we will see that our approximation has good accuracy, even for big values of the perturbation parameter. Finally, we will couple the PM and Padé methods, in order to express the results of perturbation method in a handy way.
The paper is organized as follows. In Section 2, we introduce the basic idea of the PM method. Section 3 will provide a brief introduction to the Padé approximation. For Section 4, we provide an application of the PM method. Section 5 shows an approximate solution to the Troesch equation by using Laplace-Padé approximation. Section 6 discusses the main results obtained. Finally, a brief conclusion is given in Section 7.
2. Basic idea of Perturbation Method.
Let the differential equation of one dimensional nonlinear system be in the form
Where we assume that x is a function of one variable x=x(t), L(x) is a linear operator which, in general, contains derivatives in terms of t, N(x) is a nonlinear operator, and ε is a small parameter.
Considering the nonlinear term in (1) to be a small perturbation and assuming that the solution for (1) can be written as a power series in the small parameter ε.
Substituting (2) into (1) and equating terms having identical powers of ε, we obtain a number of differential equations that can be integrated, recursively, to find the values for the functions: 
, 
,
...
3. Padé Aproximation
A Rational approximation to
on
is the quotient of two polynomials
and 
of degrees 
and
, respectively. We use the notation
to denote this quotient. ThePadé approximations to a function are given by [41,43]
The method of Padé requires that
and its derivatives be continuous at
. The polynomials used in (3) are
The polynomials in (4) and (5) are constructed so thatand
agree atand their derivatives up to
agree at
. In this case
, the approximation is just the Maclaurin expansion for
. For a fixed value of the error is smallest whenandhave the same degree or whenhas degree one higher than.
Notice that the constant coefficient ofis
. This is permissible, because it can be noted that
anddo not change when bothandare divided by the same constant. Hence the rational functionhas
unknown coefficients. Assume thatis analytic and has the Maclaurin expansion.
And from the difference 
The lower index
in the summation on the right side of (7) is chosen because the firstderivatives ofandshould agree at.
When the left side of (7) is multiplied and the coefficients of the powers of
are set equal to zero for
the result is a system of
linear equations.
and
Notice that in each equation the sum of the subscripts on the factors of each product is the same. This sum increases consecutively from 0to
. The
equations in (9) involve only the unknowns:
,
,...,
and must be solved first. Then the equations in (8) are used successively to find
,
,...,
[41].
4. Approximate solution of Troesch Equation.
The equation to solve is
where
is known as Troeschs parameter.
It is possible to find a handy solution for (10) by applying PM. Identifying terms:
andwith the PM parameter.
Since the parameter is embedded into the nonlinear operator
, we express the right hand side of (12), in terms of Taylor series expansion as it is shown
Assuming a solution for (13) in the form
and equating the terms with identical powers of , it can be solved for
,
,
,..., and so on. Later it will be seen that, a very good result is obtained, by keeping up to eighth order approximation.
In order to fulfill the boundary conditions from (10), it follows that
, 
,
,
,
,
,
,
,
,
, and so on. Thus, the results obtained are
By substituting (24)-(32) into (14), we obtain an approximate solution to (10), as it is shown
We consider as a case of study, the following values of the Troesch,s parameter:
,,
,, so that
5. An approximate solution by using Laplace-Padé transformation and PM Method
In this section we will describe the use of Laplace transform and Padé transformation [40] to deal with the truncated series (34), (35), (36) and (37) obtained by PM, in order to obtain handy approximate solutions to equation (10), keeping the same domain of the original problem [56].
First, Laplace transformation is applied for example, to series (34) and then 
is written in place of s in the equation obtained. Then Padé approximant [4/4] is applied and 1/s is written in place of
. Finally, by using the inverse Laplace transformation, we obtain the modified approximate solution.
Then, applying the same procedure to the series (35), (36) and (37), we obtain the following alternative expressions.
6. Discussion
The fact that the PM depends on a parameter which is assumed small, suggests that the method is limited. In this work, the PM method has been applied to the problem of finding an approximate solution for the nonlinear differential equation which describes the Troeschs problem. Table 1 shows the comparison between exact solution given in [42], and approximations (34), (38), ADM [44], HPM [45], HPM [46] and HAM [47] for the case. It is clear that (34), has the best accuracy and also the lowest Average Absolute Relative Error (A.A.R.E)
, followed by HAM [47], with accuracy
, despite of the fact that HPM, ADM and HAM methods are considered more general and difficult to use. Table 2 shows that for, our approximations (35) and (39) were the best. In particular PM possesses the lowest A.A.R.E 
, although cannot be considered as small.
Table 1 Comparison between (34), exact solution [42], and other reported approximate solutions, using Ɛ=0.5
Table 2 Comparison between (35), exact solution [42], and other reported approximate solutions, using
The PM method provides in general, better results for small perturbation parameters
(see (1)) and when are included the most number of terms from (2). To be precise, is a parameter of smallness, that measures how greater is the contribution of linear term
than the one of
in (1). From the approximations (34), (35), (36), (37), as well as of Figure 1, it is clear that the term proportional to 
(
), is the contribution to the approximation of the linear operator (see (15) and (24)) besides, they are the dominant terms in the approximate solutions, even for the big values of:,, and. This happens because Troesch s problem is defined in [0,1] (see (10)); in that interval
, for
. Also, the coefficients of powers: 
of the aforementioned equations (34)-(37) are small, compared with that of .
Figure 1 shows the comparison between approximate solutions (34) and (35) forand respectively, and the exact solutions given in [42]. Besides, the same Figure compares (36) 
and (37) 
with the four order Runge Kutta numerical solution of (10), for the same values of. It can be noticed that, figures are very similar in all cases, of which is clear the accuracy of results (34)-(37) as approximated solutions for (10).
Fig 1: Comparison of proposed solution (34) (solid line) for , , and ; with (solid square), (solid circles) reported in [42]; (empty square) and (diagonal-cross) calculated by using four order Runge Kutta.
We employed Laplace transform and Padé transformation to obtain the approximate solutions of equation (10), given by (38)-(41). Although some precision is lost compared with PM approximations, expressions (38)-(41) are handier and computationally more efficient than (34)-(37). In fact from Tables 1 and 2, we conclude that our PM Padé expressions are also competitive. For, (38) has an acceptable accuracy, since it's A.A.R.E 
, is better than the ones of ADM [44] and HPM [45], while for , (39) is even better, when it is compared with the other approximations, having the second best A.A.R.E 
as it had been already mentioned.
Unlike other methods, our approximate solution (33) does not depend of any adjustment parameter, for which, it is in principle, a general expression for Troesch s problem.
It is important to remark, that further research may be focused on the development of a sensitivity analysis for the solutions emanating from the perturbation method (PM), since it is possible that small perturbations on the coefficients change the accuracy of the approximate solutions.
7. Conclusion
This work showed that some nonlinear problems can be adequately approximated by using the PM method, even for large values of the perturbation parameter; as it was done for the Troesch s problem described by (10). The fact that the term proportional to x, is the dominant one in approximations: (34), (35), (36), (37), even for the big values of, contributed to the success of the method for this case and could be useful to apply it in similar cases, instead of using other sophisticated and difficult methods. Finally we showed that, it is possible to use a novel technique that coupled the PM method and the Padé-Laplace transformation to obtain, handy approximate solutions of equation (10), given by (38)-(41). In all cases, the numerical and graphical results show that the proposed solutions have good accuracy.
