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Research Article
 

Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation



U. Filobello-Nino, H. Vazquez-Leal, K. Boubaker, Y. Khan, A. Perez-Sesma, A. Sarmiento-Reyes, V.M. Jimenez-Fernandez, A. Diaz-Sanchez, A. Herrera-May, J. Sanchez-Orea and K. Pereyra-Castro
 
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ABSTRACT

Solving nonlinear ordinary differential equations is relevant because phenomena on the frontiers of modern sciences are often nonlinear in nature; therefore this article proposes Perturbation Method (PM) to solve nonlinear problems. As case study PM is employed to obtain a handy approximate solution for Gelfand’s differential equation which governing combustible gas dynamics. Comparing figures between approximate and exact solutions, it is shown that PM method result extremely efficient.

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U. Filobello-Nino, H. Vazquez-Leal, K. Boubaker, Y. Khan, A. Perez-Sesma, A. Sarmiento-Reyes, V.M. Jimenez-Fernandez, A. Diaz-Sanchez, A. Herrera-May, J. Sanchez-Orea and K. Pereyra-Castro, 2013. Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation. Asian Journal of Mathematics & Statistics, 6: 76-82.

DOI: 10.3923/ajms.2013.76.82

URL: https://scialert.net/abstract/?doi=ajms.2013.76.82
 
Received: February 28, 2013; Accepted: April 27, 2013; Published: July 18, 2013



INTRODUCTION

As it is known, Gelfand’s equation (also known as Bratu’s problem in 1D) models the chaotic dynamics in combustible gas thermal ignition. Therefore, it is important to search for accurate solutions for this equation. Unfortunately, it is difficult to solve, like many others nonlinear differential equations that appear in engineering.

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 (Chow, 1995; Holmes, 1995).

In a broad sense, it is possible to express a nonlinear differential equation in terms of one linear part and 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 (Kazemnia et al., 2008; Noorzad et al., 2008), tanh method (Evans and Raslan, 2005), exp-function (Mahmoudi et al., 2008), Adomian’s decomposition method (Kooch and Abadyan, 2011, 2012; Vanani et al., 2011; Chowdhury, 2011), homotopy perturbation method (Ganji et al., 2008, 2009; Sharma and Methi, 2011; Vazquez-Leal et al., 2012a, b; Filobello-Nino et al., 2012a, c; Khan and Wu, 2011; Mirgolbabaei and Ganji, 2009; Tolou et al., 2008), homotopy analysis method (Patel et al., 2012) and Boubaker polynomials expansion scheme (Agida and Kumar, 2010; Ghanouchi et al., 2008), variational iteration method (Saravi et al., 2013), among many others.

Although the PM method provides in general, better results for small perturbation parameters ε<<1 , it will be seen that the approximation obtained, besides to be handy has a good accuracy, even for relatively large values of the perturbation parameter (Filobello-Nino et al., 2012b).

BASIC IDEA OF PERTURBATION METHOD

Let the differential equation of one dimensional nonlinear system be in the form:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(1)

where, it is assumed 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:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(2)

After substituting (2) into (1) and equating terms having identical powers of ε, it is possible to obtain a number of differential equations that can be integrated, recursively, to find the values for the functions:

x0(t), x1(t), x2(t)...

APPROXIMATE SOLUTION OF GELFAND’S EQUATION

The equation to solve is:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(3)

where, ε is a positive parameter.

It is possible to find a handy solution for (3) by applying the PM method.

Identifying terms:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(4)

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(5)

where, prime denotes differentiation respect to x.

To solve (3), first it is required expanding the exponential term of Gelfand,s problem, resulting:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(6)

identifying ε with the PM parameter, then it is assumed a solution for (6) in the form:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(7)

Equating the terms with identical powers of ε it can be solved for y0(x), y1(x), y2(x).. and so on. Later it will be seen that, a very good handy result is obtained, by keeping up to third order approximation:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(8)

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(9)

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(10)

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(11)

Thus, the solution for the lowest order approximation is:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(12)

On the other hand the solution of (9) is given by:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(13)

The solving process for (10), leads to result:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(14)

In the same way, from (11):

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(15)

and so on.

By substituting (12-15) into (7) it is obtained a handy third order approximation for the solution of (3), as it is shown:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(16)

Considering as a case study, the value of the Gelfand’s parameter ε = 1, (16) adopts the form:

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
(17)

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 highly nonlinear Gelfand’s differential equation. This equation is relevant because it describes the dynamics in combustible gas thermal ignition. Figure 1 shows the comparison between approximation (17) with the four order Runge Kutta (RK4) numerical solution. It can be noticed that (17) sketches successfully the qualitative parabolic-like behavior of Bratu’s problem. This characteristics can be deduced from (3), because it implies that y”(x)<0 for 0≤x≤1 and hence the solution of Gelfand’s equation has to be concave downward in the same interval. As a matter of fact Fig. 2, shows that the maximum absolute error is about 5E-4 at x = 0.5, which proves the efficiency of PM method, especially because it was only considered the third-order approximation; by using four terms in series expansion of nonlinear part εey(x).

The PM method provides in general, better results for small perturbation parameters ε<<1, (Eq. 1) and when are included the most number of terms from (Eq. 2).

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
Fig. 1: Approximation of Gelfand’s equation and RK4 comparison for e = 1

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
Fig. 2: Absolute error for approximation (17) (e = 1)

Image for - Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation
Fig. 3: Approximation of Gelfand’s equation and RK4 comparison for e = 1.5

To be precise, ε is a parameter of smallness, that measure how greater is the contribution of linear term L(x) than the one of N(x) in (1). Figure 1 and 2 show a noticeable fact, that (Eq. 17) provides a good approximation as a solution of (3), despite of the fact that perturbation parameter ε = 1 cannot be considered small. Moreover, the Fig. 3 shows that (Eq. 16) is in good agreement with RK4, for ε = 1.5.

Saravi et al. (2013) employed VIM method to provide an approximate solution of (Eq. 3). The maximum absolute error reported for the VIM solution is 1.59355E-3 at x = 0.5, which shows the high accuracy of PM in comparison with VIM. In addition, VIM solution generates a more complicated solution for similar number of iterations.

Finally, our approximate solution (Eq. 16) does not depend of any adjustment parameter, for which, it is in principle, a general expression for Gelfand,s problem.

CONCLUSION

In this study, PM was presented to construct analytical approximate solutions of Gelfand’s equation in the form of rapidly convergent series. The success of the method for this case it has to be considered as a possibility to apply it in other non linear problems, instead of using other sophisticated and difficult methods. From Fig. 1-3, it is deduced that the proposed solutions are highly accurate.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support provided by the National Council for science and Technology of Mexico (CONACyT) through Grant CB-2010-01 no. 157024. The authors would like to express their gratitude to Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution to this study.

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