An Analytical Approximate Solution for the Bratu Problem by using Nonlinearities Distribution Homotopy Perturbation Method

An Analytical Approximate Solution for the Bratu Problem by using Nonlinearities Distribution Homotopy Perturbation Method

Mario A. Sandoval-Hernandez1, U. Filobello-Nino2, H. Vazquez-Leal2*, S. E. Torreblanca Bouchan1,

1. Alvarez-Gasca2, A. D. Contreras-Hernandez2, B. E. Palma-Grayeb2, Elisa De Leo Baquero1, Alexis C. Bielma Pérez1, Julio C. Vichi Mendoza1, J. Sanchez-Orea2, R. A. Callejas-Molina3,

   C. Hoyos-Reyes2

   1 CBtis 190, Av. 15 Col. Venustiano Carranza 2da Secc, 94297, Boca del Rio Ver., México.

   2 Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.

   3 Doctorado en Ciencias de la Ingeniería, Tecnológico Nacional de México en Celaya, Antonio García Cubas #600, Celaya, Guanajuato, 38010, México.

   Abstract This work introduces the nonlinearities distribution homotopy perturbation method in order to get an accurate analytical approximate solution for the relevant Bratu problem which has many applications in science and engineering. We will show that the distribution of the nonlinearities in the different iterations of the proposed method is a convenient scheme which eases the obtaining of the approximation based on elementary integrals. We will conclude that the proposed method is very convenient for practical applications.

   Keywords:-Bratu problem, nonlineardifferential equation, approximate solution.

   1. INTRODUCTION

      The Bratu problem in one dimension is called Liouville- Bratu-Gelfand and has many applications in science and engineering where physical and chemical phenomena are modeled. One of the applications of this equation is in solid fuel ignition models of the thermal combustion theory [1-2]. Gelfand's differential equation with boundary value in one dimension in planar coordinates is given by [3-8].

      u eu(x) 0. 0<x<1. (1)

      with boundary conditions (0) = (1) = 0.

      The analytical solution of (1)is given by [3, 4]

      1

      Different approximate methods have been applied to solve the Bratu problem. For example in [9] Chebyshev polynomials was employed in order to obtain an approximate solution for the proposed problem. In [10] the Adomian decomposition method was used to solve the equation. In [11] the B-spline method was proposed to solve the equation. In [12] the Leal- polynomials were proposed to find an accurate analytical approximate solution for (1). In [13] Perturbation Method was employed to get a handy approximate solution and so on.

      This research paper is organized as follows In Section 2 we introduce the idea of Homotopy Perturbation Method (HPM). For Section 3 we briefly introduce Nonlinearities Distribution Homotopy Perturbation Method (NDHPM). Additionally Section 4 presents the application of the proposed method in the search for an analytical approximate solution for Bratu problem. Besides a discussion on the results is presented in Section 5. Finally, a brief conclusion is given in Section 6.

   2. HOMOTOPY PERTURBATION METHOD

      To show how homotopy perturbation method (HPM) works, first, consider a general nonlinear differential equation as follows

      () () = 0, (5)

      () = 2ln

      cosh (( 2) 2)

      ,

      cosh ()

      4

      With the boundary conditions

      (, /) = 0, ,

      (6)

      [

      where is the solution of

      ]

      (2)

      where is a general differential operator, is a boundary

      operator, () a known analytical function and is the domain boundary for . In general terms, can be divided

      = 2 cosh ( ). (3) into two operators and , where is linear and is

      4

      There are three cases for the solution; if > then the

      nonlinear; so that (5) can be rewritten as

      problem has zero solution, if = there is one solution, and < there are two solutions, where = 3.513830719 is the critical value and satisfies the equation.

      1 = 1 2 sinh () . (4)

      4 4

      () + () () = 0.

      A homotopy can be constructed as [14, 15]

      (, ) = (1 )[() (0)] +

      [() + () ()] = 0, ,

      (7)

      (8)

      where is a homotopy parameter, which value varies continuously from 0 to 1, 0 is the first approximation for the

      epy( x) 1 py

      p2 y2

      2

      • p3 y3

      6

      …

      (16)

      solution of (5) that satisfies the boundary conditions. The

      solution of (8) can be written as power series of

      .

      = 0 + 1 + 22 + .

      After substituting (16) into (15) we obtain

      2 2 3 3

      2 2 3 3

      y p p y p y

      (9) 1

      py 2 6 …

      0, (18)

      Substituting (9) into (8) and equating identical powers of p terms, there can found values for 0, 1, 2, . when tends to (5), we obtain the approximate solution for (1) in the form

      let it be

      = 0 + 1 + 2 + .

      (10)

      y pn y (x),

      n

      n

      n0

      (19)

   3. BASIC IDEA OF NONLINEARITIES DISTRIBUTION HOMOTOPY PERTURBATION

      METHOD

      In [16] was introduced nonlinearities distribution homotopy perturbation method (NDHPM) with the purpose to reduce the complexity of solving differential equations in terms of power series.

      To carry out the procedure a homotopy is written in the form (Vazquez-Leal et al. 2012b) [16]

      (, ) = (1 )[() ( )] +

      so that after substituting (19) into (18) and equating identical powers of p terms we obtain

      0

      0

      p0 ) y 0,

      1

      1

      p1 ) y 0,

      2 0

      2 0

      p2 ) y y 0,

      p3 ) y y 2 0,

      0

      [() + (, ) (, )] = 0, .

      (11)

      and so on.

      3 2 0

      (20)

      Homotopy function (11) is essentially the same as (8), except because the non-linear operator N and the non homogeneous function , contain embedded the homotopy parameter . The rest of the procedure coincides with that explained in the last Section.

      From [14,16] we have

      The solutions of equations (20) which satisfy the initial conditions of the problem in accordance with the proposed problem are:

      0

      0

      = 0 + 1 + 22

      + .

      (12)

      p0 ) y

      Ax,

      1

      1

      When tends to 1, it is expected to get an approximate solution for (7) in the form

      = 0 + 1 + 2 + .

      p1 ) y

      x2

      cx,

      2

      2

      2

      (13)

   4. APPLICATION OF NDHPM

      p2 ) y

      Ax3

      Dx,

      6

      3

      3

      A2 x4

      Next, we will employ NDHPM in order to find an analytical approximate solution for the Bratu problem.

      p3 ) y

      Ex,

      24

      (21)

      In accordance with the proposed problem we proposed the following homotopy:

      1 p y p y epy( x) 0, (14)

      or

      and so on.

      Substituting (21) into (13)

      2 3 2 4

      2 3 2 4

      y(x) A K x x Ax A x ,

      2 6 24

      where the constant K is given by

      (22)

      y p epy( x) 0.

      (15)

      We note exponential function contain embedded the homotopy parameter .

      K c D E.

      (23)

      After using Taylor approximation for the exponential function we obtain the following approximation as follows:

      With the end to get a better precision, we will determine the value of K, from the requiring that the approximate soltion satisfy the real value for x=0.5. As a consequence

      (22) adopts the following form:

      A2

      x2

      Ax3

      A2 x4

      y(x) 384 0.0208A 0.256817 x 2

      .

      6 24

      (24)

      With the purpose to verify the accuracy of the proposed approximation we propose the value =0.5 in such a way that

      (24) assumes the form

      A2

      x2

      Ax3

      A2 x4

      y(x) 384 0.0208A 0.256817 x 4 12 48 .

      (25)

      With the end to determinate A we apply the condition y(1)=0 to (25) to generate the following algebraic equation

      7 A2 3A

      Figure 1. Comparison between numerical solution and approximate solution

      (27) for = 0.5

      On the other hand, Table 1 shows a more detailed comparison between numerical solution for (1) and

      384 48

      0.006817 0.

      (26)

      approximation (27) of this work for =0.5. The mentioned table shows that the relative error is less than one percent for

      The solution to this equation is substituted into (25) to obtain

      x2 3 4

      most of the interval. It is clear that (27), is competitive taking into account that the proposed method required to solve essentially elementary integrals and an algebraic

      y(x) 0.2590x 0.008817x

      4

      0.0002332x . (27)

      equation. From above it is very important to emphasize that, it is possible to improve the accuracy of our approximation

      Figure 1 compares (27) with the numerical solution for (1) with = 0.5.

      Although the precision of (27) is clear at sight, from Table 1 is clear the precision of the proposed solution.

   5. DISCUSION

In this work NDHPM was employed in order to find an approximate solution, for the nonlinear ordinary differential equation that describes the relevant Bratu problem that models the solid fuel ignition which rises of the thermal combustion theory. One of the great advantages of the proposed method is that it systematically distribute the nonlinearity term in the different iterations which ease the obtaining of an analytical approximate solution. The procedure consisted in ensure that the approximate solution satisfy the real value for x=0.5 and pose an algebraic equation for determining an unknown parameter A. The solution of this equation was obtained from the boundary condition (1) = 0 This solution provides the sought analytical approximation solution.

Figure 1 shows the comparison between numerical solution and approximate solution (27) for =0.5. It can be noticed that curves are in good agreement, whereby it is clear that the proposed method is potentially useful in the search for approximate solutions of nonlinear problems definite with boundary conditions.

performing more iterations from the homotopy (18).

V1. CONCLUSION

This work presented method NDHPM with the purpose to calculate an analytical approximate solution for the Bratu problem. The method basically works calculating elementary integrals after distributing the nonlinear term in the several iterations of the method in order to ease the procedure. As a matter of fact the method expressed the solution of a differential equation in terms of the solution of one or more algebraic equations. The application of the proposed method in this work showed the convenience of employ NDHPM as a practical tool with the purpose to obtain accuracy solutions for boundary value problems instead of using other more sophisticated and cumbersome procedures.

Table 1: Comparison between (27), and exact solution using =0.5

ACKNOWLEDGMENTS

Authors would like to thank Roberto Ruiz Gomez for his contribution to this project.

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