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An approximate solution to aninitial boundary value problem:Rakib–Sivashinsky equationPaulo Jorge Rebelo aa Universidade da Beira Interior, Matemática, Rua Marquês D’Ávilae Bolama, 6200-001, Covilhã, PortugalPublished online: 19 Mar 2012.

To cite this article: Paulo Jorge Rebelo (2012): An approximate solution to an initial boundaryvalue problem: Rakib–Sivashinsky equation, International Journal of Computer Mathematics, 89:7,881-889

To link to this article: http://dx.doi.org/10.1080/00207160.2012.659665

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International Journal of Computer MathematicsVol. 89, No. 7, May 2012, 881–889

An approximate solution to an initial boundary valueproblem: Rakib–Sivashinsky equation

Paulo Jorge Rebelo*

Universidade da Beira Interior, Matemática, Rua Marquês D’Ávila e Bolama, 6200-001, Covilhã, Portugal

(Received 7 September 2009; resubmitted 28 February 2011; revised version received 7 November 2011;accepted 2 December 2011 )

The construction of an approximate solution to an initial boundary value problem for the Rakib–Sivashinskyequation is of concern. The Fourier method is combined with the Adomian decomposition method inorder to provide the approximate solution. The variables are separated by the Fourier method and theapproximate solution to the nonlinear system of ordinary differential equations is obtained by the Adomiandecomposition method. One example of application is presented.

Keywords: Rakib–Sivashinsky equation; Fourier method; Adomian polynomials; initial boundary valueproblem

2009 AMS Subject Classifications: 34A34; 35K15; G.1.8; G4

1. Introduction

The Rakib–Sivashinsky equation

ut = ε2uxx − 12 u2

x + u − u, (1)

governs the weak thermal limit of an upward flame interface in a channel. In Equation (1), ε2 isa physical parameter corresponding to the Markstein length. The Markstein length measures theeffect of curvature on a flame. The larger the Markstein length, the greater the effect of curvatureon burning velocity. Usually, ε2 � 1. In Equation (1), u = u(t, x) defines an instantaneous flameprofile in dimensionless variables and u = (1/L)

∫ L0 u(t, x) dx denotes the space average. This

equation is complemented with the initial condition

u(t = 0, x) = u0(x), (2)

and with the Neumann (adiabatic) boundary conditions

ux(t, 0) = ux(t, L) = 0. (3)

*Email: rebelo@ubi.pt

ISSN 0020-7160 print/ISSN 1029-0265 online© 2012 Taylor & Francishttp://dx.doi.org/10.1080/00207160.2012.659665http://www.tandfonline.com

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The aim of this paper is to provide an approximate solution to the problem considered in [16]:

ut = ε2uxx − 1

2u2

x + u − 1

L

∫ L

0u(t, x) dx, x ∈]0, L[, t ∈ (0, T ],

u(0, x) = u0(x), x ∈]0, L[,ux = 0, x ∈ {0, L}, t > 0.

(4)

Considering that the exact solution to Equation (4) is of the form

u(t, x) = u0(t) +∞∑

k=1

uk(t) cos

(kπx

L

)(5)

which satisfies the boundary conditions, the approximate solution to Equation (4) is

u(t, x) = u0(t) +n∑

k=1

uk(t) cos

(kπx

L

). (6)

The substitution of Equation (6) in Equation (4), followed by the multiplication by cos(iπx/L),for 0 ≤ i ≤ n, and the integration with respect to the variable x leads to a nonlinear system ofordinary differential equations. An approximate solution to this system is then obtained usingthe Adomian decomposition method, described in the next section. The initial conditions for thesystem are given by the projection of the initial condition, u0(x), in the space spanned by theorthogonal basis,

B ={

cos

(kπx

L

)}n

k=0

, n ∈ N. (7)

In order to obtain a good approximation to the solution it is necessary to have a goodapproximation to the initial condition, u0(x).

This paper is divided as follows: in the next section the Adomian decomposition method isdescribed; in Section 3, the necessary steps to provide the approximate solution to problem (4)are presented; Section 4 has one example of application and the conclusions are presented inSection 5.

2. The Adomian decomposition method

Adomian [3] developed a decomposition method for solving nonlinear (stochastic) differen-tial equations using special polynomials An, usually called Adomian polynomials. The An’s aregenerated for each nonlinearity.

One of the main advantages of the Adomian polynomials is that they depend only on theknown function u0(x). Another great advantage of this method is that the algorithm is of simpleimplementation.

But usually the solutions provided by the standard Adomian decomposition method do not sat-isfy the boundary conditions.Adomian and Rach [5] and Benneouala and Cherruault [8] presenteda method that allows the standard Adomian decomposition method to solve initial boundary valueproblems for partial differential equations.

The convergence of theAdomian decomposition series has been investigated by several authors.The theoretical analysis of convergence and speed of convergence of the decomposition method

was considered in [1,2,12,13,17].

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International Journal of Computer Mathematics 883

Let us now consider the differential equation

Lu = Ru + �u, (8)

where L (linear) and R are differential operators and � is a nonlinear operator.The Adomian decomposition method decomposes the function u(t, x) in a series

u(t, x) =∞∑

n=0

un(t, x), (9)

and for a nonlinear operator �, we have the following decomposition:

�(u(t, x)) =∞∑

n=0

An, (10)

where the An, usually called the Adomian polynomial, are given by the recurrence formula

An = 1

n!dn

dλn

[�

( ∞∑n=0

λnun

)]∣∣∣∣∣λ=0

, n ≥ 0. (11)

The Adomian polynomials can be constructed as follows:

A0 = �(u0),

A1 = u1�(u0),

A2 = u2�′(u0) + 1

2 u21�

′′(u0),

A3 = u3�′(u0) + u1u2�

′′(u0) + 13!u

31�

′′′(u0)

...

Algorithms for formulating Adomian polynomials were investigated in [4,23].We also suppose the existence of the inverse operator L−1. Thus, applying L−1 to Equation (8),

we obtain the recurrence relation:

un+1 = L−1Run + L−1�un, u0 = u0(x), n ∈ N ∪ {0}, (12)

that provides a reliable approach to the solution of the problem.Now we briefly describe how to apply the Adomian decomposition method to systems of

ordinary differential equations. Let us consider a system of ordinary differential equations inthe form

Lu1 = f1(t, u1, u2, . . . , un),

Lu2 = f2(t, u1, u2, . . . , un),

...

Lun = fn(t, u1, u2, . . . , un)

(13)

with initial conditions ui(0), 1 ≤ i ≤ n, where Lu = u ≡ du/dt with inverse L−1(·) = ∫ t0 (·) dt.

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Applying the inverse operator L−1 to Equation (13), we obtain the following canonical form:

u1 = u1,0 + L−1t [ f1(t, u1, u2, . . . , un)],

u2 = u2,0 + L−1t [ f2(t, u1, u2, . . . , un)],

...

un = un,0 + L−1t [ fn(t, u1, u2, . . . , un)],

(14)

where ui,0 = ui(0) for 1 ≤ i ≤ n.Applying theAdomian decomposition method, each component of the solution of Equation (13)

can be expressed as a series of the form

ui =∞∑

j=0

ui,j, i = 0, 1, . . . , n, (15)

and the integrands on the right-hand side of Equation (14), using Equation (11), are expressed as

fi(t, u1, u2, . . . , un) =∞∑

j=0

Ai,j(ui,0, ui,1, ui,2, . . . , ui,j), 1 ≤ i ≤ n, (16)

where the Ai,j are the Adomian polynomials corresponding to the nonlinear part fi.We should note that in order to solve system (13), we obtain a system of Volterra integral

equations of the second kind, (14).In order to accelerate the convergence of the method when applied to nonlinear systems of

Volterra integral equations of second kind, we will proceed as in [7]. For an application of thisprocedure to higher order nonlinear parabolic equations, we refer the reader to [20].

In the past few years, several authors developed other methods, such as He’s Polynomials andthe Homotopy perturbation method to find approximate or exact solutions to differential equationsand integral equations. For more details on these methods, we refer the reader to the followingpapers [6,9,11,15,18,19,21,22,24] and the references therein.

3. The approximate solution

Consider the problem

ut = ε2uxx − 1

2u2

x + u − 1

L

∫ L

0u(t, x) dx, x ∈]0, L[, t ∈ (0, T ],

u(0, x) = u0(x), x ∈]0, L[,ux = 0, x ∈ {0, L}, t > 0.

(17)

We consider that the approximate solution is of the form

u(t, x) ≈ u0(t) +n∑

k=1

uk(t) cos

(kπx

L

). (18)

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International Journal of Computer Mathematics 885

Thus, using Equation (18) in Equation (17), multiplying by cos(iπx/L) for 0 ≤ i ≤ n, we obtainthe nonlinear system of ordinary differential equations:

u0(t) = −1

4

(π

L

)2 n∑k=1

k2u2k(t), (19)

ui(t) =(

1 −(

εiπ

L

)2)

ui(t) − 1

L

(π

L

)2 n∑j,k=1

jkuj(t)uk(t)

×∫ L

0sin

(jπx

L

)sin

(kπx

L

)cos

(iπx

L

)dx, (20)

for 1 ≤ i ≤ n.The initial conditions for this system are given by

u0(0) = 1

L

∫ L

0u0(x) dx,

ui(0) = 2

L

∫ L

0u0(x) cos

(iπx

L

)dx,

(21)

for 1 ≤ i ≤ n.Integrating with respect to the variable t, we obtain

u0(t) = u0(0) − 1

4

(π

L

)2 n∑k=1

k2∫ t

0u2

k(t) dt, (22)

ui(t) = ui(0) +(

1 −(

εiπ

L

)2) ∫ t

0ui(t) dt

− 1

L

(π

L

)2 n∑j,k=1

jk

[∫ L

0sin

(jπx

L

)sin

(kπx

L

)cos

(iπx

L

)dx

]×

∫ t

0uj(t)uk(t) dt,

(23)

for 1 ≤ i ≤ n.Thus, we have the following recurrence scheme:

u0,0(t) = u0(0), (24)

u0,m+1(t) = −1

4

(π

L

)2 n∑k=1

k2∫ t

0u2

k,m(t) dt,

ui,0 = ui(0), (25)

ui,m+1(t) =[

1 −(

εiπ

L

)2] ∫ t

0ui,m(t) dt

− 1

L

(π

L

)2 n∑j,k=1

jk

[∫ L

0sin

(jπx

L

)sin

(kπx

L

)cos

(iπx

L

)dx

]×

∫ t

0uj,m(t)uk,m(t) dt,

for 1 ≤ i ≤ n and m ∈ N ∪ {0}.

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The initial conditions for system (24)–(25) are given by

u0,0(t) = 1

L

∫ L

0u0(x) dx,

ui,0(t) = 2

L

∫ L

0u0(x) cos

(iπx

L

)dx,

(26)

for 1 ≤ i ≤ n.All the nonlinearities in Equations (19)–(20) (and consequently in the following systems) are

of the form �(u, υ) = uυ and �(u) = u2.Using Equation (11), we have for �(u, υ)

A0 = u0υ0,

A1 = u1υ0 + u0υ1,

A2 = u2υ0 + u1υ1 + u0υ2,

A3 = u3υ0 + u2υ1 + u1υ2 + u0υ3,

A4 = u4υ0 + u3υ1 + u2υ2 + u1υ3 + u0υ4,

...

and for �(u), it is only necessary to consider that u = υ in �(·, ·). Thus, we have

B0 = u20,

B1 = 2u1u0,

B2 = 2u2u0 + u21,

B3 = 2u3u0 + 2u1u2,

B4 = u22 + 2u1υ3 + 2u0υ4,

...

In the next section, we present one example of application.

4. Convergence of the method

In order to obtain the approximate solution to this problem, the equation is transformed in anonlinear system of ordinary differential equations,

U(T) = AU(t) + F(U(t))

with initial conditions U(0) = U0.For results concerning the existence of solution of nonlinear ordinary differential equations,

we refer the reader to [10], where this subject is mentioned. From that system, we obtain a systemof second-order Volterra integral equations. The convergence of the Adomian decompositionmethod when applied to nonlinear systems of ordinary differential equations and second-orderVolterra integral equations has been studied by several authors, namely Babolian and Biazar [7]and El-Kalla [14] where the problem of convergence is studied.

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International Journal of Computer Mathematics 887

5. Numerical simulation

All the calculations and graphics presented in this section were obtained using the symbolicsoftware Mathematica.

Let us consider the problem

ut = ε2uxx − 1

2u2

x + u − 1

π

∫ π

0u(t, x) dx, x ∈]0, π [, t ∈ (0, T ],

u(0, x) = 1 + cos(x)

100, x ∈]0, π [,

ux = 0, x ∈ {0, π}, t > 0.

(27)

For solving, consider that n = 5, ε2 = 10−2 and 0 ≤ m ≤ 5.Using Mathematica, we obtain the following approximate solution to the system (24)–(25):

u(t, x) ≈ u0(t) +5∑

k=1

uk(t) cos

(iπx

L

), (28)

where uk(t) = ∑5j=0 uk,jtj, and

u0,0(t) = 0.01000000000, u1,0(t) = 0.01000000000,

u0,1(t) = −0.000025t, u1,1(t) = 0.009999999900t,

u0,2(t) = −0.000025t2, u1,2(t) = 0.004999874900t2,

u0,3(t) = −0.0000166667t3, u1,3(t) = 0.001666416617t3,

u0,4(t) = −8.333333083330211 × 10−6t4, u1,4(t) = 0.0004164062339t4,

u0,5(t) = −3.3333331999931272 × 10−6t5, u1,5(t) = 0.00008314583074t5,

u2,0(t) = 0.00002500000000, u3,0(t) = 0,

u2,1(t) = 0.00003749999925t, u3,1(t) = 0,

u2,2(t) = 0.00002916583225t2, u3,2(t) = 1.250000000 × 10−7t2,

u2,3(t) = 0.00001562291585t3, u3,3(t) = 2.499999929 × 10−7t3,

u2,4(t) = 6.45562459 × 10−6t4, u3,4(t) = 2.604096219 × 10−7t4,

u2,5(t) = 6.45562459 × 10−6t5, u3,5(t) = 1.874788927 × 10−7t5,

u4,0(t) = 0, u5,0(t) = 0,

u4,1(t) = 0, u5,1(t) = 0,

u4,2(t) = 0, u5,2(t) = 0,

u4,3(t) = 8.33333333 × 10−10t3, u5,3(t) = 0,

u4,4(t) = 2.083333259 × 10−9t4, u5,4(t) = 6.51041667 × 10−12t4,

u4,5(t) = 2.70826649 × 10−9t5, u5,5(t) = 1.953124918 × 10−11t5.

The surface in Figure 1 is a graphical representation of the approximate solution, u(t, x) given byEquation (28).

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Figure 1. The surface shows the approximate solution u(t, x), for 0 ≤ x ≤ π , 0 ≤ t ≤ 10.

In this case, the approximate solution satisfies both initial and boundary conditions. This isa consequence of the chosen initial condition because it belongs to the space spanned by thebasis (7). If the initial condition does not belong to that space, the approximate solution does notsatisfy the initial condition but always satisfies the boundary conditions. In this case, in order toobtain a reasonable approximate solution it is necessary to obtain a very good approximation ofthe initial condition. In both cases, the approximate solution satisfies the boundary condition.

6. Conclusions

In this paper, the Fourier method and the Adomian decomposition method were used to obtain anapproximate solution to an initial boundary problem to the one-dimensional Rakib–Sivashinskyequation. The procedure is simple to implement and only with a few terms provides a reliableapproximate solution. It also avoids the difficulties and massive computational work comparedto other existing techniques.

Acknowledgment

This work was partially supported by FCT under the pluriannual founding of unit 212-CMUBI.

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