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Chaos, Solitons and Fractals 35 (2008) 738–746 www.elsevier.com/locate/chaos Numerical solution for the Falkner–Skan equation Nasser S. Elgazery * Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Heliopolis, Cairo, Egypt Accepted 22 May 2006 Abstract In this paper, an ana...

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Numerical solution for the Falkner– Skan equation Nasser Elgazery Chaos, Solitons & Fractals

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Chaos, Solitons and Fractals 35 (2008) 738–746 www.elsevier.com/locate/chaos

Numerical solution for the Falkner–Skan equation Nasser S. Elgazery

*

Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Heliopolis, Cairo, Egypt Accepted 22 May 2006

Abstract In this paper, an analysis is presented for the numerical solution of the Falkner–Skan equation. The nonlinear ordinary differential equation is solved using Adomian decomposition method (ADM). The condition at infinity will be applied to a related Pade´ approximation to the obtained series solution. By using MATHEMATICATM Adomian polynomials and Pade´ approximation of the obtained series (ADM) solution have been calculated. From the computational viewpoint, the solutions obtained thus by the ADM and shooting method are in excellent agreement with those obtained by previous works and efficient to use.  2006 Published by Elsevier Ltd.

1. Introduction Since the beginning of the 1980s, Adomian [1–5] has presented and developed a so-called decomposition method for solving linear or nonlinear problems such as ordinary differential equations. Adomian decomposition method (ADM) consists of splitting the given equation into linear and nonlinear parts, inverting the highest-order derivative operator contained in the linear operator on both sides, identifying the initial and/or boundary conditions and the terms involving the independent variable alone as initial approximation, decomposing the unknown function into a series whose components are to be determined, decomposing the nonlinear function in terms of special polynomials called Adomian polynomials, and finding the successive terms of the series solution by recurrent relation using Adomian polynomials. That is the ADM transform the system of equations into a set of recursive relations. The resulting algebraic systems of recursive relations can be easily solve by using Mathematica software. The ADM is quantitative rather than qualitative, analytic, requiring neither linearization nor perturbation, and continuous with no resort to discretization and consequent computer-intensive calculations. A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations is studied by Dogˇan and Asif [13] and a comparison of Adomian decomposition method and wavelet-Galerkin method for solving integro-differential equations is studied by El-Sayed and Abdel-Aziz [17]. Also, El-Sayed investigated the decomposition method for studying the KleinGordon equation [16]. The construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method and the construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method are studied by Wazwaz [24,25]. The ADM [21,22] has been applied to a wide class of *

Fax: +20 2 4552138. E-mail address: [email protected]

0960-0779/$ - see front matter  2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.05.040

N.S. Elgazery / Chaos, Solitons and Fractals 35 (2008) 738–746

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stochastic and deterministic problems in physics, biology and chemical reactions. Some applications [11,14,15] of this method show its advantages. Recently, Lesnic studied blow-up solutions obtained using the decomposition method [19]. In order to improve the accuracy of ADM, we used the aftertreatment technique (AT) which improves the accuracy of ADM and modifies Adomian’s series solution for general ordinary differential equations with initial conditions by using the Pade´ approximation. Generally, ADM yields the Taylor series of the true solution. Usually the AT can be used to get an analytic approximate solution which will greatly improve the convergence rate and accuracy of Adomian’s series. In this paper, we discussed oscillatory systems, so we used Laplace transformation for making the Pade´ approximation more efficient (for more details see [18]). In the present work, the system of boundary-value problem (BVP) is investigated. The shooting method is used to transform this system to the system of initial-value problem (IVP) because the AT is applicable to the (IVP) system [18]. The remaining part in this paper is organized as follows: In Section 1, we discussed the ADM solution for a model problem have exact solution in order to explain the accuracy of this method by the comparison between the exact solution, the shooting method and the present method results. Also, the Falkner–Skan equation (nonlinear ordinary differential equation) is solved numerically by using shooting method and by using the ADM in Section 2. All this calculations computed with MATHEMATICATM.

2. The ADM for a model problem Consider the boundary layer equation:   1 0 f ðgÞ ¼ 0; f 000 ðgÞ þ f ðgÞf 00 ðgÞ  ðf 0 ðgÞÞ2  M þ kp

0 6 g < 1:

ð1Þ

With the boundary conditions: f 0 ð0Þ ¼ 1 and

f ð0Þ ¼ fw ;

f 0 ð1Þ ¼ 0:

ð2Þ

This boundary layer equation is presented for magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid with suction and blowing through a porous medium where M is the magnetic parameter, kp is the permeability parameter and fw is the mass transfer parameter, which is positive for suction and negative for injection. Fortunately, the boundary value problem (1) together the boundary conditions (2) has an exact solution in the form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 1 1 2  2e2gz þ fw z ; z ¼ fw þ 4 þ 4ðM þ Þ þ fw2 : ð3Þ f  ðgÞ ¼ z kp Now, we will use the AT which improves the accuracy of the ADM and application of Laplace transformation and Pade´ approximation to the series solution derived from this method to solve the nonlinear ordinary differential Eq. (1) which can be rewritten as:   1 Lg f ðgÞ ¼ f ðgÞf 00 ðgÞ þ ðf 0 ðgÞÞ2 þ M þ f 0 ðgÞ; ð4Þ kp where Lg is considered, without loss of generality, third order differential operator. Also, we defined L1 g , without loss of generality, (inverse operator) triple integral operator as: Z gZ gZ g L1 f ðgÞdgdgdg: ð5Þ f ðgÞ ¼ g 0

0

0

Then by taken L1 g for the both sides of Eq. (4), gives     1 00 1 1 0 2 00 1 0 f ð0Þg2 þ f 0 ð0Þg þ f ð0Þ ¼ L1 f ðgÞ  ðf ðgÞf ðgÞÞ þ L ðf ðgÞÞ M þ L f ðgÞ: g g 2 kp g

ð6Þ

The ADM suggests that, the linear term f(g) be decomposed by an infinite series of components f ðgÞ ¼

1 X

fn ðgÞ;

n¼0

and the nonlinear terms f(g)f00 (g) and (f 0 (g))2 by the infinite series of the so-called Adomian polynomials

ð7Þ

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N.S. Elgazery / Chaos, Solitons and Fractals 35 (2008) 738–746

f ðgÞf 00 ðgÞ ¼ 0

2

ðf ðgÞÞ ¼

1 P

9 > An ðgÞ; > > = n¼0

1 P

Bn ðgÞ;

n¼0

ð8Þ

> > > ;

where fn(g), n P 0 is the components of f(g) that will be elegantly determined, and An(g) and Bn(g), n P 0 are Adomian polynomials that can be generated for all forms of nonlinearity [5]. We introduce new formulas that can generate the Adomian polynomials An(g) and Bn(g) as: An ðgÞ ¼

n X

fi00 ðgÞfni ðgÞ;

i¼0

Bn ðgÞ ¼

n X

0 fi0 ðgÞfni ðgÞ;

i¼0

this new formulas are in excellent agreement with those from in [5] and the modified calculation scheme was introduced in [23]. Substituting (7) and (8) into (6) gives     1 1 1 1 X X X 1 00 1 1 X 1 Lg A ðgÞ þ L fg0 ðgÞ: ð9Þ B ðgÞ þ M þ f ð0Þg2 þ f 0 ð0Þg þ f ð0Þ  L1 fn ðgÞ ¼ n n g g 2 kp n¼0 n¼0 n¼0 n¼0 Following Adomian analysis, the nonlinear Eq. (1) with the boundary conditions (2) is transformed into a set of recursive relations given by 1 f0 ðgÞ ¼ f 00 ð0Þg2 þ f 0 ð0Þg þ f ð0Þ; 2

ð10aÞ

 1 1 0 1 1 L f ðgÞ; fkþ1 ðgÞ ¼ Lg Ak ðgÞ þ Lg Bk ðgÞ þ M þ kp g k 

k P 0:

ð10bÞ

We supposed that f00 (0) = X, and for simplify the calculation we will take in this section (as an example) M = 0.5; kp = 5 and fw = 0.1. By using MATHEMATICATM, we can get to the zeroes component which is defined by f0 ðgÞ ¼

X 2 g þ g þ 0:1: 2

ð11Þ

In a like manner we obtain f1 ðgÞ ¼ g3 ð0:283333  0:0166667XÞ þ 0:0708333g4 X þ 0:00833333g5 X2 :

ð12Þ

We supposed that U1 ðgÞ ¼ f0 ðgÞ þ f1 ðgÞ; i.e., U1 ðgÞ ¼ 0:1 þ g þ

X 2 g þ g3 ð0:283333  0:0166667XÞ þ 0:0708333g4 X þ 0:00833333g5 X2 : 2

We apply Laplace transformation to U1(g) (see [18]), which yields: L½U1 ðgÞ ¼

0:1 1 5:55112  1017 þ X 1:7  0:1X 1:7X X2 þ 2þ þ þ 5 þ 6: s s s3 s4 s s

For the sake of simplicity, let s ¼ 1t , then L½U1 ðgÞ ¼ 0:1t þ t2 þ t3 X þ t4 ð1:7  0:1XÞ þ 1:7t5 X þ t6 X2 : Pade´ approximation [10] approximates a function by the ratio of two polynomials. The coefficients of the powers occurring in the polynomials are determined by the coefficients in the Taylor series expansion of the function. Generally, the Pade´ approximation can enlarge the convergence domain of the truncated Taylor series, and can improve greatly the convergence rate of the truncated Maclaurin series.

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The [P/Q] Pade´ approximation of U1(g) which has degree n is: n n is even; P ¼ 2n1 n is odd; 2 n n is even; Q ¼ 2nþ1 n is odd: 2 Notice that, we can get to the value of X by using the shooting method to solve the boundary value problem (1) together the boundary conditions (2), (X !  1.3547990800844818 at M = 0.5; kp = 5 and fw = 0.1). Then [3/3] Pade´ approximate of L[U1(g)] yields:  3 0:1t þ 1:7000026698624486t2 þ 6:411659041495708t3 : ¼ 1 þ 7:000026698624487t þ 7:664314229557027t2  0:16164405702206894t3 3 Recalling t ¼ 1s , we obtain [3/3] in terms of s:  3 ¼ 3

0:1 1:7000026698624486 6:411659041495708 þ s s2 s3 : 7:000026698624487 7:664314229557027 0:16164405702206894 1þ þ  2 3 s s s

By using inverse Laplace transformation to [3/3], we obtain the solution f(g) f ðgÞ ¼ ð0:000316741Þe5:63475g  ð0:710698Þe1:38598g þ ð0:810382Þe0:020698g :

3. The ADM for the Falkner–Skan equation The purpose of the present section is solving the Falkner–Skan equation (nonlinear ordinary differential equation) numerically by using shooting method and by using the same method in Section 1. The method of this paper differs from these previous methods [6–9,12,20] for solving the Falkner–Skan equation because it do not use a coordinate transformation to map the physical domain [0, 1] directly to another computational domain. Also, it do not impose any asymptotic conditions. The Falkner–Skan equation arises in the study of laminar boundary layers exhibiting similarity. The solutions of the one-dimensional third-order boundary-value problem described by the well-known Falkner–Skan equation are the similarity solutions of the two-dimensional incompressible laminar boundary layer equations. This is a nonlinear two-point boundary value problem for which no closed-form solutions are available. The problem is described as follows: f 000 ðgÞ þ f ðgÞf 00 ðgÞ þ bð1  ðf 0 ðgÞÞ2 Þ ¼ 0;

0 < g < 1;

ð13Þ

along with the boundary conditions: f ð0Þ ¼ 0;

f 0 ð0Þ ¼ 0

and f 0 ð1Þ ¼ 1;

ð14Þ

where, in addition to the unknown function f(g), the solution of ((13) and (14)) is characterized by the value of a = f00 (0). Now, we will use the ADM to solve the Falkner–Skan equation (13) with the boundary conditions (14). The Adomian analysis for Eqs. (13) and (14) is transformed into a set of recursive relations given by: f ðgÞ ¼

1 X

fn ðgÞ;

ð15Þ

n¼0

s.t.

f0 ðgÞ ¼ b

g3 1 00 þ f ð0Þg2 þ f 0 ð0Þg þ f ð0Þ; 3! 2

1 fkþ1 ðgÞ ¼ L1 g Ak ðgÞ þ bLg Bk ðgÞ;

k P 0;

where, An(g) and Bn(g) are the Adomian polynomials for nonlinear terms, which have the same new formulas in part 1. For simplicity the calculation we will take in this section (as an example) b = 2.0. By using MATHEMATICATM, we can get to the zeroes component which is defined by:

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N.S. Elgazery / Chaos, Solitons and Fractals 35 (2008) 738–746

1 a f0 ðgÞ ¼ g3 þ g2 : 3 2 In a like manner we obtain f1 ðgÞ ¼ 0:0015873g7  0:00555556g6 a þ 0:00833333g5 a2 : Also, we can evaluate f2(g), f3(g) and f4(g) and substitute in (15) to calculate the solution f(g) by using ADM. Now, we will use the AT which improves the accuracy of the ADM and application of Laplace transformation and Pade´ approximation to the series solution derived from this method to solve the Falkner–Skan equation (13), so we supposed that: U1 ðgÞ ¼ f0 ðgÞ þ f1 ðgÞ; i.e., 1 a U1 ðgÞ ¼ 0:0015873g7  0:00555556g6 a þ 0:00833333g5 a2  g3 þ g2 : 3 2 We apply Laplace transformation to U1(g), which yields: L½U1 ðgÞ ¼

a 2 a2 4a 8  þ  þ : s3 s4 s6 s7 s8

For the sake of simplicity, let s ¼ 1t , then L½U1 ðgÞ ¼ at3  2t4 þ a2 t6  4at7 þ 8t8 : Notice that, we can get to the value of a by using the shooting method to solve the boundary value problem (13) together the boundary conditions (14), (a ! 1.6872196469150844). Then [4/4] Pade´ approximate of L[U1(g)] yields:  4 1:68722t2 þ 0:0259397t4 ¼ : 4 1 þ 1:20076t þ 1:42336t2 þ 1:97406t4 Recalling t ¼ 1s , we obtain [4/4] in terms of s  4 ¼ 4

1:68722 0:0259397 þ s3 s4 : 1:20076 1:42336 1:97406 1þ þ þ s s2 s4

By using inverse Laplace transformation to [4/4], we obtain the solution f*(g) by using the AT which improves the accuracy of the ADM. f  ðgÞ ¼ ð0:0880334  0:318151iÞeð1:013431:08339iÞg  ð0:0880334  0:318151iÞeð1:01343þ1:08339iÞg þ ð0:0880334 þ 0:25708iÞeð0:4130530:852276iÞg þ ð0:0880334  0:25708iÞeð0:413053þ0:852276iÞg :

4. Results and discussion

In Section 1: Comparison between the exact solution, the shooting method and the present method (ADM) results for the f 0 (g) profile have been drawn in Fig. 1. Also, Table 1 represents comparison between the exact solution, the

Fig. 1. Comparison between the exact solution, the shooting method and ADM.

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N.S. Elgazery / Chaos, Solitons and Fractals 35 (2008) 738–746

Table 1 Values of f 0 (g) for the exact solution, the shooting method and the ADM results for various values of mass transfer, magnetic and permeability parameters at g = 1 fw

M

kp

The exact solution

Shooting method

ADM

0.1 0.4 0.7

0.5

5

0.25799918962082960 0.21891087492136085 0.18268352405273460

0.25800188148637143 0.21891206245286404 0.18268460084921440

0.25799912640672207 0.21891084677109696 0.18268351456133010

0.1

0.5 1.0 1.5

0.25799918962082960 0.21565352545834748 0.18379611207987157

0.25800188148637143 0.21565474930607040 0.18379723870537523

0.25799912640672207 0.21565352315318300 0.18379611193965556

0.19555195078230442 0.21809836391448925 0.23105553876403254 0.25799918962082960

0.19555320807460870 0.21809963874035168 0.23105693938934452 0.25800188148637143

0.19555195039965814 0.21809836107458060 0.23105553056713854 0.25799912640672207

0.5

1 1.5 2 5

Table 2 The error E1 between the exact solution and the ADM solution and the error E2 between the exact solution and the shooting method solution for various values of mass transfer, magnetic and permeability parameters at g = 1 fw

M

0.1 0.4 0.7

0.5

0.1

0.5 1.0 1.5

kp

E1

5

0.5

E2 8

1 1.5 2 5

6.32141 · 10 2.81503 · 108 9.4914 · 109

2.69187 · 106 1.18753 · 106 1.0768 · 106

6.32141 · 108 2.30516 · 109 1.40216 · 1010

2.69187 · 106 1.22385 · 106 1.12663 · 106

3.82646 · 1010 2.83991 · 1010 8.19689 · 109 6.32141 · 108

1.25729 · 106 1.27483 · 106 1.40063 · 106 2.69187 · 106

shooting method and the ADM results of f 0 (g) for various values of mass transfer, magnetic and permeability parameters at g = 1 (as an example). Table 2 gives the error E1 between the exact solution and the ADM solution and the error E2 between the exact solution and the shooting method solution for various values of mass transfer, magnetic and permeability parameters at g = 1. In a word, Fig. 1, Tables 1 and 2 show that the values of the ADM for f 0 (g) are in excellent agreement with the first derivative of the exact solution (3). Also, the ADM solution is better

f'

1

fw=0.1, M=0.5. kp=10 kp=1

0.8

kp=0.5 kp=0.2

0.6

kp=0.1

0.4

0.2 η

0 0

2

4

6 0

Fig. 2. Effect of porous medium kp on f distribution.

8

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N.S. Elgazery / Chaos, Solitons and Fractals 35 (2008) 738–746

f'

1

fw=0.1, kp=5

M=0.0 M=0.5 M=1.0 M=1.5

0.8

0.6

0.4

0.2 η

0 0

2

4

6

8

Fig. 3. Effect of magnetic field M on f 0 distribution.

f'

1

M=0.5, kp=5

fw=-0.4 fw=0.0 fw=0.4 fw=0.7

0.8

0.6

0.4

0.2 η

0 0

2

4

6

8

Fig. 4. Effect of surface mass transfer fw on f 0 distribution.

than the shooting method solution. Also from the physical results for the present problem Figs. 2–4 display results for the f 0 (g) profile, curves are drawn for various values of mass transfer, magnetic and permeability parameters. As shown, f 0 (g) is increasing with increasing the porous medium parameter kp. We noticed that, the effect of kp becomes smaller as kp increases. Physically, this result can be achieved when the holes of the porous medium are very large so that the resistance of the medium may be neglected. f 0 (g) decreases as the magnetic parameter M increases. Also, f 0 (g) decreases as the surface mass transfer parameter fw increases.

Table 3 The values of f(g) and f**(g) for various values of b at g = 0.5 b

f(g)

f**(g)

jf(g)  f**(g)j

2.0000 1.0000 0.5000 0.0000 0.1000 0.1200 0.1500 0.1800 0.1988

0.1698426924279085 0.1335852599638095 0.1057480196203583 0.0587574181498757 0.0420199166414075 0.0377420587432691 0.0301834174904584 0.0198334134002721 0.0047967165406654

0.17093173069507692 0.13358526544418467 0.10555299160062477 0.05864270139880591 0.04195691612559437 0.03769072712361524 0.03015031757038009 0.01981935286058759 0.00479609940435915

1.089038...