Series solution to the Falkner–Skan equation with stretching boundary

Applied Mathematics and Computation 208 (2009) 156–164

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Series solution to the Falkner–Skan equation with stretching boundary Baoheng Yao a,*, Jianping Chen b a b

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200030, China School of Resource and Environment Engineering, Liaoning Technical University, Fuxin 123000, China

a r t i c l e

i n f o

a b s t r a c t A new analytical method, namely homotopy analysis method (HAM), is applied to solve the nonlinear Falkner–Skan equation with stretching boundary and a series solution is given in this paper. The comparisons are also made among the results of the present work, Riley and Weidman’s and numerical method by fourth-order Runge–Kutta method combined with Newton–Raphson technique. It shows that the analytical approximate solution agrees well with numerical method for Falkner–Skan wedge flow (b > 0) when c P 0 and also satisfy the conclusion of Riley and Weidman that there is unique solution in this case, which shows the validity of the present work in this condition. For the case of c < 0 with a range of values of b, the analytical approximate solution gives upper solution branch of the multiple solutions of the Falkner–Skan equation with stretching boundary by numerical methods of both Riley and Weidman’s and the author’s, and the possible reasons are further analyzed. Ó 2008 Elsevier Inc. All rights reserved.

Keywords: The Falkner–Skan equation Stretching wall Approximate analytical solution Homotopy analysis method

1. Introduction The famous nonlinear Falkner–Skan equation arises from the boundary layer flow with stream-wise pressure gradient [1] is governed by 00

f 000 þ ff þ bð1  f 02 Þ ¼ 0

ð1Þ

with the boundary conditions

f ð0Þ ¼ 0;

f 0 ð0Þ ¼ c;

f 0 ðþ1Þ ¼ 1;

ð2Þ

where c is the movement velocity ratio of the plate to the mainstream, i.e. when 0 < c < 1, the speed of oncoming fluid is larger than that of the plate, while the speed of the moving plate is faster than that of oncoming fluid when c > 1, and c < 0 corresponds to the plate and the mainstream move in opposite directions. Particularly,c = 0 is for a fixed plate. Due to its strong nonlinearity in computational physics, such researchers as Hartree [2], Howarth [3], Asaithambi [4], Cebeci and Keller [5], Sher and Yakhot [6], and Liu and Chang [7] have investigated numerically the solutions of the Falkner–Skan equation with the boundary conditions of resting and impermeable wall. Besides, Liao [8,9] has applied a newly developed analytical method, namely homotopy analysis method (HAM), to solve the Falkner–Skan equation and provide an explicit, totally analytical solution with the same boundary conditions as those of above. Recently, Fang and Zhang [10] has studied analytically a special case of the Falkner–Skan equation with b = 1 with the boundary conditions of mass transfer and wall stretching. * Corresponding author. E-mail address: [email protected] (B. Yao). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.11.028

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Here, the work of Riley and Weidman [11] in 1989 should be specially emphasized, where they applied numerical techniques of both shooting method based on a variable order Adams predictor–corrector method and a second-order finite difference method to treat the problem as a two-point boundary value problem. They have explored the solution of the Falkner–Skan equation with stretching boundary for jbj 6 1 and a range of values of c, and show that there is unique solution for Falkner–Skan wedge flow (b > 0) when c P 0, however, multiple solutions of the Falkner–Skan equation with stretching boundary exist for the case of c < 0 with a range of values of b. In this paper, the author has applied the same technique as Liao, different from previous work, however, more complicated boundary conditions of f(0) = 0, f0 (0) = c, f0 (+1) = 0, which is a correspondingly stretching boundary, is concentrated on and a series solution of the Falkner–Skan equation is given. The author investigates analytically the Falkner–Skan equation with stretching boundary, which also differs from the work of Riley and Weidman. The comparisons are also made among the results of the present work, Riley and Weidman’s and numerical method by fourth-order Runge–Kutta method combined with Newton–Raphson technique. It shows that the analytical approximate solution agrees well with numerical method for Falkner–Skan wedge flow (b > 0) when c P 0 and also satisfy the conclusion of Riley and Weidman that there is unique solution in this case, which shows the validity of the present work in this condition For the case of c < 0 with a range of values of b, the analytical approximate solution gives upper solution branch of the multiple solutions of the Falkner–Skan equation with stretching boundary by numerical methods of both Riley and Weidman’s and the author’s, and the possible reasons are further analyzed. 2. Series solution to the Falkner–Skan equation by HAM Due to the boundary conditions in (2) by homotopy analysis method, a set of basis function

fgm ekng ; m; n P 0; k > 0g

ð3Þ

is chosen to express the solutions of the Falkner–Skan equation as

f ðgÞ ¼

þ1 X þ1 X

am;n gm ekng ;

ð4Þ

m¼0 n¼0

where am,n is a coefficient, and k is an auxiliary parameter which will be determined later. Thus, the solution expression of f(g) is provided, which satisfies the boundary conditions (2). 2.1. Zero-order deformation equation of HAM According to the solution expression in (4) and the boundary conditions in (2), it is led to choose the initial guess solution of Eq. (1) as

f0 ðgÞ ¼ g þ

1  ekg ðc  1Þ; k

ð5Þ

and the auxiliary linear operator L is chosen to be

L½/ðg; qÞ ¼

@3/ @2/ þk 2; 3 @g @g

ð6Þ

which has the property of

L½C 1 þ C 2 g þ C 3 ekg  ¼ 0;

ð7Þ

where C1, C2, C3 are integral constants to be determined by the boundary conditions. Eq. (1) also suggests defining a nonlinear operator N as

N½/ðg; qÞ ¼

(  2 ) @ 3 /ðg; qÞ @ 2 /ðg; qÞ @/ðg; qÞ ; þ /ð g ; qÞ þ b 1  @ g3 @ g2 @g

ð8Þ

where q 2 [0, 1] is an embedding parameter. Thus, given non-zero auxiliary parameter  h and non-zero auxiliary function H(g) as well, zero-order deformation equation of HAM is constructed as

ð1  qÞL½/ðg; qÞ  f0 ðgÞ ¼ qhHðgÞN½/ðg; qÞ

ð9Þ

with the boundary conditions

/ð0; qÞ ¼ 0;

@/ðg; qÞ jg¼0 ¼ c; @g

@/ðg; qÞ jg¼þ1 ¼ 1: @g

ð10Þ

Clearly, when q = 0, the solution to Eq. (9) is given by

/ðg; 0Þ ¼ f0 ðgÞ;

ð11Þ

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B. Yao, J. Chen / Applied Mathematics and Computation 208 (2009) 156–164

and when q = 1, because of h  – 0 and H(g) – 0, the solution is equivalent to that of Eq. (1) if the condition of

/ðg; 1Þ ¼ f ðgÞ

ð12Þ

is satisfied. Therefore, /(g; q) varies from the initial guess solution f0(g) to the exact solution f(g) for Eq. (1) continuously as q increases from 0 to 1,which is so-called deformation in topology. Therefore, /(g; q) can be expanded in Taylor series with respect to q to provide

/ðg; qÞ ¼ f0 ðgÞ þ

þ1 X

fk ðgÞqk ;

ð13Þ

k¼1

where

 1 @ k /ðg; qÞ fk ðgÞ ¼  k! @qk 

ð14Þ

:

q¼0

Note that /(g;q) is expanded in Taylor series with respect to q instead of g. Provided that the auxiliary linear operator L, the h and auxiliary functions H(g) are properly chosen so that the convergence initial guess solution f0(g), auxiliary parameters  of the above series at q = 1 is guaranteed, due to Eq. (13), the series solution of Eq. (1) can be expressed as follows:

/ðg; qÞ ¼ f0 ðgÞ þ

þ1 X

fk ðgÞ:

ð15Þ

k¼1

2.2. kth-order deformation equation of HAM For the convenience, define the vectors

~ f k ¼ ff0 ðgÞ; f1 ðgÞ; . . . ; fk ðgÞg;

ð16Þ

where k 2 N. Differentiating the zero-order deformation Eq. (9) k times with respect to q, setting q = 0, and then dividing by k!, the kth-order deformation equation is obtained by

L½fk ðgÞ  vk fk1 ðgÞ ¼ hHðgÞRk ð~ f k1 ; gÞ;

ð17Þ

which satisfies the boundary conditions

fk ð0Þ ¼ 0;

fk0 ð0Þ ¼ 0;

fk0 ðþ1Þ ¼ 0;

ð18Þ

where

vk ¼



0; k 6 1 1; k > 1

ð19Þ

;

000 f k1 ; gÞ ¼ fk1 ðgÞ þ Rk ð~

k1 X 00 0 ½fj ðgÞfk1j ðgÞ  bfj0 ðgÞfk1j ðgÞ þ bð1  vk Þ:

ð20Þ

j¼0

Let fk ðgÞ denote a special solution of equation

L½fk ðgÞ ¼ hHðgÞRk ð~ f k1 ; gÞ;

ð21Þ

then according to the property of linear operator in Eq. (7), the general solution of Eq. (17) can be expressed as

fk ðgÞ ¼ vk fk1 ðgÞ þ fk ðgÞ þ C k1 þ C k2 g þ C k3 ekg ; where

C k1 ,

C k2

and

C k3

ð22Þ

are determined by the boundary conditions in (18), i.e.

C k1 ¼ fk ð0Þ  fk0 ð0Þ=k;

C k2 ¼ 0;

C k3 ¼ fk0 ð0Þ=k:

ð23Þ

For simplicity, the auxiliary functions H(g) can be determined by

HðgÞ ¼ 1

ð24Þ

in terms of the rules of solution expression and coefficient ergodicity by homotopy analysis method [9]. Thus, strongly nonlinear Falkner–Skan equation (1) is converted into a series of linear problems as in Eq. (17), which can be easily solved by symbolic computation software such as Mathematica and Maple, etc. It is noticed that the conversion is independent of small parameters, which differs from that of perturbation technique.

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159

3. Results and analysis Liao [9] has proved in general that once a series solution given by the homotopy analysis method converges, it must be one of the solutions for considered problem, and therefore, one can obtain an analytical approximation solution by finite terms of infinite series within the admission of errors. However, if the series solution given by HAM converges very slow or the series solution has a property of chatter, one has to apply homotopy-Padé technique to further accelerate the convergence of the series solution and overcome the possible chatter of the series solution in this case, as suggested by Liao in [9]. Note that the solution series (15) contains the auxiliary parameter  h and the parameter k in the basis function, which can h curve and f00 (0)  k curve as suggested by Liao [9], where f00 (0) shows the shear be determined by plotting so-called f00 (0)   friction at the wall. For simplicity, one can fix h  = 1 unchanged and investigate the influence of parameter k. It is discovered that the parameter of k = 5 is proper according to the floor of f00 (0)  k curve by 10th-order analytical approximation solution, h = 1 are given, as shown where some fixed c is evaluated to be 1, 0, 1 and 2 respectively when the parameters of b = 2 and  in Fig. 1. 3.1. Case I. c P 0 It is discovered that 10th-order series solutions of f00 (0) by HAM agree well with numerical results by fourth-order Runge– Kutta method combined with Newton–Raphson technique for all the values of 0 6 b 6 2 when c P 0, where integral distance g1 = 10, which was discretized by 10,000 intervals, is found to be adequate with the error of f00 (0) less than 1  106, as shown in Fig. 2. Obviously, f00 (0) depends on the parameters of c and b, and the area of c in which f00 (0) agrees with numerical results, tends to decrease with the increment of b. The analytical approximate solution also agrees with the conclusion of Riley and Weidman [11] that there is unique solution for the Falkner–Skan wedge flow (b > 0) when c P 0, which shows the validity of the present work in this condition.

γ = −1 γ =0

2

f ′′(0)

1

γ =1

0 1

γ =2

2 2

4

6

λ

8

10

12

14

Fig. 1. f0 (0)  k curve of 10th-order analytical approximation solution with  h = 1 and b = 2 (solid line: c = 1; dotted line: c = 0; double dotted line: c = 1; dashed line: c = 2).

0 20

f ′′(0)

40 60

β =2

80

β =1

β =0

100 120 140 10

20 γ

30

40

Fig. 2. The comparison of 10th-order HAM analytical approximation solution of f00 (0) with numerical results when  h = 1, k = 5 and 0 6 c 6 40 (solid line: b = 0; dotted line: b = 1; dashed line: b = 2; filled circles: numerical results).

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B. Yao, J. Chen / Applied Mathematics and Computation 208 (2009) 156–164

Furthermore, the influences of b and c on the solution curves of f0 (g)  g are further studied, where some fixed b is evaluated to be 0, 1 and 2 respectively when the parameter of c = 5 is fixed unchanged, some fixed c is evaluated to be 0, 1 and 5 respectively when the parameter of b = 1 is fixed, and some fixed b is evaluated to be 2 when the parameter of c = 10 is fixed unchanged as well. It is found that the 10th-order analytical solutions by HAM agree well with numerical results by fourthorder Runge–Kutta method combined with Newton–Raphson technique, as shown in Figs. 3–5.

5

f ′ (η)

4 β = 0,1,2

3 2 1

1

2 η

3

4

Fig. 3. f0 (g)  g curve of 10th-order analytical approximation solution with  h = 1 and k = 5 when c = 5 (solid line: b = 0; dotted line: b = 1; dashed line: b = 2; filled circles: numerical results).

5

f ′ (η)

4 3

γ =5

2 1

γ =1 γ =0 1

2

3

η

4

Fig. 4. f0 (g)  g curve of 10th-order analytical approximation solution with  h = 1 and k = 5 when b = 1 (solid line: c = 0; dotted line: c = 1; dashed line: c = 5; filled circles: numerical results).

10

f ′ (η)

8 6 4 2 0 1

2

3

η

4

5

6

7

Fig. 5. f0 (g)  g curve of 10th-order analytical approximation solution with  h = 1, b = 2 and k = 5 when c = 10 (solid line: 10th-order analytical approximation solution; filled circles: numerical results).

161

B. Yao, J. Chen / Applied Mathematics and Computation 208 (2009) 156–164 Table 1 The comparison of HAM analytical approximation solutions of f00 (0) with numerical results with h = 1 and k = 5 when b = 0.

c

5th-order

10th-order

15th-order

20th-order

25th-order

Numerical results

0.2 0 0.5 1.0 3.0 5.0 8.0 10.0 15.0

0.372060 0.389688 0.287185 0 2.49233 6.29061 13.1749 18.3491 34.0257

0.439676 0.466892 0.332080 0 2.36646 5.97203 13.0309 18.3969 34.8547

0.436655 0.468833 0.328822 0 2.40445 5.94971 12.9803 18.4193 34.8744

0.434488 0.469471 0.328738 0 2.39700 5.97074 12.9563 18.4315 34.7645

0.432928 0.469563 0.328736 0 2.39663 5.97647 12.9441 18.4388 34.7648

0.4302 or 0.0222 0.4696 0.3287 0 2.3973 5.9725 12.935 18.453 34.79

Table 2 The comparison of HAM analytical approximation solutions of f00 (0) with numerical results with h = 1 and k = 5 when b = 1.

c

5th-order

10th-order

15th-order

20th-order

25th-order

Numerical results

1.0 0.3 0.2 0 0.5 1.0 3.0 5.0 8.0 10.0 15.0

1.29709 1.41575 1.37290 1.24884 0.741052 0 4.43772 10.2070 20.9746 29.9975 58.1732

1.32832 1.42931 1.37470 1.23079 0.708524 0 4.32030 10.2617 21.5723 30.8940 56.7176

1.32947 1.42752 1.37366 1.23245 0.713750 0 4.28057 10.2676 21.7356 30.680 57.1123

1.32897 1.42754 1.37390 1.23266 0.713282 0 4. 27317 10.2679 21.7383 30.5776 57.1140

1.32885 1.42758 1.37389 1.23258 0.713296 0 4.27439 10.2672 21.6968 30.6754 56.9737

1.3288 or 0.0 1.4276 1.3739 1.2326 0.7133 0 4.2765 10.265 21.685 30.655 57.056

Table 3 The comparison of HAM analytical approximation solutions of f00 (0) with numerical results with h = 1 and k = 5 when b = 2.

c

5th-order

10th-order

15th-order

20th-order

25th-order

Numerical results

0.3 0.2 0 0.5 1.0 3.0 5.0 8.0 10.0

2.05433 1.96881 1.75408 0.999698 0 5.51703 12.8795 28.0534 40.5253

1.98623 1.89893 1.68647 0.962484 0 5.58088 13.2860 28.2674 39.2896

1.98716 1.90004 1.68711 0.960121 0 5.59497 13.3905 27.9577 39.8238

1.98721 1.90006 1.68719 0.960380 0 5.59998 13.3928 28.1710 39.5455

1.98724 1.90010 1.68721 0.960413 0 5.60213 13.3702 28.0633 39.6867

1.9872 1.9001 1.6872 0.96042 0 5.6042 13.362 28.093 39.642

The comparison of series solution of f00 (0) by HAM with numerical results by fourth-order Runge–Kutta method combined with Newton–Raphson technique is made as well, where the parameter of c varies from 0 to 15 with some fixed b is evaluated to be 0, 1 and 2 respectively, as shown in Tables 1–3. Especially, homotopy-Padé technique is applied to further accelerate the convergence of the series solution and overcome the ”chatter” of the series solution by HAM, as shown in

Table 4 The comparison of [m, m] homotopy-Padé analytical approximation solutions of f00 (0) with numerical results with h = 1 and k = 5 when b = 0.

c

[5, 5]

[10, 10]

[12, 12]

Numerical results

0.2 0 0.5 1.0 3.0 5.0 8.0 10.0 15.0

0.43358 0.46736 0.32735 0 2.3858 5.9541 12.922 18.445 34.788

0.43021 0.46960 0.32874 0 2.3973 5.9725 12.935 18.453 34.790

0.43024 0.46960 0.32874 0 2.3973 5.9725 12.935 18.453 34.790

0.4302 or 0.0222 0.4696 0.3287 0 2.3973 5.9725 12.935 18.453 34.79

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B. Yao, J. Chen / Applied Mathematics and Computation 208 (2009) 156–164

Table 5 The comparison of [m, m] homotopy-Padé analytical approximation solutions of f00 (0) with numerical results with h = 1 and k = 5 when b = 1.

c

[5, 5]

[10, 10]

[12, 12]

Numerical results

1.0 0.3 0.2 0 0.5 1.0 3.0 5.0 8.0 10.0 15.0

1.3276 1.4258 1.3720 1.2306 0.7123 0 4.2713 10.2592 21.6825 30.6538 57.0558

1.3288 1.4276 1.3739 1.2326 0.7133 0 4.2765 10.2648 21.6848 30.6547 57.0558

1.3288 1.4276 1.3739 1.2326 0.7133 0 4.2765 10.2648 21.6848 30.6547 57.0558

1.3288 or 0.0 1.4276 1.3739 1.2326 0.7133 0 4.2765 10.265 21.685 30.655 57.056

Table 6 The comparison of [m, m] homotopy-Padé analytical approximation solutions of f00 (0) with numerical results with h = 1 and k = 5 when b = 2.

c

[5, 5]

[10, 10]

[12, 12]

Numerical results

0.3 0.2 0 0.5 1.0 3.0 5.0 8.0 10.0

1.9862 1.8991 1.6864 0.96007 0 5.6024 13.3605 28.0927 39.6416

1.9872 1.9001 1.6872 0.96042 0 5.6042 13.3618 28.0930 39.6417

1.9872 1.9001 1.6872 0.96042 0 5.6042 13.3618 28.0930 39.6417

1.9872 1.9001 1.6872 0.96042 0 5.6042 13.362 28.093 39.642

Tables 4–6. Here, the author only provides the results of [5, 5], [10, 10], [12, 12] homotopy-Padé approximation results, which agrees well with numerical results, and further verifies the validity of the present work in this case. 3.2. Case II. c < 0 As pointed out by Riley and Weidman [11], multiple solutions of the Falkner–Skan equation with stretching boundary exist for the case of c < 0 with a range of values of b. The numerical results by fourth-order Runge–Kutta method combined with Newton–Raphson technique in present work also verify it, as shown in Tables 1–3, where c is evaluated to be 0.2 for b = 0, and 1, 0.3 and 0.2 for b = 1, and 0.3 and 0.2 for b = 2, respectively. Homotopy-Padé technique is also applied to further accelerate the convergence of the series solution and overcome the ‘‘chatter” of the series solution by HAM in this case, as shown in Tables 4–6, which also agrees well with numerical results. Although homotopy analysis method provides the freedom to choose different initial guess solution f0(g) and auxiliary parameters  h to express the series solution, it is found that the series solution given by homotopy analysis method in this

1.5 1.25

f ′ (η )

1 0.75 0.5 0.25 0 2

4

6

8

10

η Fig. 6. f0 (g)  g curve of 20th-order analytical approximation solution comparisons with numerical results for upper and lower solution branch when  = 1, k = 5, b = 0 and c = 0.2 (solid line: 20th-order analytical approximation solution; filled circles: numerical results for upper and lower solution h branches).

B. Yao, J. Chen / Applied Mathematics and Computation 208 (2009) 156–164

163

1.5 1.25

β = 0,1,2

f ′ (η)

1 0.75 0.5 0.25 0 2

4

η

6

8

Fig. 7. f0 (g)  g curve of analytical approximation solution with  h = 1 and k = 5 when c = 0.2 (solid line: 20th-order analytical approximation solution with b = 0; dotted line: 10th-order analytical approximation solution with b = 1; dashed line: 10th-order analytical approximation solution with b = 2; filled circles: numerical results for upper solution branch when b = 0, and numerical results for b = 1 and b = 2).

paper provides correspondingly the upper solution branch of numerical results for the case c < 0, as shown in Fig. 6, where f0 (g)  g curve of 20th-order analytical approximation solution comparisons with numerical results for upper and lower solution branch are given when  h = 1, k = 5, b = 0 and c = 0.2. Moreover, Fig. 7 shows the influence of parameterb on h = 1, k = 5 and c = 0.2, which also agrees well with numerthe f0 (g)  g curve, where b is evaluated to be 0, 1 and 2, with  ical results. Possible reason attributes to the fact that the chosen basis function fgm ekng ; m; n P 0; k > 0g to express f(g) can guarantee the solution to have the asymptotic property of f0 (+1) ? 1 with exponential mode, however, as pointed out by Liao [9], more proofs should be required for numerical methods of both present work and Riley and Weidman’s to support multiple solutions of the Falkner–Skan equation to have the asymptotic property of f0 (+1) ? 1 with exponential mode, because numerical methods can handle only discrete limited area instead of infinity. Thus, the proof of f0 (+1) ? 1 with exponential mode is beyond the capability of numerical methods. This leaves an open topic whether multiple solutions of the Falkner– Skan equation with stretching boundary by numerical methods have the asymptotic property of f0 (+1) ? 1 with exponential mode. If it is true, how can we obtain analytically multiple solutions of the Falkner–Skan equation with stretching boundary? Otherwise, what is the asymptotic property of f0 (+1) ? 1 for the Falkner–Skan equation with stretching boundary? 4. Conclusions A new analytical method, namely homotopy analysis method (HAM), is applied to solve the nonlinear Falkner–Skan equation with stretching boundary and a series solution is given in this paper. The influences of b and c on the solution are further studied. The comparisons are also made among the results of the present work, Riley and Weidman’s and numerical method by fourth-order Runge–Kutta method combined with Newton–Raphson technique. It shows that the analytical approximate solution agrees well with numerical method for Falkner–Skan wedge flow (b > 0) when c P 0 and also satisfy the conclusion of Riley and Weidman that there is unique solution in this case, which shows the validity of the present work in this condition. For the case of c < 0 with a range of values of b, the analytical approximate solution gives upper solution branch of the multiple solutions of the Falkner–Skan equation with stretching boundary by numerical methods of both Riley and Weidman’s and the author’s, and the possible reasons are further analyzed. An open topic is left whether multiple solutions of the Falkner–Skan equation with stretching boundary by numerical methods have the asymptotic property of f0 (+1) ? 1 with exponential mode. If it is true, how can we obtain analytically multiple solutions of the Falkner–Skan equation with stretching boundary? Otherwise, what is the asymptotic property of f0 (+1) ? 1 for the Falkner–Skan equation with stretching boundary? Acknowledgement The author expresses sincere thanks to anonymous reviewers for their valuable comments and provision of the reference of Riley and Weidman to improve the present work. References [1] V.M. Falkner, S.W. Skan, Some approximate solutions of the boundary layer equations, Philos. Mag. 12 (1931) 865–896. [2] D.H. Hartree, On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer, Proc. Camb. Philos. Soc. 33 (Part II) (1937) 223–239. [3] L. Howarth, On the solution of the laminar boundary layer equations, Proc. Roy. Soc. London A 164 (1938) 547–579.

164 [4] [5] [6] [7] [8] [9] [10] [11]

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