Research Article Numerical Solution of the Blasius Viscous ...downloads.hindawi.com/archive/2016/9014354.pdf · Research Article Numerical Solution of the Blasius Viscous Flow Problem

Research ArticleNumerical Solution of the Blasius Viscous Flow Problem byQuartic B-Spline Method

Hossein Aminikhah and Somayyeh Kazemi

Department of Applied Mathematics Faculty of Mathematical Sciences University of Guilan PO Box 1914 Rasht 41938 Iran

Correspondence should be addressed to Hossein Aminikhah hosseinaminikhahgmailcom

Received 1 March 2016 Accepted 28 March 2016

Academic Editor Jose A Tenereiro Machado

Copyright copy 2016 H Aminikhah and S KazemiThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

A numerical method is proposed to study the laminar boundary layer about a flat plate in a uniform stream of fluid The presentedmethod is based on the quartic B-spline approximations with minimizing the error 119871

2-norm Theoretical considerations are

discussed The computed results are compared with some numerical results to show the efficiency of the proposed approach

1 Introduction

One of the well-known equations arising in fluid mechanicsand boundary layer approach is Blasius differential equationThe classical Blasius [1] equation is a third-order nonlineartwo-point boundary value problem which describes two-dimensional incompressible laminar flow over a semi-infiniteflat plate at high Reynolds number

2119891101584010158401015840

(119909) + 119891 (119909) 11989110158401015840(119909) = 0 119909 ge 0 (1)

with boundary conditions

119891 (0) = 0

1198911015840(0) = 0

lim119909rarrinfin

1198911015840(119909) = 1

(2)

where the prime denotes the derivatives with respect to 119909 Inaddition to the unknown function 119891 the solution of (1) and(2) is characterized by the value of 120572 = 119891

10158401015840(0) Blasius [1] in

1908 found the exact solution of boundary layer equation overa flat plate A highly accurate numerical solution of Blasiusequation has been provided by Howarth [2] who obtainedthe initial slope 120572 = 119891

10158401015840

ex(0) = 0332057 Liu and Chang[3] have developed a new numerical technique they havetransformed the governing equation into a nonlinear second-order boundary value problem by a new transformation

technique and then they have solved it by the Lie groupshooting method He [4 5] gave a solution in a family ofpower series with parameter 119901 by means of the perturbationmethod for solving this equation Bender et al [6] proposeda simple approach using 120575-expansion to obtain accuratetotally analytical solution of viscous fluid flow over a flatplate Aminikhah [7] used LTNHPM to obtain an analyticalapproximation to the solution of nonlinear Blasius viscousflow equation Recently the fixed point method (FPM) [8]which is based on the fixed point concept in functional anal-ysis is adopted to acquire the explicit approximate analyticalsolution to the nonlinear differential equation Finally efforts[9 10] have been made to obtain the solution at the surfaceboundary and changed the problem from a boundary valuedifferential equation into an initial value one The solution inthe entire domain however still requires computation

In this paper the quartic B-spline approximations areemployed to construct the numerical solution for solving theBlasius equationTheunknowns are obtainedwith using opti-mizationTheproposedmethod is applied to the problem andthe computed results are compared with those of Howarthrsquosmethod to demonstrate its efficiency

2 Description of the Method

Let there be a uniformpartition of an interval [0 119871] as follows0 = 119909

0lt 1199091lt sdot sdot sdot lt 119909

119873minus1lt 119909119873

= 119871 where ℎ = 119909119895+1

minus 119909119895

119895 = 0 1 119873 minus 1 The quartic B-splines are defined upon

Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2016 Article ID 9014354 6 pageshttpdxdoiorg10115520169014354

2 International Journal of Engineering Mathematics

an increasing set of 119873 + 1 knots over the problem domainplus 8 additional knots outside the problem domain the 8additional knots are positioned as

119909minus4

lt 119909minus3

lt 119909minus2

lt 119909minus1

lt 1199090

119909119873

lt 119909119873+1

lt 119909119873+2

lt 119909119873+3

lt 119909119873+4

(3)

The quartic B-splines119861119895(119909) 119895 = minus2 minus1 119873+1 at the knots

119909119895are defined as [12]

119861119895(119909) =

1

24ℎ4

(119909 minus 119909119895minus2

)4

119909119895minus2

le 119909 lt 119909119895minus1

ℎ4+ 4ℎ3(119909 minus 119909

119895minus1) + 6ℎ

2(119909 minus 119909

119895minus1)2

+ 4ℎ (119909 minus 119909119895minus1

)3

minus 4 (119909 minus 119909119895minus1

)4

119909119895minus1

le 119909 lt 119909119895

11ℎ4+ 12ℎ

3(119909 minus 119909

119895) minus 6ℎ

2(119909 minus 119909

119895)2

minus 12ℎ (119909 minus 119909119895)3

+ 6 (119909 minus 119909119895)4

119909119895le 119909 lt 119909

119895+1

ℎ4+ 4ℎ3(119909119895+2

minus 119909) + 6ℎ2(119909119895+2

minus 119909)2

+ 4ℎ (119909119895+2

minus 119909)3

minus 4 (119909119895+2

minus 119909)4

119909119895+1

le 119909 lt 119909119895+2

(119909119895+3

minus 119909)4

119909119895+2

le 119909 lt 119909119895+3

0 otherwise

(4)

And the set 119861minus2 119861minus1 119861

119873+1 of quartic B-splines forms a

basis over the region 0 le 119909 le 119871Let 119878(119909) be the quartic B-spline function at the nodal

pointsThen approximate solution of (1) can be written as [13]

119878 (119909) =

119873+1

sum

119895=minus2

119862119895119861119895(119909) (5)

where 119861119895(119909) are the quartic B-spline functions and 119862

119895are the

unknown coefficients Each B-spline covers the five elementsso that an element is covered by five B-splines The values of119861119895(119909) and its derivatives are tabulated in Table 1Then from (5) we have

1198781015840(119909) =

119873+1

sum

119895=minus2

1198621198951198611015840

119895(119909)

11987810158401015840(119909) =

119873+1

sum

119895=minus2

11986211989511986110158401015840

119895(119909)

119878101584010158401015840

(119909) =

119873+1

sum

119895=minus2

119862119895119861101584010158401015840

119895(119909)

(6)

Using Table 1 in (5)-(6) we obtained

119878 (119909119895) = (

1

24)119862119895minus2

+ (11

24)119862119895minus1

+ (11

24)119862119895

+ (1

24)119862119895+1

1198781015840(119909119895) = (

minus1

6ℎ)119862119895minus2

+ (minus1

2ℎ)119862119895minus1

+ (1

2ℎ)119862119895

+ (1

6ℎ)119862119895+1

11987810158401015840(119909119895) = (

1

2ℎ2)119862119895minus2

+ (minus1

2ℎ2)119862119895minus1

+ (minus1

2ℎ2)119862119895

+ (1

2ℎ2)119862119895+1

119878101584010158401015840

(119909119895) = (

minus1

ℎ3)119862119895minus2

+ (3

ℎ3)119862119895minus1

+ (minus3

ℎ3)119862119895

+ (1

ℎ3)119862119895+1

(7)

where 119895 = 0 1 119873 Substituting (7) into (1) and (2) and byassuming initial slope 11989110158401015840(0) = 0332057 similar to Howarthwe have

119878 (1199090) = 0

1198781015840(1199090) = 0

11987810158401015840(1199090) = 0332057

119878101584010158401015840

(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895) = 0 119895 = 0 1 119873

(8)

Then we get a system of (119899 + 4) nonlinear equations in the(119899 + 4) unknowns 119862

minus2 119862minus1 119862

119873+1

In order to solve system (8) we direct attention to that119878(119909) and its derivatives satisfied in (1) and (2) and initial slope11989110158401015840(0) = 0332057 approximately and then we have

119878 (1199090) asymp 0

1198781015840(1199090) asymp 0

11987810158401015840(1199090) asymp 0332057

119878101584010158401015840

(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895) asymp 0 119895 = 0 1 119873

(9)

International Journal of Engineering Mathematics 3

Table 1 Coefficient of quartic B-splines and its derivatives at knots119909119895

119909 119909119895minus2

119909119895minus1

119909119895

119909119895+1

119909119895+2

119909119895+3

119861119895(119909) 0

1

24

11

24

11

24

1

240

1198611015840

119895(119909) 0

1

6ℎ

1

2ℎminus

1

2ℎminus

1

6ℎ0

11986110158401015840

119895(119909) 0

1

2ℎ2minus

1

2ℎ2minus

1

2ℎ2

1

2ℎ20

119861101584010158401015840

119895(119909) 0

1

ℎ3minus

3

ℎ3

3

ℎ3minus

1

ℎ30

Therefore the error vector 119864 of the approximation can bewritten as

1198641= 119878 (119909

0)

1198642= 1198781015840(1199090)

1198643= 11987810158401015840(1199090) minus 0332057

119864119895+4

= 119878101584010158401015840

(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895) 119895 = 0 1 119873

(10)

Now wewish tominimize the error norm the 1198712-norm such

that

1198712= radic

119873+4

sum

119895=1

1198642

119895=0 (11)

By solving (11) we can get the values of 119862119895

for119895 = minus2 minus1 119873 + 1 Now with substituting the values119862minus2 119862minus1 1198620 119862

119873+1into (7) the approximate value of 119878(119909)

and its derivatives will be ensuredFor calculating the truncation error of this method from

(7) the following relationships can be obtained [14]

ℎ [1198781015840(119909119895minus2

) + 111198781015840(119909119895minus1

) + 111198781015840(119909119895) + 1198781015840(119909119895+1

)]

= 4 [119878 (119909119895+1

) + 3119878 (119909119895) minus 3119878 (119909

119895minus1) minus 119878 (119909

119895minus2)]

ℎ211987810158401015840(119909119895)

= 2 [119878 (119909119895+1

) minus 2119878 (119909119895) + 119878 (119909

119895minus1)]

minusℎ

2[1198781015840(119909119895+1

) minus 1198781015840(119909119895minus1

)]

ℎ3119878101584010158401015840

(119909119895)

= 12 [119878 (119909119895+1

) minus 119878 (119909119895minus1

)]

minus 3ℎ [1198781015840(119909119895+1

) + 61198781015840(119909119895) + 1198781015840(119909119895minus1

)]

(12)

and then we have

1198781015840(119909119895) = 1198911015840(119909119895) +

ℎ4

720119891(5)

(119909119895) + 119874 (ℎ

6)

11987810158401015840(119909119895) = 11989110158401015840(119909119895) minus

ℎ4

240119891(6)

(119909119895) + 119874 (ℎ

6)

119878101584010158401015840

(119909119895) = 119891101584010158401015840

(119909119895) minus

ℎ2

12119891(5)

(119909119895) +

ℎ4

240119891(7)

(119909119895)

+ 119874 (ℎ6)

(13)

Therefore truncation error is defined as follows

119890 (119909119895) = [119878

101584010158401015840(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895)] minus [119891

101584010158401015840(119909119895)

+1

2119891 (119909119895) 11989110158401015840(119909119895)] = [119878

101584010158401015840(119909119895) minus 119891101584010158401015840

(119909119895)]

+1

2[119878 (119909119895) 11987810158401015840(119909119895) minus 119891 (119909

119895) 11989110158401015840(119909119895)]

= [minusℎ2

12119891(5)

(119909119895) +

ℎ4

240119891(7)

(119909119895) + 119874 (ℎ

6)]

+ [minusℎ4

480119891 (119909119895) 119891(6)

(119909119895) +

ℎ6

12096119891 (119909119895) 119891(8)

(119909119895)

+ 119874 (ℎ8)] 119895 = 1 2 119873

(14)

Thus for 119895 = 1 2 119873 we have

119890 (119909119895) = [minus

119891(5)

(119909119895)

12] ℎ2

+ [

2119891(7)

(119909119895) minus 119891 (119909

119895) 119891(6)

(119909119895)

480] ℎ4

+ 119874 (ℎ6) 119895 = 1 2 119873

(15)

and for 119909 = 0 we have

119890 (1199090) = [119878

101584010158401015840(1199090) +

1

2119878 (1199090) 11987810158401015840(1199090)]

minus [119891101584010158401015840

(1199090) +

1

2119891 (1199090) 11989110158401015840(1199090)]

(16)

Since 119891(0) = 119878(0) = 0 we have the following result

119890 (0) = 119878101584010158401015840

(0) minus 119891101584010158401015840

(0)

= [minus119891(5)

(0)

12] ℎ2+ [

119891(7)

(0)

240] ℎ4+ 119874 (ℎ

6)

(17)

3 Numerical Results

In this section approximation values of 119891(119909) 1198911015840(119909) and

11989110158401015840(119909) based on proposed methods for some values of 119909


Table 2 Comparison of 119891(119909) between some numerical results and our results using different values of ℎ

119909 DTM [11] LTNHPM [7] Howarth (RK4) Proposed methodℎ = 02 ℎ = 01 ℎ = 001

00 000000 000000 000000 000000 000000 00000002 000664 000664 000664 000664 000664 00066404 002656 002656 002656 002656 002656 00265606 005973 005973 005974 005973 005973 00597308 010611 010611 010611 010609 010610 01061110 016557 016557 016557 016554 016556 01655712 023795 023795 023795 023790 023794 02379514 032298 032298 032298 032290 032296 03229816 042032 042032 042032 042021 042029 04203218 052952 052952 052952 052936 052948 05295220 065002 065002 065003 064982 064997 06500222 078119 078119 078120 078093 078113 07811924 092229 092228 092230 092197 092221 09222926 107250 107250 107252 107212 107241 10725028 123098 123098 123099 123054 123087 12309830 139681 139681 139682 139631 139668 13968132 156909 156909 156911 156854 156896 15690934 174695 174695 174696 174635 174680 17469536 192952 192952 192954 192889 192936 19295238 211603 211602 211605 211536 211586 21160340 230574 230575 230576 230506 230557 23057442 249804 249805 249806 249733 249786 24980444 269236 269242 269238 269164 269218 26923646 288824 288859 288826 288751 288806 28882448 308532 308718 308534 308458 308513 30853250 328327 329272 328329 328252 328308 328327



00 000000 000000 000000 000000 000000 00000002 006641 006641 006641 006640 006641 00664104 013276 013276 013277 013275 013276 01327606 019894 019894 019894 019890 019893 01989408 026471 026471 026471 026465 026469 02647110 032978 032978 032979 032969 032976 03297812 039378 039378 039378 039365 039374 03937814 045626 045626 045627 045610 045622 04562616 051676 051676 051676 051656 051671 05167618 057476 057476 057477 057452 057470 05747620 062976 062977 062977 062950 062970 06297622 068131 068131 068132 068102 068124 06813124 072898 072898 072899 072868 072891 07289826 077245 077245 077246 077215 077238 07724528 081151 081151 081152 081122 081144 08115130 084604 084604 084605 084577 084598 08460432 087608 087608 087609 087584 087602 08760834 090176 090176 090177 090155 090171 09017636 092333 092333 092333 092315 092329 09233338 094112 094112 094112 094098 094108 09411240 095552 095553 095552 095541 095549 09555242 096696 096704 096696 096688 096694 09669644 097587 097639 097587 097581 097586 09758746 098268 098564 098269 098264 098267 09826848 098779 100322 098779 098775 098778 09877950 099154 106671 099155 099151 099153 099154




00 033206 033206 033206 033206 033206 03320602 033198 033198 033199 033195 033197 03319804 033147 033147 033147 033140 033145 03314706 033008 033008 033008 032997 033005 03300808 032739 032739 032739 032725 032735 03273910 032301 032301 032301 032284 032297 03230112 031659 031659 031659 031641 031654 03165914 030786 030786 030787 030767 030782 03078616 029666 029666 029667 029648 029662 02966618 028293 028293 028293 028276 028289 02829320 026675 026675 026675 026662 026672 02667522 024835 024835 024835 024826 024833 02483524 022809 022809 022809 022805 022808 02280926 020645 020645 020646 020647 020646 02064528 018401 018401 018401 018408 018402 01840130 016136 016136 016136 016148 016139 01613632 013913 013913 013913 013928 013917 01391334 011788 011788 011788 011805 011792 01178836 009809 009809 009809 009827 009813 00980938 008013 008014 008013 008030 008017 00801340 006423 006436 006424 006438 006427 00642342 005052 005131 005052 005064 005055 00505244 003897 004360 003897 003906 003899 00389746 002948 005446 002948 002954 002950 00294848 002187 014672 002187 002190 002188 00218750 001591 059821 001591 001591 001591 001591

0 1 2 3 4 5 6 7 8 9

x

Proposed methodHowarth

8

7

6

5

4

3

2

1

0

minus1

f(x)

Figure 1 The comparison of 119891(119909) between our results andHowarthrsquos results with ℎ = 001

are presented These values for ℎ = 02 01 001 havebeen calculated and these results are compared with theHowarth results Tables 2 3 and 4 are made to comparebetween the present results and results given by Howarth forapproximation values of 119891(119909) 119891

1015840(119909) and 119891

10158401015840(119909) respectively

In Figures 1 2 and 3 one can also see the comparison betweenour results and Howarthrsquos results

0 1 2 3 4 5 6 7 8 9

x


1

09

08

07

06

05

04

03

02

01

0

df(x)

Figure 2 The comparison of 1198911015840(119909) between our results and

Howarthrsquos results with ℎ = 001

4 Conclusion

In this survey the quartic B-spline approximations are usedto solve the Blasius equation This method led to a systemof nonlinear equations The unknowns are obtained by min-imizing the error norm The computed results are comparedwith those of DTM LTNHPM and Howarthrsquos methods to


0 1 2 3 4 5 6 7 8 9

x


035

03

025

02

015

01

005

0

d2f(x)



demonstrate the validity and applicability of the techniqueThis method is simple in applicability and the results showthat the solutions will become more accurate with reducingstep size The computations associated with the examples inthis paper were performed using MATLAB R2015a

Competing Interests

The authors of the paper do not have a direct financialrelation that might lead to ldquocompeting interestsrdquo for any ofthe authors

References

[1] H Blasius ldquoGrenzschichten in flussigkeiten mit kleiner rei-bungrdquo Zeitschrift fur Angewandte Mathematik und Physik vol56 pp 1ndash37 1908

[2] L Howarth ldquoOn the solution of the laminar boundary layerequationsrdquo Proceedings of the Royal Society A MathematicalPhysical and Engineering Sciences vol 164 no 919 pp 547ndash5791938

[3] C-S Liu and J-R Chang ldquoThe Lie-group shooting methodfor multiple-solutions of Falkner-Skan equation under suction-injection conditionsrdquo International Journal of Non-LinearMechanics vol 43 no 9 pp 844ndash851 2008

[4] J H He ldquoApproximate analytical solution of Blasiusrsquo equationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 3 no 4 pp 260ndash263 1998

[5] J-H He ldquoA simple perturbation approach to Blasius equationrdquoApplied Mathematics and Computation vol 140 no 2-3 pp217ndash222 2003

[6] C M Bender K A Milton S S Pinsky and J L M SimmonsldquoA new perturbative approach to nonlinear problemsrdquo Journalof Mathematical Physics vol 30 no 7 pp 1447ndash1455 1989

[7] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusrsquo viscous flow equation by LTNHPMrdquo ISRN

Mathematical Analysis vol 2012 Article ID 957473 10 pages2012

[8] D Xu and X Guo ldquoFixed point analytical method for nonlineardifferential equationsrdquo Journal of Computational and NonlinearDynamics vol 8 no 1 Article ID 011005 2013

[9] L Wang ldquoA new algorithm for solving classical Blasius equa-tionrdquo Applied Mathematics and Computation vol 157 no 1 pp1ndash9 2004

[10] R Fazio ldquoBlasius problem and Falkner-Skan model Topferrsquosalgoritm and its extensionrdquo Computers amp Fluids vol 73 pp202ndash209 2013

[11] S A Lal and N M Paul ldquoAn accurate taylors series solutionwith high radius of convergence for the Blasius function andparameters of asymptotic variationrdquo Journal of Applied FluidMechanics vol 7 no 4 pp 557ndash564 2014

[12] C Deboor A Practical Guide to Splines Springer BerlinGermany 1978

[13] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989

[14] J Goh A A Majid and A I M Ismail ldquoA quartic B-spline forsecond-order singular boundary value problemsrdquo Computersand Mathematics with Applications vol 64 no 2 pp 115ndash1202012

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an increasing set of 119873 + 1 knots over the problem domainplus 8 additional knots outside the problem domain the 8additional knots are positioned as

119909minus4

lt 119909minus3

lt 119909minus2

lt 119909minus1

lt 1199090

119909119873

lt 119909119873+1

lt 119909119873+2

lt 119909119873+3

lt 119909119873+4

(3)

The quartic B-splines119861119895(119909) 119895 = minus2 minus1 119873+1 at the knots

119909119895are defined as [12]

119861119895(119909) =

1

24ℎ4

(119909 minus 119909119895minus2

)4

119909119895minus2

le 119909 lt 119909119895minus1

ℎ4+ 4ℎ3(119909 minus 119909

119895minus1) + 6ℎ

2(119909 minus 119909

119895minus1)2

+ 4ℎ (119909 minus 119909119895minus1

)3

minus 4 (119909 minus 119909119895minus1

)4

119909119895minus1

le 119909 lt 119909119895

11ℎ4+ 12ℎ

3(119909 minus 119909

119895) minus 6ℎ

2(119909 minus 119909

119895)2

minus 12ℎ (119909 minus 119909119895)3

+ 6 (119909 minus 119909119895)4

119909119895le 119909 lt 119909

119895+1

ℎ4+ 4ℎ3(119909119895+2

minus 119909) + 6ℎ2(119909119895+2

minus 119909)2

+ 4ℎ (119909119895+2

minus 119909)3

minus 4 (119909119895+2

minus 119909)4

119909119895+1

le 119909 lt 119909119895+2

(119909119895+3

minus 119909)4

119909119895+2

le 119909 lt 119909119895+3

0 otherwise

(4)

And the set 119861minus2 119861minus1 119861

119873+1 of quartic B-splines forms a

basis over the region 0 le 119909 le 119871Let 119878(119909) be the quartic B-spline function at the nodal

pointsThen approximate solution of (1) can be written as [13]

119878 (119909) =

119873+1

sum

119895=minus2

119862119895119861119895(119909) (5)

where 119861119895(119909) are the quartic B-spline functions and 119862

119895are the

unknown coefficients Each B-spline covers the five elementsso that an element is covered by five B-splines The values of119861119895(119909) and its derivatives are tabulated in Table 1Then from (5) we have

1198781015840(119909) =

119873+1

sum

119895=minus2

1198621198951198611015840

119895(119909)

11987810158401015840(119909) =

119873+1

sum

119895=minus2

11986211989511986110158401015840

119895(119909)

119878101584010158401015840

(119909) =

119873+1

sum

119895=minus2

119862119895119861101584010158401015840

119895(119909)

(6)

Using Table 1 in (5)-(6) we obtained

119878 (119909119895) = (

1

24)119862119895minus2

+ (11

24)119862119895minus1

+ (11

24)119862119895

+ (1

24)119862119895+1

1198781015840(119909119895) = (

minus1

6ℎ)119862119895minus2

+ (minus1

2ℎ)119862119895minus1

+ (1

2ℎ)119862119895

+ (1

6ℎ)119862119895+1

11987810158401015840(119909119895) = (

1

2ℎ2)119862119895minus2

+ (minus1

2ℎ2)119862119895minus1

+ (minus1

2ℎ2)119862119895

+ (1

2ℎ2)119862119895+1

119878101584010158401015840

(119909119895) = (

minus1

ℎ3)119862119895minus2

+ (3

ℎ3)119862119895minus1

+ (minus3

ℎ3)119862119895

+ (1

ℎ3)119862119895+1

(7)

where 119895 = 0 1 119873 Substituting (7) into (1) and (2) and byassuming initial slope 11989110158401015840(0) = 0332057 similar to Howarthwe have

119878 (1199090) = 0

1198781015840(1199090) = 0

11987810158401015840(1199090) = 0332057

119878101584010158401015840

(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895) = 0 119895 = 0 1 119873

(8)

Then we get a system of (119899 + 4) nonlinear equations in the(119899 + 4) unknowns 119862

minus2 119862minus1 119862

119873+1

In order to solve system (8) we direct attention to that119878(119909) and its derivatives satisfied in (1) and (2) and initial slope11989110158401015840(0) = 0332057 approximately and then we have

119878 (1199090) asymp 0

1198781015840(1199090) asymp 0

11987810158401015840(1199090) asymp 0332057

119878101584010158401015840

(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895) asymp 0 119895 = 0 1 119873

(9)



119909 119909119895minus2

119909119895minus1

119909119895

119909119895+1

119909119895+2

119909119895+3

119861119895(119909) 0

1

24

11

24

11

24

1

240

1198611015840

119895(119909) 0

1

6ℎ

1

2ℎminus

1

2ℎminus

1

6ℎ0

11986110158401015840

119895(119909) 0

1

2ℎ2minus

1

2ℎ2minus

1

2ℎ2

1

2ℎ20

119861101584010158401015840

119895(119909) 0

1

ℎ3minus

3

ℎ3

3

ℎ3minus

1

ℎ30


1198641= 119878 (119909

0)

1198642= 1198781015840(1199090)

1198643= 11987810158401015840(1199090) minus 0332057

119864119895+4

= 119878101584010158401015840

(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895) 119895 = 0 1 119873

(10)


that

1198712= radic

119873+4

sum

119895=1

1198642

119895=0 (11)






ℎ [1198781015840(119909119895minus2

) + 111198781015840(119909119895minus1

) + 111198781015840(119909119895) + 1198781015840(119909119895+1

)]

= 4 [119878 (119909119895+1

) + 3119878 (119909119895) minus 3119878 (119909

119895minus1) minus 119878 (119909

119895minus2)]

ℎ211987810158401015840(119909119895)

= 2 [119878 (119909119895+1

) minus 2119878 (119909119895) + 119878 (119909

119895minus1)]

minusℎ

2[1198781015840(119909119895+1

) minus 1198781015840(119909119895minus1

)]

ℎ3119878101584010158401015840

(119909119895)

= 12 [119878 (119909119895+1

) minus 119878 (119909119895minus1

)]

minus 3ℎ [1198781015840(119909119895+1

) + 61198781015840(119909119895) + 1198781015840(119909119895minus1

)]

(12)

and then we have

1198781015840(119909119895) = 1198911015840(119909119895) +

ℎ4

720119891(5)

(119909119895) + 119874 (ℎ

6)

11987810158401015840(119909119895) = 11989110158401015840(119909119895) minus

ℎ4

240119891(6)

(119909119895) + 119874 (ℎ

6)

119878101584010158401015840

(119909119895) = 119891101584010158401015840

(119909119895) minus

ℎ2

12119891(5)

(119909119895) +

ℎ4

240119891(7)

(119909119895)

+ 119874 (ℎ6)

(13)


119890 (119909119895) = [119878

101584010158401015840(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895)] minus [119891

101584010158401015840(119909119895)

+1

2119891 (119909119895) 11989110158401015840(119909119895)] = [119878

101584010158401015840(119909119895) minus 119891101584010158401015840

(119909119895)]

+1

2[119878 (119909119895) 11987810158401015840(119909119895) minus 119891 (119909

119895) 11989110158401015840(119909119895)]

= [minusℎ2

12119891(5)

(119909119895) +

ℎ4

240119891(7)

(119909119895) + 119874 (ℎ

6)]

+ [minusℎ4

480119891 (119909119895) 119891(6)

(119909119895) +

ℎ6

12096119891 (119909119895) 119891(8)

(119909119895)

+ 119874 (ℎ8)] 119895 = 1 2 119873

(14)

Thus for 119895 = 1 2 119873 we have

119890 (119909119895) = [minus

119891(5)

(119909119895)

12] ℎ2

+ [

2119891(7)

(119909119895) minus 119891 (119909

119895) 119891(6)

(119909119895)

480] ℎ4

+ 119874 (ℎ6) 119895 = 1 2 119873

(15)


119890 (1199090) = [119878

101584010158401015840(1199090) +

1

2119878 (1199090) 11987810158401015840(1199090)]

minus [119891101584010158401015840

(1199090) +

1

2119891 (1199090) 11989110158401015840(1199090)]

(16)


119890 (0) = 119878101584010158401015840

(0) minus 119891101584010158401015840

(0)

= [minus119891(5)

(0)

12] ℎ2+ [

119891(7)

(0)

240] ℎ4+ 119874 (ℎ

6)

(17)

3 Numerical Results






00 000000 000000 000000 000000 000000 00000002 000664 000664 000664 000664 000664 00066404 002656 002656 002656 002656 002656 00265606 005973 005973 005974 005973 005973 00597308 010611 010611 010611 010609 010610 01061110 016557 016557 016557 016554 016556 01655712 023795 023795 023795 023790 023794 02379514 032298 032298 032298 032290 032296 03229816 042032 042032 042032 042021 042029 04203218 052952 052952 052952 052936 052948 05295220 065002 065002 065003 064982 064997 06500222 078119 078119 078120 078093 078113 07811924 092229 092228 092230 092197 092221 09222926 107250 107250 107252 107212 107241 10725028 123098 123098 123099 123054 123087 12309830 139681 139681 139682 139631 139668 13968132 156909 156909 156911 156854 156896 15690934 174695 174695 174696 174635 174680 17469536 192952 192952 192954 192889 192936 19295238 211603 211602 211605 211536 211586 21160340 230574 230575 230576 230506 230557 23057442 249804 249805 249806 249733 249786 24980444 269236 269242 269238 269164 269218 26923646 288824 288859 288826 288751 288806 28882448 308532 308718 308534 308458 308513 30853250 328327 329272 328329 328252 328308 328327



00 000000 000000 000000 000000 000000 00000002 006641 006641 006641 006640 006641 00664104 013276 013276 013277 013275 013276 01327606 019894 019894 019894 019890 019893 01989408 026471 026471 026471 026465 026469 02647110 032978 032978 032979 032969 032976 03297812 039378 039378 039378 039365 039374 03937814 045626 045626 045627 045610 045622 04562616 051676 051676 051676 051656 051671 05167618 057476 057476 057477 057452 057470 05747620 062976 062977 062977 062950 062970 06297622 068131 068131 068132 068102 068124 06813124 072898 072898 072899 072868 072891 07289826 077245 077245 077246 077215 077238 07724528 081151 081151 081152 081122 081144 08115130 084604 084604 084605 084577 084598 08460432 087608 087608 087609 087584 087602 08760834 090176 090176 090177 090155 090171 09017636 092333 092333 092333 092315 092329 09233338 094112 094112 094112 094098 094108 09411240 095552 095553 095552 095541 095549 09555242 096696 096704 096696 096688 096694 09669644 097587 097639 097587 097581 097586 09758746 098268 098564 098269 098264 098267 09826848 098779 100322 098779 098775 098778 09877950 099154 106671 099155 099151 099153 099154




00 033206 033206 033206 033206 033206 03320602 033198 033198 033199 033195 033197 03319804 033147 033147 033147 033140 033145 03314706 033008 033008 033008 032997 033005 03300808 032739 032739 032739 032725 032735 03273910 032301 032301 032301 032284 032297 03230112 031659 031659 031659 031641 031654 03165914 030786 030786 030787 030767 030782 03078616 029666 029666 029667 029648 029662 02966618 028293 028293 028293 028276 028289 02829320 026675 026675 026675 026662 026672 02667522 024835 024835 024835 024826 024833 02483524 022809 022809 022809 022805 022808 02280926 020645 020645 020646 020647 020646 02064528 018401 018401 018401 018408 018402 01840130 016136 016136 016136 016148 016139 01613632 013913 013913 013913 013928 013917 01391334 011788 011788 011788 011805 011792 01178836 009809 009809 009809 009827 009813 00980938 008013 008014 008013 008030 008017 00801340 006423 006436 006424 006438 006427 00642342 005052 005131 005052 005064 005055 00505244 003897 004360 003897 003906 003899 00389746 002948 005446 002948 002954 002950 00294848 002187 014672 002187 002190 002188 00218750 001591 059821 001591 001591 001591 001591

0 1 2 3 4 5 6 7 8 9

x


8

7

6

5

4

3

2

1

0

minus1

f(x)



1015840(119909) and 119891

10158401015840(119909) respectively


0 1 2 3 4 5 6 7 8 9

x


1

09

08

07

06

05

04

03

02

01

0

df(x)



4 Conclusion



0 1 2 3 4 5 6 7 8 9

x


035

03

025

02

015

01

005

0

d2f(x)




Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 119909119895minus2

119909119895minus1

119909119895

119909119895+1

119909119895+2

119909119895+3

119861119895(119909) 0

1

24

11

24

11

24

1

240

1198611015840

119895(119909) 0

1

6ℎ

1

2ℎminus

1

2ℎminus

1

6ℎ0

11986110158401015840

119895(119909) 0

1

2ℎ2minus

1

2ℎ2minus

1

2ℎ2

1

2ℎ20

119861101584010158401015840

119895(119909) 0

1

ℎ3minus

3

ℎ3

3

ℎ3minus

1

ℎ30


1198641= 119878 (119909

0)

1198642= 1198781015840(1199090)

1198643= 11987810158401015840(1199090) minus 0332057

119864119895+4

= 119878101584010158401015840

(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895) 119895 = 0 1 119873

(10)


that

1198712= radic

119873+4

sum

119895=1

1198642

119895=0 (11)






ℎ [1198781015840(119909119895minus2

) + 111198781015840(119909119895minus1

) + 111198781015840(119909119895) + 1198781015840(119909119895+1

)]

= 4 [119878 (119909119895+1

) + 3119878 (119909119895) minus 3119878 (119909

119895minus1) minus 119878 (119909

119895minus2)]

ℎ211987810158401015840(119909119895)

= 2 [119878 (119909119895+1

) minus 2119878 (119909119895) + 119878 (119909

119895minus1)]

minusℎ

2[1198781015840(119909119895+1

) minus 1198781015840(119909119895minus1

)]

ℎ3119878101584010158401015840

(119909119895)

= 12 [119878 (119909119895+1

) minus 119878 (119909119895minus1

)]

minus 3ℎ [1198781015840(119909119895+1

) + 61198781015840(119909119895) + 1198781015840(119909119895minus1

)]

(12)

and then we have

1198781015840(119909119895) = 1198911015840(119909119895) +

ℎ4

720119891(5)

(119909119895) + 119874 (ℎ

6)

11987810158401015840(119909119895) = 11989110158401015840(119909119895) minus

ℎ4

240119891(6)

(119909119895) + 119874 (ℎ

6)

119878101584010158401015840

(119909119895) = 119891101584010158401015840

(119909119895) minus

ℎ2

12119891(5)

(119909119895) +

ℎ4

240119891(7)

(119909119895)

+ 119874 (ℎ6)

(13)


119890 (119909119895) = [119878

101584010158401015840(119909119895) +

1

2119878 (119909119895) 11987810158401015840(119909119895)] minus [119891

101584010158401015840(119909119895)

+1

2119891 (119909119895) 11989110158401015840(119909119895)] = [119878

101584010158401015840(119909119895) minus 119891101584010158401015840

(119909119895)]

+1

2[119878 (119909119895) 11987810158401015840(119909119895) minus 119891 (119909

119895) 11989110158401015840(119909119895)]

= [minusℎ2

12119891(5)

(119909119895) +

ℎ4

240119891(7)

(119909119895) + 119874 (ℎ

6)]

+ [minusℎ4

480119891 (119909119895) 119891(6)

(119909119895) +

ℎ6

12096119891 (119909119895) 119891(8)

(119909119895)

+ 119874 (ℎ8)] 119895 = 1 2 119873

(14)

Thus for 119895 = 1 2 119873 we have

119890 (119909119895) = [minus

119891(5)

(119909119895)

12] ℎ2

+ [

2119891(7)

(119909119895) minus 119891 (119909

119895) 119891(6)

(119909119895)

480] ℎ4

+ 119874 (ℎ6) 119895 = 1 2 119873

(15)


119890 (1199090) = [119878

101584010158401015840(1199090) +

1

2119878 (1199090) 11987810158401015840(1199090)]

minus [119891101584010158401015840

(1199090) +

1

2119891 (1199090) 11989110158401015840(1199090)]

(16)


119890 (0) = 119878101584010158401015840

(0) minus 119891101584010158401015840

(0)

= [minus119891(5)

(0)

12] ℎ2+ [

119891(7)

(0)

240] ℎ4+ 119874 (ℎ

6)

(17)

3 Numerical Results






00 000000 000000 000000 000000 000000 00000002 000664 000664 000664 000664 000664 00066404 002656 002656 002656 002656 002656 00265606 005973 005973 005974 005973 005973 00597308 010611 010611 010611 010609 010610 01061110 016557 016557 016557 016554 016556 01655712 023795 023795 023795 023790 023794 02379514 032298 032298 032298 032290 032296 03229816 042032 042032 042032 042021 042029 04203218 052952 052952 052952 052936 052948 05295220 065002 065002 065003 064982 064997 06500222 078119 078119 078120 078093 078113 07811924 092229 092228 092230 092197 092221 09222926 107250 107250 107252 107212 107241 10725028 123098 123098 123099 123054 123087 12309830 139681 139681 139682 139631 139668 13968132 156909 156909 156911 156854 156896 15690934 174695 174695 174696 174635 174680 17469536 192952 192952 192954 192889 192936 19295238 211603 211602 211605 211536 211586 21160340 230574 230575 230576 230506 230557 23057442 249804 249805 249806 249733 249786 24980444 269236 269242 269238 269164 269218 26923646 288824 288859 288826 288751 288806 28882448 308532 308718 308534 308458 308513 30853250 328327 329272 328329 328252 328308 328327



00 000000 000000 000000 000000 000000 00000002 006641 006641 006641 006640 006641 00664104 013276 013276 013277 013275 013276 01327606 019894 019894 019894 019890 019893 01989408 026471 026471 026471 026465 026469 02647110 032978 032978 032979 032969 032976 03297812 039378 039378 039378 039365 039374 03937814 045626 045626 045627 045610 045622 04562616 051676 051676 051676 051656 051671 05167618 057476 057476 057477 057452 057470 05747620 062976 062977 062977 062950 062970 06297622 068131 068131 068132 068102 068124 06813124 072898 072898 072899 072868 072891 07289826 077245 077245 077246 077215 077238 07724528 081151 081151 081152 081122 081144 08115130 084604 084604 084605 084577 084598 08460432 087608 087608 087609 087584 087602 08760834 090176 090176 090177 090155 090171 09017636 092333 092333 092333 092315 092329 09233338 094112 094112 094112 094098 094108 09411240 095552 095553 095552 095541 095549 09555242 096696 096704 096696 096688 096694 09669644 097587 097639 097587 097581 097586 09758746 098268 098564 098269 098264 098267 09826848 098779 100322 098779 098775 098778 09877950 099154 106671 099155 099151 099153 099154




00 033206 033206 033206 033206 033206 03320602 033198 033198 033199 033195 033197 03319804 033147 033147 033147 033140 033145 03314706 033008 033008 033008 032997 033005 03300808 032739 032739 032739 032725 032735 03273910 032301 032301 032301 032284 032297 03230112 031659 031659 031659 031641 031654 03165914 030786 030786 030787 030767 030782 03078616 029666 029666 029667 029648 029662 02966618 028293 028293 028293 028276 028289 02829320 026675 026675 026675 026662 026672 02667522 024835 024835 024835 024826 024833 02483524 022809 022809 022809 022805 022808 02280926 020645 020645 020646 020647 020646 02064528 018401 018401 018401 018408 018402 01840130 016136 016136 016136 016148 016139 01613632 013913 013913 013913 013928 013917 01391334 011788 011788 011788 011805 011792 01178836 009809 009809 009809 009827 009813 00980938 008013 008014 008013 008030 008017 00801340 006423 006436 006424 006438 006427 00642342 005052 005131 005052 005064 005055 00505244 003897 004360 003897 003906 003899 00389746 002948 005446 002948 002954 002950 00294848 002187 014672 002187 002190 002188 00218750 001591 059821 001591 001591 001591 001591

0 1 2 3 4 5 6 7 8 9

x


8

7

6

5

4

3

2

1

0

minus1

f(x)



1015840(119909) and 119891

10158401015840(119909) respectively


0 1 2 3 4 5 6 7 8 9

x


1

09

08

07

06

05

04

03

02

01

0

df(x)



4 Conclusion



0 1 2 3 4 5 6 7 8 9

x


035

03

025

02

015

01

005

0

d2f(x)




Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












00 000000 000000 000000 000000 000000 00000002 000664 000664 000664 000664 000664 00066404 002656 002656 002656 002656 002656 00265606 005973 005973 005974 005973 005973 00597308 010611 010611 010611 010609 010610 01061110 016557 016557 016557 016554 016556 01655712 023795 023795 023795 023790 023794 02379514 032298 032298 032298 032290 032296 03229816 042032 042032 042032 042021 042029 04203218 052952 052952 052952 052936 052948 05295220 065002 065002 065003 064982 064997 06500222 078119 078119 078120 078093 078113 07811924 092229 092228 092230 092197 092221 09222926 107250 107250 107252 107212 107241 10725028 123098 123098 123099 123054 123087 12309830 139681 139681 139682 139631 139668 13968132 156909 156909 156911 156854 156896 15690934 174695 174695 174696 174635 174680 17469536 192952 192952 192954 192889 192936 19295238 211603 211602 211605 211536 211586 21160340 230574 230575 230576 230506 230557 23057442 249804 249805 249806 249733 249786 24980444 269236 269242 269238 269164 269218 26923646 288824 288859 288826 288751 288806 28882448 308532 308718 308534 308458 308513 30853250 328327 329272 328329 328252 328308 328327



00 000000 000000 000000 000000 000000 00000002 006641 006641 006641 006640 006641 00664104 013276 013276 013277 013275 013276 01327606 019894 019894 019894 019890 019893 01989408 026471 026471 026471 026465 026469 02647110 032978 032978 032979 032969 032976 03297812 039378 039378 039378 039365 039374 03937814 045626 045626 045627 045610 045622 04562616 051676 051676 051676 051656 051671 05167618 057476 057476 057477 057452 057470 05747620 062976 062977 062977 062950 062970 06297622 068131 068131 068132 068102 068124 06813124 072898 072898 072899 072868 072891 07289826 077245 077245 077246 077215 077238 07724528 081151 081151 081152 081122 081144 08115130 084604 084604 084605 084577 084598 08460432 087608 087608 087609 087584 087602 08760834 090176 090176 090177 090155 090171 09017636 092333 092333 092333 092315 092329 09233338 094112 094112 094112 094098 094108 09411240 095552 095553 095552 095541 095549 09555242 096696 096704 096696 096688 096694 09669644 097587 097639 097587 097581 097586 09758746 098268 098564 098269 098264 098267 09826848 098779 100322 098779 098775 098778 09877950 099154 106671 099155 099151 099153 099154




00 033206 033206 033206 033206 033206 03320602 033198 033198 033199 033195 033197 03319804 033147 033147 033147 033140 033145 03314706 033008 033008 033008 032997 033005 03300808 032739 032739 032739 032725 032735 03273910 032301 032301 032301 032284 032297 03230112 031659 031659 031659 031641 031654 03165914 030786 030786 030787 030767 030782 03078616 029666 029666 029667 029648 029662 02966618 028293 028293 028293 028276 028289 02829320 026675 026675 026675 026662 026672 02667522 024835 024835 024835 024826 024833 02483524 022809 022809 022809 022805 022808 02280926 020645 020645 020646 020647 020646 02064528 018401 018401 018401 018408 018402 01840130 016136 016136 016136 016148 016139 01613632 013913 013913 013913 013928 013917 01391334 011788 011788 011788 011805 011792 01178836 009809 009809 009809 009827 009813 00980938 008013 008014 008013 008030 008017 00801340 006423 006436 006424 006438 006427 00642342 005052 005131 005052 005064 005055 00505244 003897 004360 003897 003906 003899 00389746 002948 005446 002948 002954 002950 00294848 002187 014672 002187 002190 002188 00218750 001591 059821 001591 001591 001591 001591

0 1 2 3 4 5 6 7 8 9

x


8

7

6

5

4

3

2

1

0

minus1

f(x)



1015840(119909) and 119891

10158401015840(119909) respectively


0 1 2 3 4 5 6 7 8 9

x


1

09

08

07

06

05

04

03

02

01

0

df(x)



4 Conclusion



0 1 2 3 4 5 6 7 8 9

x


035

03

025

02

015

01

005

0

d2f(x)




Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












00 033206 033206 033206 033206 033206 03320602 033198 033198 033199 033195 033197 03319804 033147 033147 033147 033140 033145 03314706 033008 033008 033008 032997 033005 03300808 032739 032739 032739 032725 032735 03273910 032301 032301 032301 032284 032297 03230112 031659 031659 031659 031641 031654 03165914 030786 030786 030787 030767 030782 03078616 029666 029666 029667 029648 029662 02966618 028293 028293 028293 028276 028289 02829320 026675 026675 026675 026662 026672 02667522 024835 024835 024835 024826 024833 02483524 022809 022809 022809 022805 022808 02280926 020645 020645 020646 020647 020646 02064528 018401 018401 018401 018408 018402 01840130 016136 016136 016136 016148 016139 01613632 013913 013913 013913 013928 013917 01391334 011788 011788 011788 011805 011792 01178836 009809 009809 009809 009827 009813 00980938 008013 008014 008013 008030 008017 00801340 006423 006436 006424 006438 006427 00642342 005052 005131 005052 005064 005055 00505244 003897 004360 003897 003906 003899 00389746 002948 005446 002948 002954 002950 00294848 002187 014672 002187 002190 002188 00218750 001591 059821 001591 001591 001591 001591

0 1 2 3 4 5 6 7 8 9

x


8

7

6

5

4

3

2

1

0

minus1

f(x)



1015840(119909) and 119891

10158401015840(119909) respectively


0 1 2 3 4 5 6 7 8 9

x


1

09

08

07

06

05

04

03

02

01

0

df(x)



4 Conclusion



0 1 2 3 4 5 6 7 8 9

x


035

03

025

02

015

01

005

0

d2f(x)




Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9

x


035

03

025

02

015

01

005

0

d2f(x)




Competing Interests


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Numerical Solution of the Blasius Viscous ...downloads.hindawi.com/archive/2016/9014354.pdf · Research Article Numerical Solution of the Blasius Viscous Flow Problem