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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
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
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
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
Proposed methodHowarth
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
6 International Journal of Engineering Mathematics
0 1 2 3 4 5 6 7 8 9
x
Proposed methodHowarth
035
03
025
02
015
01
005
0
d2f(x)
Figure 3 The comparison of 11989110158401015840(119909) between our results and
Howarthrsquos results with ℎ = 001
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
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
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
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
Proposed methodHowarth
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
6 International Journal of Engineering Mathematics
0 1 2 3 4 5 6 7 8 9
x
Proposed methodHowarth
035
03
025
02
015
01
005
0
d2f(x)
Figure 3 The comparison of 11989110158401015840(119909) between our results and
Howarthrsquos results with ℎ = 001
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
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
Proposed methodHowarth
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
6 International Journal of Engineering Mathematics
0 1 2 3 4 5 6 7 8 9
x
Proposed methodHowarth
035
03
025
02
015
01
005
0
d2f(x)
Figure 3 The comparison of 11989110158401015840(119909) between our results and
Howarthrsquos results with ℎ = 001
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
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
Proposed methodHowarth
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
6 International Journal of Engineering Mathematics
0 1 2 3 4 5 6 7 8 9
x
Proposed methodHowarth
035
03
025
02
015
01
005
0
d2f(x)
Figure 3 The comparison of 11989110158401015840(119909) between our results and
Howarthrsquos results with ℎ = 001
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
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
Proposed methodHowarth
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
6 International Journal of Engineering Mathematics
0 1 2 3 4 5 6 7 8 9
x
Proposed methodHowarth
035
03
025
02
015
01
005
0
d2f(x)
Figure 3 The comparison of 11989110158401015840(119909) between our results and
Howarthrsquos results with ℎ = 001
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
6 International Journal of Engineering Mathematics
0 1 2 3 4 5 6 7 8 9
x
Proposed methodHowarth
035
03
025
02
015
01
005
0
d2f(x)
Figure 3 The comparison of 11989110158401015840(119909) between our results and
Howarthrsquos results with ℎ = 001
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