Analytical Solutions of the Blasius Equation
by Perturbation Iteration Method
MEHMET PAKDEMIRLI
Department of Mechanical Engineering
Manisa Celal Bayar University
45140, Muradiye, Manisa,
TURKEY
Abstract:- The Blasius equation is treated by employing the Perturbation Iteration method. Analytical solutions
are derived for different perturbation iteration algorithms. Solutions are contrasted with the numerical solution
obtained by an adaptive step size Runge-Kutta algorithm. It is found that the series type perturbation iteration
solution better represents the behavior inside the boundary layer whereas the exponentially decaying
perturbation iteration solution better represents the real solution outside the boundary layer. A composite
expansion uniting both solutions and valid over the whole region is constructed using the gamma interval
functions. The composite analytical solution is indistinguishable from the numerical one and can replace the
numerical solution in calculations.
Key-Words:- Perturbation Methods, Perturbation-Iteration Algorithm, Boundary Layer Theory, Blasius
Equation, Analytical Solutions
Received: March 29, 2022. Revised: November 7, 2023. Accepted: December 10, 2023. Published: December 31, 2023.
1 Introduction
Due to their complexity, Navier-Stokes equations
are hard to solve analytically except for some
restricted boundary conditions. An excellent
approximation of the Navier-Stokes equation was
proposed [1] by introducing the boundary layer
concept which led to vast applications in
technology especially in the field of aerodynamics.
The partial differential equations were transformed
into an ordinary differential equation, namely the
Blasius equation, via similarity transformations and
an approximate series solution to the equation were
given [1]. For boundary layer type of problems, in
the vicinity of the boundary, a sharp divergence
from the global solution exists. The solution inside
the boundary layer is usually called the inner
solution and the solution outside the boundary
layer, the outer solution in the context of
perturbation analysis. The outer solution is usually
a solution converging to a simple form whereas the
boundary layer solution exists in the vicinity of the
boundary with a sharp deviation from the outer
solution. The two different characteristics of the
solutions make it hard to establish an analytical
solution throughout the whole domain.
Due to its fundamental importance, Blasius
equation, a third order nonlinear ordinary
differential equation, attracted the attention of
many researchers. A vast number of analytical and
numerical techniques were employed in search of
solutions. Blasius equation is solved by the
Variational Iteration Method (VIM) and its variants
[2-4], Adomian Decomposition Method [5, 6],
Parameter Iteration method [7], reproducing kernel
method [8], semi analytic iterative method [9],
Iteration perturbation method [10], an analytical
self-consistent method [11], Homotopy
Perturbation method [12] and a variant of it [13],
Quasi linearization method [14], Adomian Kamal
Transform method [15], the Differential Transform
Method [16], the Optimal Homotopy Asymptotic
Method [17], Sinc-Collocation Method [18].
Bougoffa and Wazwaz [19] assumed an initial
exact solution which does not satisfy the boundary
conditions and found an analytical iterative
solution. Numerical solutions of the equation were
presented by applying the Crocco-Wang
transformation [20]. For a theoretical mathematical
study of the generalized Blasius equation, see [21].
The Perturbation Iteration method (PIM) is
applied to Blasius equation for the first time in this
work. PIM is a systematic way of producing
perturbation iteration algorithms, PIA(n,m) where n
represents the number of correction terms in the
perturbation expansion and m represents the order
of the derivatives in the Taylor expansions. The
method is developed originally for nonlinear
algebraic equations [22] and the formalism is later
PROOF
DOI: 10.37394/232020.2023.3.14
Mehmet Pakdemirli
E-ISSN: 2732-9941
100
Volume 3, 2023
applied to differential equations [23,24]. In the last
decade, the method has been successfully applied
to many mathematical models arising from physical
problems [25-48]. A general convergence analysis
of PIM as well as an error analysis is given in [49].
In this work, it is found that the PIA(1,1)
solution describes well the real solution inside the
boundary layer and the PIA(1,2) solution describes
the real solution well in the far end of the boundary
layer and outside the boundary layer. Hence, a
composite analytical solution is constructed using
gamma interval functions which is
indistinguishable from the numerical solution
within the whole domain.
2 Boundary Layer Equations and
Reduction
The boundary layer equations for a fluid passing
over a flat plate are [1]
(1)
(2)
with the no-slip boundary conditions at the plate
and the condition at infinity being
, , (3)
where and represents the x and y
components of the velocity, U is the inviscid
incompressible velocity outside the boundary layer
and is the kinematic viscosity. Blasius
transformed the equations into a nonlinear ordinary
differential equation via the similarity
transformations
,
,
(4)
which leaded to the well-known Blasius equation
(5)
with the transformed boundary conditions for the
flow over a flat plate
,
,
. (6)
From the similarity transformations, it is obvious
that the x component of the velocity is directly
related to
and the y component is related to
and
. Analytical and numerical solutions
will be presented in the following sections.
3. PIA(1,1) Solution
The general functional form of the Blasius equation
is
(7)
where
is added in front of the nonlinear term as a
book-keeping parameter which will be eliminated
in the final iteration equation. In the PIA(1,1)
algorithm, only one correction term in the
perturbation expansion and first order derivatives in
the Taylor expansion are considered. Hence,
(8)
where is the n’th iteration solution,
is the
correction term at the n’th iteration. Substituting (8)
into (7) and expanding the Taylor series up to first
order derivatives in the vicinity of
=0 yields the
iteration algorithm
(9)
Since
, ,
,
,
, and
, substituting all into (9) yields the
iteration algorithm
. (10)
The boundary value problem is transformed into an
initial value problem
,
,
(11)
for practical purposes. Starting from an initial
trivial solution
(12)
the consecutive iteration solutions which satisfy the
conditions (11) are
(13)
(14)
(15)
and the derivative of the third iteration is
(16)
Higher iterations are not considered, as the
denominators of the coefficients become
extensively large and the aim is to seek a simple
solution.
The PIA(1,1) in fact generates a series type
solution which can be deduced by substituting
(17)
into the original equation and equating to zero the
coefficients of like powers
,
(18)
The initial conditions require , and
. Other coefficients can be calculated up to
PROOF
DOI: 10.37394/232020.2023.3.14
Mehmet Pakdemirli
E-ISSN: 2732-9941
101
Volume 3, 2023
any arbitrary order of . The series solution up to
is
(19)
which is also the solution given by Blasius [2].
Comparing (15) and (19), one realizes that there is
a discrepancy in the coefficient of the last term.
This stems from the termination of the iteration at
n=3. However, since the terms higher than are
totally ignored in the series expansion, with the
previous experience for such solutions, PIA(1,1)
solution may include the effects of such terms in
the last term and may slightly perform better than
the series solution. By using Variational Iteration
Method, Wazwaz [3] calculated higher order terms
also.
To test the analytical solutions, an adaptive step
size numerical solution employing Runge-Kutta
method is used. The Blasius equation is expressed
as a system of first order equations by defining
,
,
,
(20)
(21)
(22)
and the boundary value problem is converted to an
initial value problem
, , (23)
The specific value of which makes the solution
satisfy the condition is determined by
the shooting technique
0.3320573 (24)
Within the tolerances of our numerical algorithm,
there is no need to take a more precise value. As
mentioned earlier,
determines the x component
of the velocity whereas
and determines the y
component. Hence, comparisons of these quantities
are given in Figures 1 and 2.
Fig. 1: Comparisons of PIA(1,1) and series
solutions with the numerical solution for
Fig. 2: Comparisons of PIA(1,1) and series
solutions with the numerical solution for
From both figures, the analytical solutions start
diverging after . The series solution is slightly
more divergent than the PIA(1,1) solution as
expected. From Figure 2, the boundary layer
reaches the fully developed state after .
Excellent match is observed in the interval [0, 3]
but the solutions are not reliable as one proceeds to
higher values.
For a general convergence and error analysis,
refer to [49] for details.
4 PIA(1,2) Solution
For constructing the PIA(1,2) algorithm, one takes
one correction term in the perturbation expansion
similar to the previous case, i.e. Eq. (8), but
expands the Taylor series up to second order
derivatives
(25)
After evaluating the derivatives at and using
(8), one has the iteration equation
,
(26)
The boundary value problem need not be
transformed as in the PIA(1,1) case, that is
,
,
. (27)
Starting with an initial constant solution
(28)
the first iteration solution is
PROOF
DOI: 10.37394/232020.2023.3.14
Mehmet Pakdemirli
E-ISSN: 2732-9941
102
Volume 3, 2023
(29)
which satisfies the conditions (27) for any . The
first derivative is
(30)
For the second iteration,
. (31)
Upon substituting , the equation is a variable
coefficient linear third order equation, hard to
solve. Instead, just for the coefficients at the left-
hand side of the equation, one may assume a
simplification by taking instead of . The final
solution satisfying the boundary conditions is
(32)
with its derivative
(33)
Note that α remains arbitrary in the first and second
iteration solutions. It can be selected so that the
match between the numerical results is better. In
Figures 3 and 4, the first and second iteration
solutions are contrasted with the numerical ones.
Fig. 3: Comparisons of first and second iteration
PIA(1,2) with the numerical solution for
Fig. 4: Comparisons of first and second iteration
PIA(1,2) with the numerical solution for
For the first iteration , and for the second
iteration . From both figures, it is evident
that the second iteration solution does not perform
better than the first iteration. One reason might be
that, in obtaining the second iteration, a simplifying
assumption about the coefficients of the equation is
made. On the contrary, there are no blow ups in the
whole domain of interest as in the PIA(1,1) and
series solution cases and the solution is admissible
albeit with some small fractional errors. The errors
introduced are mainly inside the boundary layer.
Outside the boundary layer, the solution matches
better with the numerical solution.
The convergence and error analysis of PIM is
an important issue and the problem is addressed in
detail in [49].
5 A Composite Solution
The aim is to construct an analytical solution
which can replace the numerical solution and is
valid throughout the whole domain of interest.
Based on the results presented in the previous
sections, PIA(1,1) solution will be taken in the
interval [0 ] and PIA(1,2) solution will be taken
in the interval []. The reason for this choice is
that PIA(1,1) is more successful in representing the
behavior inside the boundary layer and PIA(1,2) is
more successful outside it. For a smooth connection
at the intermediate junction point , the functions,
the first and second derivatives should be equated.
At , from (15), they are
(34)
(35)
(36)
The first iteration PIA(1,2) is sufficient for
calculations and the form of the solution is
PROOF
DOI: 10.37394/232020.2023.3.14
Mehmet Pakdemirli
E-ISSN: 2732-9941
103
Volume 3, 2023
(37)
which is called the outer solution valid as
approaches infinity. Equating the function, first and
second derivatives of the outer solution to the inner
solution at the intermediate point, i.e. Eqs. (34)-
(36), the coefficients are calculated
,
,
. (38)
To write the composite solution as a single
expression, one may define a new gamma interval
function with properties
(39)
The composite solution and its first derivative are
expressed as single analytical expressions
(40)
(41)
The composite expansions (40) and (41) are
contrasted with the numerical solutions in Figures 5
and 6. In the calculations, =3, ,
0.3320573 are selected.
Fig. 5: Comparisons of the composite expansion
with the numerical solution for
Fig. 6: Comparisons of the composite expansion
with the numerical solution for
From the figures, the composite solution is almost
indistinguishable from the numerical solution.
Hence, instead of the discrete numerical solution,
the composite solution can be used safely for a
continuous expression of the solution.
6 Concluding Remarks
Perturbation iteration method is used for solving
the Blasius equation for the first time. Iteration
algorithms PIA(1,1) and PIA(1,2) are employed in
search of approximate analytical solutions. It is
found that PIA(1,1) better represents the numerical
solutions within the boundary layer adjacent to the
plate whereas PIA(1,2) better represents the
numerical solutions at the far edge of the boundary
layer and outside the boundary layer. Hence a
composite expansion valid throughout the whole
domain is constructed by combining both solutions.
Most of the series solutions presented in the
literature diverge at the edge and outside the
boundary layer because the nature of the solutions
is different inside and outside of the boundary
layer. The composite expansion has the advantage
of representing the solutions precisely in both
domains. As a final conclusion, the continuous
composite expansion can safely replace the discrete
numerical solutions.
References:
[1] Schilichting H., Boundary Layer Theory,
McGraw Hill, New York, 2004.
[2] He J., Approximate analytical solution of
Blasius equation, Communications in Nonlinear
Science and Numerical Simulation, Vol. 4, No. 1,
1999, pp. 75-78.
PROOF
DOI: 10.37394/232020.2023.3.14
Mehmet Pakdemirli
E-ISSN: 2732-9941
104
Volume 3, 2023
[3] Wazwaz A. M., The variational iteration
method for solving two forms of Blasius equation
on a half-infinite domain, Applied Mathematics and
Computation, Vol. 188, 2007, pp. 485-491.
[4] Sajid M., Abbas Z., Ali N. and Javed T., A
hybrid variational iteration method for Blasius
equation, Applications and Applied Mathematics:
An International Journal, Vol. 10, No. 1, 2015,
223-229.
[5] Wang L., A new algorithm for solving classical
Blasius equation, Applied Mathematics and
Computation, Vol. 157, 2004, No. 1-9.
[6] Abbasbandy S., A numerical solution of Blasius
equation by Adomian’s decomposition method and
comparison with homotopy perturbation method,
Chaos, Solitons and Fractals, Vol. 31, 2007, pp.
257-260.
[7] Lin J., A new approximate iteration solution of
Blasius equation, Communications in Nonlinear
Science and Numerical Simulation, Vol. 4, No. 2,
1999, pp. 91-94.
[8] Akgül A., A novel method for the solution of
Blasius equation in semi-infinite domains, An
International Journal of Optimization and Control:
Theories and Applications, Vol. 7, No. 2, 2017,
225-233.
[9] Selamat M. S., Halmi N. A. and Ayob N. A., A
semi analytic iterative method for solving two
forms of Blasius equation, Journal of Academia,
Vol. 7, No. 2, 2019, pp. 76-85.
[10] He J. H., A simple perturbation approach to
Blasius equation, Applied Mathematics and
Computation, Vol. 140, 2003, pp. 217-222.
[11] Yasuri A. K., An analytical self-consistent
method for differential forms of the Blasius
equation, Mathematical Methods in the Applied
Sciences, Vol. 46, 2023, pp. 5836-5849.
[12] Ganji D. D., Babazadeh H., Noori F., Pirouz
M. M.and Janipour M., An application of
homotopy perturbation method for nonlinear
Blasius equation to boundary layer flow over a flat
plate, International Journal of Nonlinear Science,
Vol. 7, No. 4, 2009, pp. 399-404.
[13]Aminikhah H., An analytical approximation for
solving nonlinear Blasius equation by NHPM,
Numerical Methods for Partial Differential
Equations, Vol. 26, No. 6, 2010, pp. 1291-1299.
[14] Delkhosh M. and Cheraghian H., An efficient
hybrid method to solve nonlinear differential
equations in applied sciences, Computational and
Applied Mathematics, Vol. 41, Article No. 322,
2022.
[15] Khandelwal R., Kumawat P. and Khandelwal
Y., Solution of the Blasius equation by using
Adomian Kamal Transform, International Journal
of Applied and Computational Mathematics, Vol. 5,
Article No. 20, 2019.
[16] Peker H. A., Karaoğlu O. and Oturanç G., The
differential transformation method and pade
approximant for a form of Blasius equation,
Mathematical and Computational Applications,
Vol. 16, No. 2, 2011, pp. 507-513.
[17] Marinca V. and Herişanu N., The optimal
homotopy asymptotic method for solving Blasius
equation, Applied Mathematics and Computation,
Vol. 231, 2014, pp. 134-139.
[18] Parand K., Dehghan M. and Pirkhedri A.,
Sinc-collocation method for solving the Blasius
equation, Physics Letters A, Vol. 373, 2009, pp.
4060-4065.
[19] Bougoffa L and Wazwaz A. M., New
approximate solutions of the Blasius equation,
International Journal of Numerical Methods for
Heat and Fluid Flow, Vol. 25, No. 7, 2015, pp.
1590-1599.
[20] Asaithambi A., Numerical solution of the
Blasius equation with Crocco-Wang
transformation, Journal of Applied Fluid
Mechanics, Vol. 9, No. 5, 2016, pp. 2595-2603.
[21] Makhfi A. and Bebbouchi R., On the
generalized Blasius equation, Afrika Matematika,
Vol. 31, 2020, pp. 803-811.
[22] Pakdemirli M., Boyacı H., Generation of root
finding algorithms via perturbation theory and
some formulas, Applied Mathematics and
Computation, Vol. 184, No. 2, 2007, pp. 783-788.
[23] Aksoy Y. and Pakdemirli M., New
Perturbation-Iteration Solutions for Bratu-type
Equations, Computers & Mathematics with
Applications, Vol. 59, 2010, pp. 2802-2808.
[24] Aksoy Y., Pakdemirli M., Abbasbandy, S,
Boyacı, H., New Perturbation-Iteration Solutions
for Nonlinear Heat Transfer Equations.
International Journal of Numerical Methods for
Heat & Fluid Flow, Vol. 22, 2012, pp. 814-828.
[25] Al Saif A. J. and Harfash A. J., A comparison
between the reduced differential transform method
and perturbation-iteration algorithm for solving two
dimensional unsteady incompressible Navier-
Stokes equations, Journal of Applied Mathematics
and Physics, Vol. 6, 2018, pp. 2518-2543.
[26] Al-Saif A. J. and Harfash A. J., Perturbation
Iteration Algorithm for solving heat and mass
transfer in the unsteady squeezing flow between
parallel plates, Journal of Applied Computational
Mechanics, Vol. 5, No. 4, 2019, pp. 804-815.
[27] Bahşi M. M. and Çevik M., Numerical
solution of Pantograph type delay differential
equations using Perturbation Iteration Algorithms,
Journal of Applied Mathematics, Vol. 2015, 2015.
PROOF
DOI: 10.37394/232020.2023.3.14
Mehmet Pakdemirli
E-ISSN: 2732-9941
105
Volume 3, 2023
[28] Bildik N. Optimal Perturbation-Iteration
method for solving Telegraph equations,
International Journal of Applied Physics and
Mathematics, Vol. 7, No. 3, 2017, pp. 165-172.
[29] Bildik N. and Deniz S., A new efficient
method for solving delay differential equations and
a comparison with other methods, The European
Physical Journal Plus, Vol. 132, Article No. 51,
2017.
[30] Bildik N. and Deniz S., New analytic
approximate solutions to the generalized
regularized long wave equations, Bulletin of
Korean Mathematical Society, Vol. 55, No. 3,
2018, pp. 749-762.
[31] Bildik N. and Deniz S., Solving the Burgers
and long wave equations using the new
perturbation iteration technique, Numerical
Methods for Partial Differential Equations, Vol.
34, 2018, pp. 1489-1501.
[32] Bildik N. and Deniz S., Comparative study
between optimal homotopy asymptotic
method and perturbation iteration technique for
different types of Nonlinear equations, Iranian
Journal of Science and Technology, Transactions
A-Science, Vol. 42, 2018, 647-654.
[33] Deniz S. and Bildik N., A new analytical
technique for solving Lane-Emden type
equations arising in astrophysics, Bulletin of
Belgium Mathematical Society-Simon Stevin, Vol.
24, No. 2, 2017, pp. 305-320.
[34] Deniz S. and Bildik N., Optimal Perturbation-
Iteration method for Bratu type problems,
Journal of King Saud University-Science, Vol. 30,
2018, pp. 91-99.
[35] Gahamanyi M., Ntganda J. M. and Hapgar M.
S. D., Perturbation Iteration Method for
solving mathematical model of glucose and insulin
in diabetic human during physical activity.
Open Journal of Applied Sciences, Vol. 6, 2016,
pp. 826-838.
[36] Harfash A. J. and Al-Saif A. J., MHD flow of
fourth grade fluid solve by PIA algorithm,
Journal of Advanced Research in Fluid Mechanics
and Thermal Sciences, Vol. 59, No. 2, 2019, pp.
220-231.
[37] Khalid M., Sultana M., Zaidi F. and Khan J., A
solution of a water quality model in a uniform
stream channel using new iterative method,
International Journal of Computer Applications,
Vol. 115, No. 6, 2015, pp. 1-4.
[38] Khalid M., Sultana M., Zaidi F. and Arshaul
U., An effective perturbation-iteration algorithm for
solving Riccati differential equations, International
Journal of Computer Applications, Vol. 111, No.
10, 2015, pp. 1-5.
[39] Khalid M., Sultana M., Zaidi F. and Uroosa
A., Solving linear and non-linear Klein-
Gordon equations by new perturbation iteration
transform method, TWMS Journal of Applied
Engineering Mathematics, Vol. 6, No. 1, 2016, pp.
115-125.
[40] Pakdemirli M., Perturbation–iteration method
for strongly nonlinear vibrations, Journal of
Vibration and Control, Vol. 23, No. 6, 2017, pp.
959-969.
[41] Singh R. P. and Reddy Y. N., Perturbation
Iteration method for solving differential
difference equations having boundary layer,
Communications in Mathematics and Applications
Vol. 11, No. 4, 2020, pp. 617-633.
[42] Srivastava H. M., Deniz S. and Saad K. M.,
An efficient semi-analytical method for solving the
generalized regularized long wave equations with a
new fractional derivative operator, Journal of King
Saud University-Science, Vol. 33, Article No.
101345, 2021.
[43] Şenol M., Alquran M. and Kasmaci H. D., On
the comparison of perturbation-iteration algorithms
and residual power series method to solve
fractional Zakharov-Kuznetsov equation,
Results in Physics, Vol. 9, 2018, pp. 321-327.
[44] Şenol M., Atpinar S., Zararsız Z., Salahshour
S. and Ahmadian A., Approximate solution of time
fractional fuzzy partial differential equations,
Computational and Applied Mathematics, Vol. 38,
Article No. 18, 2019.
[45] Şenol M. and Dolapçı I. T., On the
Perturbation Iteration Algorithm for fractional
differential equations. Journal of King Saud
University-Science, Vol. 28, 2016, pp. 69-74.
[46] Şenol M. and Kasmaci H. D., Perturbation-
Iteration Algorithm for systems of fractional
differential equations and convergence analysis,
Progress in Fractional Differentiation and
Applications, Vol. 3, No. 4, 2017, pp. 271-279.
[47] Tasbozan O., Şenol M., Kurt A. and Özkan O.,
New solutions of fractional Drinfeld-
Sokolov-Wilson system in shallow water waves,
Ocean Engineering, Vol. 161, 2018, pp. 62-68.
[48] Yıldız V., Pakdemirli M. and Aksoy Y.,
Parallel plate flow of a third- grade fluid and a
Newtonian fluid with variable viscosity. Zeitschrift
fur Naturforschung A, Vol. 71, No. 7, 2016, pp.
595-606.
[49] Bildik N., General Convergence Analysis for
the Perturbation Iteration Technique, Turkish
Journal of Mathematics and Computer Science,
Vol. 6, 2017, pp. 1-9.
PROOF
DOI: 10.37394/232020.2023.3.14
Mehmet Pakdemirli
E-ISSN: 2732-9941
106
Volume 3, 2023
PROOF
DOI: 10.37394/232020.2023.3.14
Mehmet Pakdemirli
E-ISSN: 2732-9941
107
Volume 3, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
