# Approximate Analytical Solutions Using Hyperbolic Functions for the Generalized Blasius Problem

Authors: Beong In Yun
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We propose simple forms of approximate analytical solutions for the generalized Blasius problem based on the given boundary conditions and some known properties of the solution. The efficiency of the proposed solutions is shown for various cases. As a result, one can see that the solutions are uniformly accurate over the whole region.


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## Body

## 1. Introduction

We consider the following generalized Blasius problem corresponding to two-dimensional laminar viscous flow over a thin plate:(1.1)Nf(x):=f′′′(x)+αf(x)f′′(x)=0,0≤x<∞
for α>0, subject to the boundary conditions
(1.2)f(0)=f′(0)=0,f′(∞)=β,
where β>0. We call the solution f(x) the Blasius function. Up to now many analytical methods, for example, Adomian decomposition method [1–3], variational iteration method [4–11], and homotopy analysis method [12–15] have been proposed. In addition, numerical solutions were given in [16–18].For the special case ofα=1/2 and β=1, the Blasius problem was completely reviewed by Boyd [19] with several known properties of the Blasius function f(x). In the recent work [20], the author proposed simple approximate analytical solutions which result in good uniform approximations to the exact solution f(x).In this paper, we extend the method developed in [20] to the generalized Blasius problem (1.1). Based on the given boundary conditions and known properties of the Blasius function f(x), we propose three approximate analytical solutions which consist of the hyperbolic cosine and tangent functions. From the results of the numerical experiments, we can observe that for every cases of α and β the presented approximate solutions are efficient and available over the whole region. In particular, the proposed three-term approximate solution results in the relative errors less than 0.033% and 0.065% for approximation to the exact solution and its derivative, respectively. In addition, using the known properties of the Blasius function, we apply the proposed approach to the homotopy perturbation method [6–8]. Numerical results show the validity of the obtained approximate solution.


## 2. Uniformly Accurate Analytical Solutions

We can see that for arbitraryα and β, the Blasius function f(x) satisfies the following properties [21]:(i)
κ(α;β):=f′′(0)=αβ3κ0,(ii)
B(α;β):=limx→∞{f(x)-βx}=β/αB0,where κ0 and B0 are the constants which correspond to the case of α=1 and β=1 as follows [19, 22]:
(2.1)κ0=0.4695999883⋯,B0=-1.2167806216⋯.For the special case ofα=1/2 and β=1, in the recent work [20], the author introduced the single term approximate solution:
(2.2)h(x)=1blog[cosh(bx)]
and the two term approximate solution
(2.3)g(x)=1blog[cosh(bx)]+ctanh4(rx),r>0,
where the constants b and c are determined from the known properties of the Blasius function as follows:
(2.4)g′′(0)=κ02,limx→∞{g(x)-βx}=2B0.
The parameter r is chosen by minimizing the L2-norm of the residual function Ng(x) over the whole region [0,∞).In this paper, we extend the aforementioned idea to the generalized Blasius problem given in (1.1) and (1.2). First, referring to the single term approximate solution h(x) in (2.2), we modify it as
(2.5)hα,β(x)=βblog[cosh(bx)].
It is straightforward to see that the function hα,β(x) has the first and second derivatives,
(2.6)hα,β′(x)=βtanh(bx),hα,β′′(x)=bβsech2(bx)
and all of the boundary conditions in (1.2) are satisfied. Moreover, taking hα,β′′(0)=κ(α;β), from the property (i) we have
(2.7)b=κ(α;β)β=αβκ0.For the cases ofα=β=1 and α=1, β=3, Figure 1 shows graphs of the proposed approximate solution hα,β(x) and its first derivative hα,β′(x) compared with those of the numerical solution fα,β(x) obtained by using the software Mathematica. We, in this work, regard fα,β(x) as the exact solution.Graphs of the presented solutionhα,β(x) and its derivative (dotted lines) compared with those of the numerical solution fα,β(x) (solid lines).
(a)
α
=
β
=
1
(b)
α
=
1, β=3We can see that the functionhα,β(x) has rather a similar behavior with the numerical solution. Table 1 includes maximum values of the relative percentage errors of the presented approximate solution hα,β(x) and its derivative. Therein, for a function u, M[m]u is defined as
(2.8)M[m]u=Maxj{|E[m]u(xj)|},m=0,1,
where xj are sample points selected as xj=(0.2)j, j=1,2,…,50, and
(2.9)E[m]u(xj)=u(m)(xj)-fα,β(m)(xj)fα,β(m)(xj)×100(%).
The table shows that hα,β(x) uniformly approximates fα,β(x) with the maximum relative errors less than 8.3% and 10.3% for approximation to fα,β(x) and its derivative, respectively. In addition, this tendency of the proposed solution hα,β(x) seems to be independent of the selected values of α(=1/2,1) and β(=1/2,1,2,3).Table 1
Maximum relative% errors of the presented approximate solution hα,β(x) and its derivative for the sample points xj=(0.2)j, j=1,2,…,50.
α
β
M
[
0
]
h
α
,
β
M
[
1
]
h
α
,
β
1
/
2
1
/
2
8.24
10.2
1
8.24
10.2
2
8.24
10.2
3
8.23
10.2
1
1
/
2
8.24
10.2
1
8.24
10.2
2
8.24
10.2
3
8.24
10.1In order to improve the error ofhα,β(x), we employ the two term approximate solution
(2.10)gα,β(x)=βblog[cosh(bx)]+ctanh4(rx),
where b is given in (2.7) and c and r are unknowns to be determined later. We can see that gα,β(x) satisfies the boundary conditions in (1.2) and the properties (i) as well, that is,
(2.11)gα,β(0)=gα,β′(0)=0,gα,β′(∞)=β,gα,β′′(0)=αβ3κ0.
To find the appropriate value of the constant c, we note that from (2.7) and (2.10)
(2.12)limx→∞{gα,β(x)-βx}=-βακ0log2+c.
Then, taking a constraint limx→∞{gα,β(x)-βx}=B(α,β), from the property (ii) we can determine the value of c as follows:
(2.13)c=B(α,β)+βακ0log2=βα{B0+log2κ0}.For determination of the optimal value of the parameterr in the approximate solution gα,β(x), we consider minimization of the residual function
(2.14)Ngα,β(x)=gα,β′′′(x)+αgα,β(x)gα,β′′(x)
over the interval 0<x<∞ in the L2-norm sense, that is, minimization of
(2.15)∥Ngα,β∥22=∫0∞‍{Ngα,β(x)}2dx
with respect to the parameter r included therein. In practice, using the software Mathematica, we can obtain numerical optimal values of r in gα,β(x), denoted by r*=r*(α,β), for each given α and β. Numerical results for the values of r* and the maximum relative percentage errors of gα,β(x) with r* are given in Table 2 for each α=1/2,1 and β=1/2,1,2,3. The table shows that gα,β(x) uniformly approximates the numerical solution fα,β(x) with the maximum relative errors less than 0.44% and 0.97% for approximation to fα,β(x) and its derivative, respectively. As a result, one can see that the two term approximate solution gα,β(x) with r=r* well improves the single term approximation hα,β(x).Table 2
Numerical optimal values of the parameterr and maximum relative % errors of the two-term approximate solution gα,β(x) for each value of α and β.
α
β
r
*
M
[
0
]
g
α
,
β
M
[
1
]
g
α
,
β
1
/
2
1
/
2
0.21475516
0.434
0.965
1
0.30370966
0.434
0.962
2
0.42951032
0.434
0.965
3
0.52604056
0.433
0.955
1
1
/
2
0.30370966
0.434
0.962
1
0.42951032
0.434
0.965
2
0.60741932
0.434
0.962
3
0.74393370
0.433
0.933For further improvement of the approximate solutions given above, we propose another approximate analytical solution as(2.16)kα,β(x)=βblog[cosh(bx)]+c2{tanh4(rx)+tanh8(rx)},
where b and c are as given in (2.7) and (2.13). Similarly to the case of gα,β(x), the optimal value r* of the parameter r in (2.16) should be determined by the minimization of ∥Nkα,β∥22. Table 3 includes numerical results for the values of r* and the maximum relative percentage errors for each α=1/2,1 and β=1/2,1,2,3. One can see that kα,β(x) uniformly approximates fα,β(x) with the maximum relative errors less than 0.033% and 0.065% for approximation to fα,β(x) and its derivative, respectively. Therefore, the three-term approximate solution kα,β(x) with r=r* highly improves the previous approximate solutions hα,β(x) and gα,β(x).Table 3
Numerical optimal values of the parameterr and maximum relative % errors of the three-term approximate solution kα,β(x) for each value of α and β.
α
β
r
*
M
[
0
]
k
α
,
β
M
[
1
]
k
α
,
β
1
/
2
1
/
2
0.25243528
0.0321
0.0649
1
0.35699739
0.0314
0.0647
2
0.50487056
0.0321
0.0646
3
0.61833763
0.0320
0.0647
1
1
/
2
0.35699740
0.0314
0.0647
1
0.50487056
0.0321
0.0646
2
0.71399479
0.0311
0.0647
3
0.87446146
0.0310
0.0649Figure2 includes graphs of the relative percentage errors of the proposed approximate analytical solutions gα,β(x) and kα,β(x) and those of their derivatives gα,β′(x) and kα,β′(x), where the optimal values r=r* given in Tables 2 and 3 are used.Relative% errors of the presented approximate solutions (in upper row) and those of their derivatives (in lower row). Thin lines indicate the two term approximate solution gα,β(x), and thick lines indicate the three-term approximate solution kα,β(x).
(a)
α=β=1
(b)
α
=
1, β=3


## 3. Application to the Homotopy Perturbation Method

In this section, we consider application of the presented approach to the homotopy perturbation method [6] which is composed of coupling iteration method and perturbation method.First, we take an iteration formula for the original equation (1.1) as
(3.1)ϕn+1′′′(x)+αϕn(x)ϕn+1′′(x)=0,n≥0.
Setting an initial approximate solution
(3.2)ϕ0(x)=aα
for a constant a>0 and substituting it into (3.1), we have
(3.3)ϕ1′′′(x)+aϕ1′′(x)=0.
Referring to the boundary conditions in (1.2), we have a solution
(3.4)ϕ1(x)=βx-βa(1-e-ax)
which satisfies ϕ1(0)=ϕ1′(0)=0 and ϕ1′(∞)=β.Substitutingϕ1(x) into (3.1), we obtain
(3.5)ϕ2′′′(x)+aϕ2′′(x)+α{βx-βa(1-e-ax)-aα}ϕ2′′(x)=0.
If we embed an artificial parameter ϵ, then it follows that
(3.6)ϕ2′′′(x)+aϕ2′′(x)+ϵα{βx-βa(1-e-ax)-aα}ϕ2′′(x)=0.
Suppose the solution of this equation can be expressed as
(3.7)ϕ2(x)=ϕ2[0](x)+ϵϕ2[1](x),
then we have the following two equations:
(3.8)(ϕ2[0](x))′′′+a(ϕ2[0](x))′′=0
with ϕ2[0](0)=(ϕ2[0])′(0)=0,(ϕ2[0])′(∞)=β and
(3.9)(ϕ2[1](x))′′′+a(ϕ2[1](x))′′+α{βx-βa(1-e-ax)-aα}(ϕ2[0](x))′′=0
with ϕ2[1](0)=(ϕ2[1])′(0)=0,(ϕ2[1])′(∞)=0.One can see that the solution of (3.8) is
(3.10)ϕ2[0](x)=βx-βa(1-e-ax).
Substitution of ϕ2[0](x) into (3.9) results in
(3.11)(ϕ2[1](x))′′′+a(ϕ2[1](x))′′=-αβ{aβx-β(1-e-ax)-a2α}e-ax.
Assume that the approximate solution of (3.11) can be expressed as
(3.12)ϕ2[1](x)=-2Ce-ax+Ce-2ax+C
for some constant C.By settingϵ=1 in (3.7), from (3.10) and (3.12) we obtain
(3.13)ϕ2(x)=βx-βa(1-e-ax)-2Ce-ax+Ce-2ax+C
with ϕ2(0)=ϕ2′(0)=0 and ϕ2′(∞)=β. To determine the unknown constants C and a we take the conditions limx→∞{ϕ2(x)-βx}=B(α,β) and ϕ2′′(0)=κ(α;β) given in (ii) and (i), respectively. Then it follows that
(3.14)C=βa+B(α,β),aβ+2Ca2=κ(α;β)
which results in
(3.15)a=14B(α,β){-3β+9β2+8κ(α;β)B(α,β)}.On the other hand, if we take an additional termtanh4(rx) such as
(3.16)ϕ~2(x)=βx-βa(1-e-ax)-2Ce-ax+Ce-2ax+C+Ctanh4(rx),
then we have(3.17)C=12{βa+B(α,β)},a=1B(α,β){-β+β2+κ(α;β)B(α,β)}.
The value of r is taken by minimization of the L2-norm of the residual function Nϕ~2(x) as defined in (2.14).Numerical results for the maximum relative percentage errors of the homotopy perturbation method (3.13) and the modified method (3.16) are included in Table 4. The table shows that both the solutions ϕ2(x) and ϕ~2(x) uniformly approximate the exact solution fα,β(x), and that the maximum relative errors of ϕ~2(x) are less than 8.09% and 8.96% for approximation to fα,β(x) and its derivative, respectively. However, though the modified solution ϕ~2(x) improves ϕ2(x) based on the homotopy perturbation method, its accuracy is not comparable with the three-term approximate solution kα,β(x) in (2.16).Table 4
Maximum relative% errors of the approximate solutions ϕ2(x) and ϕ~2(x) based on the homotopy perturbation method.
α
β
M
[
0
]
ϕ
2
M
[
1
]
ϕ
2
M
[
0
]
ϕ
~
2
M
[
1
]
ϕ
~
2
1
/
2
1
/
2
19.2
21.8
8.08
8.95
1
19.2
21.8
8.07
8.93
2
19.2
21.8
8.07
8.95
3
19.2
21.8
8.06
8.96
1
1
/
2
19.2
21.8
8.07
8.93
1
19.2
21.8
8.07
8.95
2
19.2
21.8
8.06
8.88
3
19.2
21.7
8.09
8.91


## 4. Conclusions

In this paper, we have presented three forms of approximate analytical solutions for the generalized Blasius problem (1.1) and (1.2). The presented solutions uniformly approximate the exact solution on the whole interval 0≤x<∞, regardless of the values of α and β. Particularly, the three-term approximate solution kα,β(x) in (2.16) with the parameter r=r* given in Table 3 results in the relative error less than 0.033%. However, it should be pointed out that there will be room for further improvement if more properties of the exact solution, like (i) and (ii) in Section 2, are informed. In addition, employing the known properties (i) and (ii) of the generalized Blasius problem, we have explored the homotopy perturbation method for application of the presented approach. From the numerical results one can see that the presented three-term approximate solution kα,β(x) gives superior results in accuracy.



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*Source: 581453-2012-11-25.xml*