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Multiple solutions of problems in fluid mechanics by
predictor optimal homotopy asymptotic method
A. K. Alomari1 , N. Ratib Anakira2,∗, and I. Hashim2,3
1
Department of Mathematics, Faculty of Science,
Yarmouk University, 211-63 Irbid, Jordan
2
Centre for Modelling and Data Analysis, School of Mathematical Sciences,
Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia
3
Department of Mathematics, Faculty of Science,
King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia
October 10, 2014
Abstract
A new algorithm based on the standard optimal homotopy asymptotic method
namely the predictor optimal homotopy asymptotic method (POHAM) is proposed
to predict the multiplicity of the solutions of nonlinear differential equations with
boundary conditions. This approach is successfully implemented to obtain multiple
solutions of a mixed convection flow model in a vertical channel and a nonlinear
model arising in heat transfer. A new technique for obtaining the initial guess to
accelerate the convergence of the series solution is presented.
Keywords: Multiple solutions, Nonlinear differential equations, Optimal homotopy
asymptotic method, Predictor optimal homotopy asymptotic method
∗
Corresponding author. Email: alanaghreh [email protected], Tel: 0060189189664
1
1
Introduction
The study of the existence of multiple solutions for nonlinear differential equations has
drawn a lot of attention in the last few years [1, 2]. Multiple solution for nonlinear
boundary value problems via homotopy analysis method was investigated by several papers for example, Liao [3, 4], Li and Liao [5], Xu and Liao [6]. Moreover, Abbasbandy
and Shivanian [7] presented a kind of analytical method namely predictor homotopy
analysis method (PHAM) to predict the multiplicity of the solutions of nonlinear differential equations with boundary conditions. Also, Alomari et al. [8] used the PHAM to
find the multiple solutions of fractional boundary value problem (FBVP).
In the last few years, optimal homotopy asymptotic method (OHAM) has been introduced and developed by Marinca and Herisanu [9, 10, 11, 12], and has been applied
successfully to many strongly nonlinear problems [13, 14, 15, 16]. Iqbal et al. [17] obtained OHAM solutions to linear and nonlinear Klein-Gordan equations, Anakira et al.
[18] used OHAM for solving singular two-point boundary value problems and Alomari et
al. [19] employed OHAM to obtain approximate solution of nonlinear system of boundary value problems arising in fluid flow problems. Moreover, Idrees et al. [20, 21] and
Mabood et al. [22, 23] have applied OHAM effectively to different higher-order boundary
values problems.
The main aim of this paper is to present a type of analytical method namely the
predictor optimal homotopy asymptotic method (POHAM) to predict the multiplicity of
the solutions of the nonlinear differential equations with boundary conditions by using
one auxiliary linear operator, one auxiliary function and one initial guess. This paper
is composed as follows: In Section 2, the structure of POHAM is formulated for finding multiple solutions of nonlinear differential equations. In Section 3, we present two
numerical examples and finally, we give the conclusion of this study in Section 4.
2
The predictor optimal homotopy asymptotic method (POHAM)
Consider the following differential equation:
L(y(x)) + g(x) + N (y(x)) = 0,
x∈Ω
(1)
with boundary conditions:
(
∂y
B y,
∂η
)
= 0,
2
x ∈ Γ,
(2)
where L is the linear operator, N is the linear or nonlinear operator, g(x) is a known
function, B is a boundary operator and Γ is the boundary of the domain Ω. The key
step of POHAM depends on the fact that the boundary-value problem (1) and (2) should
be transcribed to an equivalent problem so that the boundary condition (2) involves an
unknown parameter, the so-called prescribed parameter δ, and is decomposed into
B
′
(
∂y
y, δ,
∂η
)
x ∈ Γ,
= 0,
y(α) = β,
(3)
where y(α) = β is the forcing condition that is resulted from the original condition (2)
and B ′ is the remaining boundary operator that contains the prescribed parameter δ. As
it will be noticed, the parameter δ with the help of convergence-controller parameter Ci ’s
will play a significant role to realize the multiplicity of solutions. Now, we construct the
POHAM on equations (1) and (3). By using a homotopy map h(v(x, δ; q); q) : Ω×[0, 1] →
ℜ which satisfies
(1 − q)[L(v(x, δ; q)) − y0 (x; δ)] = H(q)[L(v(x, δ; q)) + g(x) + N (v(x, δ; q))],
)
(
∂v(x, δ; q)
′
= 0,
B v(x, δ; q),
∂x
(4)
where x ∈ ℜ, q ∈ [0, 1] is an embedding parameter, H(q) is a nonzero auxiliary function
for q ̸= 0, H(0) = 0 and v(x, δ; q) is an unknown function. Obviously, when q = 0 and
q = 1, it holds that v(x, δ; 0) = y0 (x; δ) and v(x, δ; 1) = y(x) respectively. Thus, as q
varies from 0 to 1, the solution v(x, δ; q) approaches from y0 (x; δ) to y(x), where y0 (x; δ)
is the initial guess that satisfies the linear operator and the boundary conditions
B
′
(
dy0
y0 ,
dx
)
= 0.
(5)
Next, we choose the auxiliary function H(q) in the form
H(q) = qC1 + q 2 C2 + q 3 C3 + · · · ,
(6)
where C1 , C2 , C3 , . . . are constants which will be determined later. To obtain an approximate solution, we expand v(x, δ; q) in Taylor’s series about q in the following manner,
v(x, δ; q) = y0 (x, δ) +
∞
∑
yk (x, δ)q k .
(7)
k=1
Substituting (7) into (4) and equating the coefficient of like powers of p, we obtain the
→
following linear equations, by defining the vector −
y = {y0 (x), y1 (x), . . . y (x)}. The
k
k
first- and second-order problems are given as
L(y1 (x, δ)) + g(x) = C1 N0 (y0 (x, δ)),
3
B
′
(
dy1
y1 ,
dx
)
= 0,
(8)
and
L(y2 (x, δ)) − L(y1 (x, δ)) = C2 N0 (y0 (x, δ)) + C1 [L (y1 (x) + N1 (y0 (x, δ) , y1 (x, δ))] ,
(
)
dy2
′
B y2 ,
= 0.
(9)
dx
The general governing equations for yk (x, δ) are
L(yk (x, δ)) − L(yk−1 (x, δ)) = Ck N0 (y0 (x, δ)) +
k−1
∑
Ci [L(yk−i (x, δ))
i=1
(
B ′ yk ,
dyk
dx
−
+Nk−i (→
y k−1 (x, δ))],
)
= 0,
(10)
where k = 2, 3, . . . and
−
Nm (→
y m−1 , x, δ) =
∑
n 1
∂ m−1 N [ ∞
n=0 yn (x, δ)q ] .
(m − 1)!
∂q m−1
q=0
(11)
The convergence of the series (7) depends upon the auxiliary convergent control parameters C1 , C2 , C3 , . . .. If it is convergent at q = 1, then
∞
∑
v(x, Ci ) = y0 (x, δ) +
yk (x, C1 , C2 , . . . , Ck ).
(12)
k=1
Thus the mth-order solution is given by
y˜(x, δ, C1 , C2 , C3 , . . . , Cm ) = y0 (x, δ) +
m
∑
yi (x, δ, C1 , C2 , . . . , Ci ).
(13)
i=1
Substituting (12) into (1) yields the following residual
R (x, δ, C1 , C2 , C3 , . . . , Cm ) = L(˜
y (x, δ, C1 , C2 , C3 , . . . , Cm )) + g (x)
+N (˜
y (x, δ, C1 , C2 , C3 , . . . , Cm )).
(14)
If R = 0, then y˜ will be the exact solution. Generally, such a case will not arise for
nonlinear problems, but we can minimize the functional
∫
b
J(C1 , C2 , C3 , . . . , Cm ) =
R2 (x, δ, C1 , C2 , C3 , . . . , Cm )dx,
(15)
a
where a and b are two values, depending on the problem. The unknown convergent
control parameters Ci (i = 1, 2, 3, . . . , m) and the so-called prescribed parameter δ can
be identified from the conditions
∂J
∂J
∂J
=
= ··· =
= 0,
∂C1
∂C2
∂Cm
y˜(α) = β.
(16)
With those known convergent control parameters, the approximate solution (of order m)
is determined. Note that the force condition plays a significant role to determine the
multiple solutions.
4
3
Applications
In this Section, we will present two examples to demonstrate the effectiveness and high
precision of present method.
3.1
Example 1
The first example comes from a mixed convection flow problem in a vertical channel
studied by Barletta et al. [25]. The two walls are assumed to be isothermal. Furthermore,
the effect of viscous dissipation is taken into account and the Boussinesq approximation
is adopted. By employing appropriate dimensionless quantities, the governing equations
were reduced to the following non-linear fourth-order ordinary differential equation for
the dimensionless velocity field (cf. [25, 7])
E
d4 y
=
4
dx
16
(
dy
dx
)2
,
(17)
subject to the boundary conditions
∫
y(1) = y ′ (0) = y ′′′ (0) = 0,
1
y(x)dx = 1,
(18)
0
where y is the dimensionless velocity, x is the transversal coordinate and E is a constant.
In the case E = 0, Eqs. (17) and (18) are easily solved and admit a unique solution
3
y(x) = (1 − x2 ).
2
(19)
It was shown by perturbation method and a numerical method that Eqs. (17) and
(18) admit multiple solutions to any given E in the interval (−∞, 0) ∪ (0, Emax ) where
Emax = 228.128 (cf. [24, 25]).
To find out the multiple solutions, we consider Eqs. (17) and (18) and suppose that
y ′′ (0)
= δ, where δ is a prescribed parameter that plays an important role in recognizing
the multiplicity of solutions, so that the problem becomes
d4 y
E
=
4
dx
16
(
dy
dx
)2
,
(20)
y ′′ (0) = δ,
(21)
subject to the boundary conditions
y(1) = y ′ (0) = y ′′′ (0) = 0,
with additional force condition
∫
1
y(x)dx = 1.
0
5
(22)
According to POHAM, we chose the linear and nonlinear operator as follows, respectively,
L[v(x, δ, q)] =
∂ 4 v(x, δ, q)
,
∂x4
N [v(x, δ, q)] =
E
16
(
∂v(x, δ, q)
∂x
)2
.
(23)
Under the rule of the solution expression and according to conditions (21) it is easy to
choose
δ
y0 (x, δ) = (x2 − 1),
2
(24)
as initial guess. The first-order problem obtained from Eq. (8) as follow
(4)
−Ex2 δ 2 C1
,
16
y1′ (0) = y1′′′ (0) = y1 (1) = y1′′ (0) = 0.
y1 (x, δ) =
(25)
From Eq. (9) the second-order problem is defined as
Ex2 δ 2 C2 ExδC1 ′
−
y1 (x, δ) + y1′′′′ (x, δ) + C1 y1′′′′ (x, δ),
16
8
y2′ (0) = y2′′′ (0) = y2 (1) = y2′′ (0) = 0.
y2′′′′ (x, δ) = −
(26)
By using Eq. (10) for k = 3 and k = 4, the third- and fourth-order problems are given,
respectively, as follows
1
1
Ex2 δ 2 C3 − ExδC2 y1′ (x, δ)
16
8
1
1
′ 2
− EC1 (y1 ) (x, δ) − ExδC1 y2′ (x, δ)
16
8
−C2 y1′′′′ (x, δ) − C1 y2′′′′ (x, δ),
y3′′′′ (x, δ) = y2′′′′ (x, δ) −
y3′ (0) = y3′′′ (0) = y3 (1) = y3′′ (0) = 0.
(27)
and
Ex2 δ 2 C4
16
ExδC3 ′
EC2 ′ 2
ExδC2 ′
−
y1 (x) −
(y ) (x, δ) −
y2 (x, δ)
8
16 1
16
ExδC1 ′
EC1 ′
−
y (x, δ)y2′ (x, δ) −
y3 (x, δ) + C3 y1′′′′ (x, δ),
8 1
8
y4′ (0) = y4′′′ (0) = y4 (1) = y4′′ (0) = 0.
y4′′′′ (x, δ) = C2 y2′′′′ (x, δ) + y3′′′′ (x, δ) + C1 y3′′′′ (x, δ) −
(28)
By substituting the solution of Eqs. (25)–(28) into Eq. (13), yields the fourth-order
approximate solutions (m = 4) for Eqs. (20) and (21):
y˜(x, δ) = y0 (x, δ) + y1 (x, δ) + y2 (x, δ) + y3 (x, δ) + y4 (x, δ).
(29)
By substituting the fourth-order approximate solution (29) into Eq. (14) yields the
residual error and the functional J, respectively:
R˜
y (x, δ) = y˜′′′′ (x, δ) −
6
E ′
(˜
y (x, δ))2 .
16
(30)
Table 1: Values of Ci ’s corresponding to the prescribed parameter δ for Example 1.
E = −20
E = 20
δ1 = −3.09773
δ2 = −161.726
δ1 = −2.92300
δ2 = 170.039
C1
−1.63557 × 10−15
−0.738509
−1.07453
−0.438751
C2
5.54432 × 10−15
−0.695483
0.005063
−0.592496
C3
10−15
−0.617731
−0.000767
0.259233
−1.14066
−0.836292
−0.088768
−0.518427
1.05665 ×
C4
∫
J(C1 , C2 , C3 , C4 ) =
1
R2 (x, δ, C1 , C2 , C3 , C4 )dx,
(31)
0
with the additional force condition
∫
1
y(x)dx = 1.
(32)
0
Now, to be specific, we consider two case consisting of E = 20 and E = −20, with
a prescribed parameter δ as a function of the convergence controller parameter. The
values of the convergent control parameters Ci ’s are obtained based on the values of the
prescribed parameter δ as presented in Table 1. We plot the multiple (dual) solutions in
Fig. 1. It is worth to mention here that Fig. 1(a) indicating existence of two solutions
for E = 20 so that, u′′ (0) = −3.08411 for the first branch of the solution and u′′ (0) =
−161.726 for the second branch of the solution. The same procedure was done for case
E = −20. As we see in Fig. 1(b). It is obviously that the results being obtained by using
fourth terms POHAM approximate solution is to an extent identical compared with the
25th terms PHAM approximate solution [7]. This means that the POHAM solution
reveals very good agreement with PHAM solution in few terms which prove the validity
and the efficiency of our procedure in solving strongly nonlinear problems.
3.2
Example 2
Our second example comes from the heat conduction problem for a fin with heat transfer
coefficient varying as a power-law function of temperature considered by Chang [27]. The
rate of heat transfer on a solid surface can be enhanced by fins. In the mathematical
modelling, Chang [27] considered a straight fin of finite length and uniform cross-section
area. The fin surface is exposed to a prescribed ambient temperature. The dimensionless
equation for the one-dimensional steady state heat conduction is given as (cf. [27, 7]),
d2 y
− N 2 y n+1 = 0,
dx2
7
(33)
(a)
1.0
∆ = -3.08411
yH0L = 1.51086
y4 HxL
y4 H0L
0.5
0.0
∆ = -161.726
-0.5
yH0L = 14.248
-1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
(b)
1.0
∆ = -2.92300
yH0L = 1.49072
y4 H0L
y4 HxL
0.5
0.0
∆ = 170.039
yH0L = -16.1591
-0.5
-1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
Figure 1: Multiple solutions of Eqs. (20) and (21) for (a) E = 20 and (b) E = −20.
8
subject to the boundary conditions
dy
(0) = 0,
dx
y(1) = 1,
(34)
where N is the convective-conductive parameter. It was shown that the problem (33)
and (34), when −4 ≤ n ≤ −2, either admit multiple solutions or does not admit any
solution for a given convective-conductive parameter N [26]. In particular, suppose that
N=
2
5
and n = −4 then the Eq. (33) is converted into
4
= 0,
25
(35)
y(0) = δ,
(36)
y ′′ (x)y 3 (x) −
subject to the boundary conditions
y ′ (0) = 0,
where δ is the temperature of the fin tip and is determined by the rule of multiplicity of
the solutions. With the additional force condition
y(1) = 1.
(37)
The exact multiple (dual) solutions of this problem is given by
5 √ 2
λ y (x) − λ2 = x,
2
(38)
where y(0) = λ is represented by two values: λ = 0.4472135954 for the first branch of
the solution and λ = 0.8944271909 for the second branch of the solution. The linear
operator can be defined as
L[v(x, δ, q)] =
∂ 2 v(x, δ, q)
.
∂x2
(39)
It is straightforward to choose
y0 (x) = δ
(40)
as the initial guess. The first four order problems are as follows:
First-order problem given by Eq. (8):
4C1
,
25
y1 (0) = 0, y1′ (0) = 0.
y1′′ (x, δ) = −
(41)
Second-order problem given by Eq. (9):
y2′′ (x, δ) = y1′′ (x, δ) + δ 3 C1 y1′′ (x, δ) −
y2 (0) = 0,
y2′ (0) = 0.
9
4C2
,
25
(42)
Third-order problem obtained from Eq. (10) for k = 3:
y3′′ (x, δ) = δ 3 C2 y1′′ (x, δ) + 3δ 2 C1 y1 (x, δ)y1′′ (x, δ)
4C3
+y2′′ (x, δ) + δ 3 C1 y2′′ (x, δ) −
,
25
y3 (0) = 0, y3′ (0) = 0.
(43)
Fourth-order problem obtained from Eq. (10) for k = 4
y4′′ (x, δ) = δ 3 C3 y1′′ (x, δ) + 3δ 2 C2 y1 (x, δ)y1′′ (x, δ)
+3δC1 (y1 (x, δ))2 y1′′ (x, δ)
+3δ 2 C1 y2 (x, δ)y1′′ (x, δ)
+δ 3 C2 y2′′ (x, δ) + 3δ 2 C1 y1 (x, δ)y2′′ (x, δ)
4C4
+y3′′ (x, δ) + δ 3 C1 y3′′ (x, δ) −
,
25
y4 (0) = 0, y4′ (0) = 0.
(44)
The fourth-order POHAM approximate analytical solution obtained by substituting the
solutions of Eqs. (41), (42), (43), and (44) into Eq. (13) for m = 4
y˜(x, δ) = y0 (x, δ) + y1 (x, δ) + y2 (x, δ) + y3 (x, δ) + y4 (x, δ).
(45)
By using Eq. (45) into Eq. (14), yields the residual error and the functional J, respectively:
4
R˜
y (x, δ) = y˜′′ (x, δ)˜
y 3 (x, δ) − ,
25
∫
b
J(C1 , C2 , C3 , C4 ) =
(46)
R2 (x, δ, C1 , C2 , C3 , C4 )dx,
(47)
a
with the additional force condition
y(1) = 1.
(48)
The optimal values of the convergent control parameters Ci ’s obtained corresponding to
the values of the prescribed parameter δ1 = 0.44721 and δ2 = 0.89442 are as shown in
Table 2. Obviously, multiple (dual) solutions exist and are plotted in Fig. 2. It wroth
to mention her that Fig. 2 indicating existence of two solutions, θ(0) = 0.44721 for the
first branch of the solution and θ(0) = 0.89442 for the second branch of the solution.
We notice that POHAM does not yield good agreement with the exact solutions for
both cases shown in Fig. 2. To improve the accuracy we can use a new initial guess.
We use the standard homotopy analysis method (HAM) for two terms and employ the
Least Squares method to define the optimal values of ¯h =
1
.
δ3
This yields the following
new initial guess
y0 (x, δ) = δ +
2x2
2x4
−
.
25δ 3 625δ 7
10
(49)
Table 2: The values of Ci ’s corresponding to the prescribed parameter δ for Example 2.
1.0
δ1 = 0.44721
δ2 = 0.89442
C1
−1.89861
−0.794786
C2
−11.7257
−0.486958
C3
−84.2776
14.4166
C4
133.886
−12.3558
Exact solution
POHAM solution
0.9
∆ = 0.89442
yHxL
0.8
0.7
∆ = 0.44721
0.6
0.5
0.0
0.2
0.4
0.6
0.8
1.0
x
Figure 2: Comparison between the fourth-order POHAM approximate solutions and the
exact multiple (dual) solutions (38) for Example 2.
11
Table 3: The values of Ci ’s corresponding to the new initial guess and the prescribed
parameter δ for Example 2.
1.0
δ1 = 0.44721
δ2 = 0.89442
C1
−2.53856
−0.828156
C2
2.24311
−0.402523
C3
−1.8665
0.157102
Exact solution
POHAM solution
0.9
∆ = 0.89442
yHxL
0.8
∆ = 0.44721
0.7
0.6
0.5
0.0
0.2
0.4
0.6
0.8
1.0
x
Figure 3: Comparison between the third-order POHAM approximate solutions and the
exact multiple (38) solutions for Example 2.
With this new initial guess and taking only three terms in POHAM series solution we
obtaine δ1 = 0.44721 and δ2 = 0.89442 as shown in Table 3. Fiq. 3 demonstrate the
good accuracy of the POHAM multiple solutions. In this regards, it is very important
to notify that this results which is obtained by using three terms POHAM approximate
solution compared with 40th terms PHAM approximate solution [7]. This implies that
this method could be a useful and effective in solving nonlinear differential equations and
it can be converge to the exact solution in few terms.
4
conclusions
In this work, a new algorithm called the predictor optimal homotopy asymptotic method
(POHAM) is employed to find approximate solutions of nonlinear differential equations.
POHAM has been shown to be a reliable method for obtaining multiple (dual) solutions
of nonlinear differential equations which arise in fluid mechanics. General framework
for the multiple solutions is given without any need to perturbation methods, special
discretization or transformation. The validity and applicability of this procedure is in12
dependent whether there exists a small parameter in the governing equations or not.
5
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this
paper.
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15
List of Tables
1
Values of Ci ’s corresponding to the prescribed parameter δ for Example 1.
2
The values of Ci ’s corresponding to the prescribed parameter δ for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
7
11
The values of Ci ’s corresponding to the new initial guess and the prescribed
parameter δ for Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . .
12
List of Figures
1
Multiple solutions of Eqs. (20) and (21) for (a) E = 20 and (b) E = −20.
2
Comparison between the fourth-order POHAM approximate solutions and
the exact multiple (dual) solutions (38) for Example 2. . . . . . . . . . . .
3
8
11
Comparison between the third-order POHAM approximate solutions and
the exact multiple (38) solutions for Example 2. . . . . . . . . . . . . . . .
16
12