A particle swarm optimization algorithm and its application in
hydrodynamic equations
Jianhui Zhou1, Shuzhong Zhao1, Lixi Yue1, Yannan lu1, and Xinyi Si2 1College of Qian’an, Hebei United University, Tangshan, China
2College of Water Conservancy and Hydropower Engineering, Hehai University, Nanjing, China
Keywords: Particle swarm optimization algorithm, Direct search method, Ordinary differential equations, Parameter estimation
Abstract. In fluid mechanics, how to solve multiple solutions in ordinary differential equations is
always a concerned and difficult problem. A particle swarm optimization algorithm combining with
the direct search method (DSPO) is proposed for solving the parameter estimation problems of the
multiple solutions in fluid mechanics. This algorithm has improved greatly in precision and the
success rate. In this paper, multiple solutions can be found through changing accuracy and search
coverage and multi-iterations of computer. Parameter estimation problems of the multiple solutions
of ordinary differential equations are calculated, and the result has great accuracy and this method is
practical.
1. Introduction
In fluid mechanics, the multiple solutions of boundary value problems are important topics and have
received considerable attentions. Many researchers had put forward some methods to solve such
problems[1]. In general, a shooting method based on the fourth order Runge-Kutta scheme is effective
numerically, which have been used to solve a lot of problems. In order to solve such problems, the
boundary value problems are transformed into initial value problems by introducing new unknown
parameters, which are decided by the boundary conditions. However, it is difficult to find all the
unknown parameters which satisfy the boundary conditions, especially for the problems that the
equation has multiple solutions. Later, the Homotopy Analysis Method is proposed by Liao SJ [2,3],
which also is an efficient analytical method and has been used to solve many problems with multiple
solutions[4,5,6]. However, the HAM suffers from a number of restrictive measures, such as the
requirement that the solution sought ought to conform to the so-called rule of solution expression and
the rule of coefficient ergodicity. By great search ability of NRNA-GA, many problems could be
solved directly.
PSO was first introduced by Kennedy and Eberhart. The algorithm is driven by the social behavior
of a bird flock and can be viewed as a population-based stochastic optimization algorithm. PSO has
been applied to many fields. Several modifications in the PSO algorithm had been done by various
researchers[7,8,9,10,11,12]. In this paper, multiple solutions in ordinary differential equation are very
similar, so we need to improve accuracy of the solution. In order to find multiple solutions in ordinary
differential equation, we combine a particle swarm optimization algorithm with the direct search
method. At last, comparing with the previous result we find that the calculated results of DSPO is
feasible.
2. Procedure of DPSO with Runge-Kutta method
Let x and v denote a particle’s position and velocity in a search space. The i th particle can be
represented as ( )1 2, ~i i i iDx x x x= in the D -dimensional search space. The best previous position of
the i th particle is recorded and represented as ( )1 2, ~i i i iDp p p p= .The index of the best particle in
the group, i.e., the particle with the smallest function value, is represented by, while the velocity of the
i th particle is represented as ( )1 2, ~g g g gDP p p p= . The velocity and position of each particle can be
manipulated according to the following equations:
Advanced Materials Research Vol. 510 (2012) pp 472-477Online available since 2012/Apr/25 at www.scientific.net© (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.510.472
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( ) ( )( )1 1 2 2id id id id gd idv w v c r p x c r P x= + − + − (1)
id id idx x v= + (2)
where w , 1c and 2c are positive constants known as acceleration coefficients; and 1r and 2r are
two random numbers within the range [0, 1].
The procedure of DPSO and Runge-Kutta method can be summarized as follows.
Step 1. Initial positions and velocities for all agents are generated. The current position of each
particle is stored as the objective-function value iP calculated through Runge-Kutta method. A iP
with the best value is designated as gP , and this value is stored.
Step 2. The next position for each particle is generated using Eqs. (1) and (2).
Step 3. The objective-function value is calculated through Runge-Kutta method for the new
positions of each particle. If an agent achieves a better position, its iP value is replaced by the current
value.
Step 4. A gP value is selected from the new set of iP values. If the new gP value is better than the
previous gP value, the previous gP value is replaced by the new gP value, carry out the direct search
method for gP , the best searching result will be saved.
Step 5. Repeat step 2 to step 4 until the termination criteria are met, and the solution is found.
Step 6. If the absolute value of this solution and previous solution is more than ξ (ξ denotes the smallest distance of the two solutions, in this paper, we suppose that it is 0.0001), then this solution
would be saved, the number of solution found, iteration would be ended or go to step 1 with changing
the search coverage and precision of s and t .
3. Parameter estimation of the multiple solutions of ordinary differential equations
Example 1 A parameter estimation problem can be written in the following standard form.
In classical works, the power-law kinematic viscosity was only applied in momentum equations of
non-Newtonian fluids while the thermal conductivity k is still treated as a constant. However,
practical situations require variable physical properties. To describe the heat transfer properly, the
thermal conductivity for non-Newtonian fluids was assumed power-law dependence on the velocity
gradient by Pop et al[13,14] and Liancun Zheng et al [15,16]. Botong Li et al have already solved this
problem[17].
In order to obtain numerical solutions, we directly transfer the problem into a system of first-order
equations using variables ,F U and V , respectively:
F U′ = (3)
U V′ = (4)
21 nV F Vn
−′ = − ⋅ (5)
W Y′ = (6)
Advanced Materials Research Vol. 510 473
(1 ) 21 n
pr
nY N F Y V ZH V
n
−− ′ = − ⋅ ⋅ − ⋅
(7)
The corresponding boundary conditions are:
(0) 0, (0) 0, (0) 0F U W= = = (8)
( ) 1, ( ) 1U W+∞ = +∞ = (9)
We introduce the parameter s as:
(0) , (0)V s Y t= = (10)
Then, the problem is to find the parameter s and the Eqs.(3)-(8), (10)to satisfy the boundary conditions (9), which equals to solve the minimum of the Eq.(11)
, ,min ( , ) ( ) ( ) ( ) ( )s t s tf s t U U W W= +∞ − +∞ + +∞ − +∞ (11)
In this paper, n , prN and ZH are constant, , , , ,F U V W Y are functions of x , the s and t could be
calculated by IRNA-GA combining Runge-Kutta method, which are shown in Table 1,Fig.1, Fig.2
and Fig.3.
Table 1. The vaule of n , prN and ZH .
s t
0.5, 2, 1prn N ZH= = = 0.43576714732586 0.72964065003433
0.8, 2, 1prn N ZH= = = 0.44843213550011 0.76557564660105
1, 2, 1prn N ZH= = = 0.46961165789273 0.80732433051041
0.8, 2, 0.1prn N ZH= = = 0.44848961686098 0.5840354767184
0.8, 2, 0.5prn N ZH= = = 0.44845146985194 0.6647164008297
0.8, 2, 1prn N ZH= = = 0.44851370322896 0.76567173289385
0.8, 2, 2prn N ZH= = = 0.44851370322896 0.96746627601782
0.8, 2, 3prn N ZH= = = 0.44852368963149 1.16932936598764
0.8, 2, 4prn N ZH= = = 0.44852368963149 1.37111467154955
0.5, 5, 1prn N ZH= = = 0.43576714732586 0.87644769970245
0.5, 10, 1prn N ZH= = = 0.43573662928206 1.03163195239185
0.5, 15, 1prn N ZH= = = 0.4357366292820 1.14438086518654
0.5, 20, 1prn N ZH= = = 0.43576714732586 1.23588921950103
474 Machinery, Materials Science and Engineering Applications, MMSE2012
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
1.4
x
W
n=0.5
n=0.8
n=1
Fig. 1. The W Function of x when2, 1prN ZH= =
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
1.4
x
W
ZH=1
ZH=2
ZH=3
ZH=4
Fig. 2. The W function of x when 0.8, 2prn N= =
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
1.4
x
W
Npr=5
Npr=10
Npr=15
Npr=20
Fig. 3. The W function of x when 0.5, 1n ZH= =
Example 2 A parameter estimation problem can be written in the following standard form
( )( )2
2.5
1 21 0
11 1 s
f
mF FF F
mρϕ ϕ ϕ
ρ
′′′ ′′ ′+ + − = + − − +
(12)
Advanced Materials Research Vol. 510 475
, ,(0) 0 '(0) ( ) 1F F Fλ ′= = +∞ = (13)
In order to obtain numerical solutions, we transfer the problem (1-1) and (1-2) into a system of
first-order equations by denoting the , , ,F F F F′ ′′ ′′′ using variables ,F U and V , respectively
F U′ = (14)
U V′ = (15)
( )( )2
2.5
1 21 0
11 1 s
f
mV FV U
mρϕ ϕ ϕ
ρ
′ + + − = + − − +
(0, )x∈ +∞ (16)
The corresponding boundary conditions are:
(0) 0, (0)F U λ= = (17)
( ) 1U +∞ = (18)
We introduce the parameter s as:
(0)V s= (19)
Then, the problem is to find the parameter s and the Eqs.(14)-(17), (19) to satisfy the boundary conditions (18), which equals to solve the minimum of the Eq. (20)
min ( ) ( ) ( )sf s U U= +∞ − +∞ (20)
In this paper, ϕ , m and λ are constant, , ,F U V are functions of x , the s could be calculated by
IRNA-GA combining Runge-Kutta method, which are shown in Table 2[18]and Fig.4.
Table 2 The values of (0)f ′′ for various values of m when 0λ = and 0ϕ = .
m Rosenhead Watanabe Yih Wang Nor
Azizah
Yacob
Present results
0 / 0.46960 0.469600 / 0.469599 0.4695999879552
1/11 / 0.65498 0.654979 / 0.654994 0.65499368837452
0.2 / 0.80213 0.802125 / 0.802126 0.80212559336846
1/3 / 0.92765 0.927653 / 0.927680 0.92768003952868
0.5 / / / / 1.038903 1.0389034801719
1 1.232588 / 1.232588 1.232588 1.232588 1.23258765117093
476 Machinery, Materials Science and Engineering Applications, MMSE2012
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
2
RF
′′(0)
first solution
second solution
Fig. 4. The values of (0)f ′′ for various values of λ when 0m = and 0ϕ = .
4. Conclusions
In this paper, the DPSO combining with Runge-Kutta method is applied to two examples and the
result is satisfying. So the method is practical and could be a new method for solving multiple
solutions of ordinary differential equations.
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Advanced Materials Research Vol. 510 477
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