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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