A New Approach to Solve Blasius Equation using Parameter Identification of Nonlinear Functions based on the Bees Algorithm (BA)
In this paper, a new approach is introduced to solve
Blasius equation using parameter identification of a nonlinear
function which is used as approximation function. Bees Algorithm
(BA) is applied in order to find the adjustable parameters of
approximation function regarding minimizing a fitness function
including these parameters (i.e. adjustable parameters). These
parameters are determined how the approximation function has to
satisfy the boundary conditions. In order to demonstrate the
presented method, the obtained results are compared with another
numerical method. Present method can be easily extended to solve a
wide range of problems.
[1] A. Malek, R.S. Beidokhti, Numerical solution for high order differential
equations using a hybrid neural networkÔÇöOptimization method.
Applied Mathematics and Computation. 2006; 183: 260-271.
[2] H. Lee, I.S. Kang, Neural algorithms for solving differential equations,
Journal of Computational Physics 1990; 91: 110-131.
[3] A.J. Meade Jr, A.A. Fernandez, The numerical solution of linear
ordinary differential equations by feedforward neural networks,
Mathematical and Computer Modelling. 1994; 19 (12): 1-25.
[4] I.E. Lagris, A. Likas, D.I. Fotiadis. Artificial neural networks for solving
ordinary and partitial differential equations. IEEE Transactions on
Neural Networks. 1998; 9 (5): 987-1000.
[5] J.A. Khan, R.M.A. Zahoor, I.M. Qureshi, Swarm intelligence for the
problem of non-linear ordinary differential equations and its application
to well known Wessinger's equation. European Journal of scientific
research. 2009; 34(4): 514-525.
[6] Z.Y. Lee, Method of bilaterally bounded to solution blasius equation
using particle swarm optimization. Applied Mathematics and
Computation 2006; 179: 779-786.
[7] Pham DT, Ghanbarzadeh A, Koc E, Otri S, Rahim S, Zaidi M. Technical
Note: Bees Algorithm. Cardiff: Cardiff University, Manufacturing
Engineering Centre; 2005.
[8] Von Frisch K. Bees: Their Vision, Chemical Senses and Language.
Revised Edition ed. NY: Ithaca, Cornell University Press; 1976.
[9] Seeley TD. The Wisdom of the Hive: The Social Physiology of Honey
Bee Colonies. Cambridge, Massachusetts: Harvard University Press;
1996.
[10] Camazine S, Deneubourg JL, Franks NR, Sneyd J, Theraulaz G,
Bonabeau E. Self-Organization in Biological Systems. Princeton:
Princeton University Press; 2003.
[11] Bonabeau E, Dorigo M, Theraulaz G. Swarm Intelligence: from Natural
to Artificial Systems. New York: Oxford University Press; 1999.
[12] Pham DT, Ghanbarzadeh A, Koc E, Otri S, Rahim S, Zaidi M. The Bees
Algorithm, A Novel Tool for Complex Optimisation Problems. in 2nd
Int Virtual Conf on Intelligent Production Machines and Systems
(IPROMS);2006, pp.454-459.
[13] Pham DT, Ghanbarzadeh A, Koc E, Otri S. Application of the Bees
Algorithm to the Training of Radial Basis Function Networks for
Control Chart Pattern Recognition. in 5th CIRP International Seminar on
Intelligent Computation in Manufacturing Engineering (CIRP ICME
'06). Ischia, Italy; 2006, pp. 711-716.
[14] M.A.Behrang, E.Assareh, M.R.Assari, A.Ghanbarzadeh, Total energy
demand estimation in Iran using Bees Algorithm. Energy Sources, Part
B: Economics, Planning, and Policy. Accepted. DOI:
10.1080/15567240903502594.
[15] Using Bees Algorithm and Artificial Neural Network to Forecast World
Carbon Dioxide Emission. Energy Sources, Part A: Recovery,
Utilization, and Environmental Effects. Accepted. DOI:
10.1080/15567036.2010.493920.
[16] E. Assareh, M.A.Behrang, M.R.Assari, A.Ghanbarzadeh. Application of
particle swarm optimization (PSO) and genetic algorithm (GA)
techniques on demand estimation of oil in Iran. Energy 35 (2010) 5223-
5229.
[17] M.A. Behrang., E. Assareh, M.R. Assari, M.R., and A. Ghanbarzadeh.
Assessment of electricity demand in Iran's industrial sector using
different intelligent optimization techniques. Applied Artificial
Intelligence 2011; 25: 292-304. doi:10.1080/08839514.2011.559572
[18] M.A. Behrang, E. Assareh, A.R. Noghrehabadi, and A. Ghanbarzadeh.
New sunshine-based models for predicting global solar radiation using
PSO (particle swarm optimization) technique. Energy 2011; 36: 3036-
3049. doi:10.1016/j.energy.2011.02.048.
[1] A. Malek, R.S. Beidokhti, Numerical solution for high order differential
equations using a hybrid neural networkÔÇöOptimization method.
Applied Mathematics and Computation. 2006; 183: 260-271.
[2] H. Lee, I.S. Kang, Neural algorithms for solving differential equations,
Journal of Computational Physics 1990; 91: 110-131.
[3] A.J. Meade Jr, A.A. Fernandez, The numerical solution of linear
ordinary differential equations by feedforward neural networks,
Mathematical and Computer Modelling. 1994; 19 (12): 1-25.
[4] I.E. Lagris, A. Likas, D.I. Fotiadis. Artificial neural networks for solving
ordinary and partitial differential equations. IEEE Transactions on
Neural Networks. 1998; 9 (5): 987-1000.
[5] J.A. Khan, R.M.A. Zahoor, I.M. Qureshi, Swarm intelligence for the
problem of non-linear ordinary differential equations and its application
to well known Wessinger's equation. European Journal of scientific
research. 2009; 34(4): 514-525.
[6] Z.Y. Lee, Method of bilaterally bounded to solution blasius equation
using particle swarm optimization. Applied Mathematics and
Computation 2006; 179: 779-786.
[7] Pham DT, Ghanbarzadeh A, Koc E, Otri S, Rahim S, Zaidi M. Technical
Note: Bees Algorithm. Cardiff: Cardiff University, Manufacturing
Engineering Centre; 2005.
[8] Von Frisch K. Bees: Their Vision, Chemical Senses and Language.
Revised Edition ed. NY: Ithaca, Cornell University Press; 1976.
[9] Seeley TD. The Wisdom of the Hive: The Social Physiology of Honey
Bee Colonies. Cambridge, Massachusetts: Harvard University Press;
1996.
[10] Camazine S, Deneubourg JL, Franks NR, Sneyd J, Theraulaz G,
Bonabeau E. Self-Organization in Biological Systems. Princeton:
Princeton University Press; 2003.
[11] Bonabeau E, Dorigo M, Theraulaz G. Swarm Intelligence: from Natural
to Artificial Systems. New York: Oxford University Press; 1999.
[12] Pham DT, Ghanbarzadeh A, Koc E, Otri S, Rahim S, Zaidi M. The Bees
Algorithm, A Novel Tool for Complex Optimisation Problems. in 2nd
Int Virtual Conf on Intelligent Production Machines and Systems
(IPROMS);2006, pp.454-459.
[13] Pham DT, Ghanbarzadeh A, Koc E, Otri S. Application of the Bees
Algorithm to the Training of Radial Basis Function Networks for
Control Chart Pattern Recognition. in 5th CIRP International Seminar on
Intelligent Computation in Manufacturing Engineering (CIRP ICME
'06). Ischia, Italy; 2006, pp. 711-716.
[14] M.A.Behrang, E.Assareh, M.R.Assari, A.Ghanbarzadeh, Total energy
demand estimation in Iran using Bees Algorithm. Energy Sources, Part
B: Economics, Planning, and Policy. Accepted. DOI:
10.1080/15567240903502594.
[15] Using Bees Algorithm and Artificial Neural Network to Forecast World
Carbon Dioxide Emission. Energy Sources, Part A: Recovery,
Utilization, and Environmental Effects. Accepted. DOI:
10.1080/15567036.2010.493920.
[16] E. Assareh, M.A.Behrang, M.R.Assari, A.Ghanbarzadeh. Application of
particle swarm optimization (PSO) and genetic algorithm (GA)
techniques on demand estimation of oil in Iran. Energy 35 (2010) 5223-
5229.
[17] M.A. Behrang., E. Assareh, M.R. Assari, M.R., and A. Ghanbarzadeh.
Assessment of electricity demand in Iran's industrial sector using
different intelligent optimization techniques. Applied Artificial
Intelligence 2011; 25: 292-304. doi:10.1080/08839514.2011.559572
[18] M.A. Behrang, E. Assareh, A.R. Noghrehabadi, and A. Ghanbarzadeh.
New sunshine-based models for predicting global solar radiation using
PSO (particle swarm optimization) technique. Energy 2011; 36: 3036-
3049. doi:10.1016/j.energy.2011.02.048.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:63127", author = "E. Assareh and M.A. Behrang and M. Ghalambaz and A.R. Noghrehabadi and A. Ghanbarzadeh", title = "A New Approach to Solve Blasius Equation using Parameter Identification of Nonlinear Functions based on the Bees Algorithm (BA)", abstract = "In this paper, a new approach is introduced to solve
Blasius equation using parameter identification of a nonlinear
function which is used as approximation function. Bees Algorithm
(BA) is applied in order to find the adjustable parameters of
approximation function regarding minimizing a fitness function
including these parameters (i.e. adjustable parameters). These
parameters are determined how the approximation function has to
satisfy the boundary conditions. In order to demonstrate the
presented method, the obtained results are compared with another
numerical method. Present method can be easily extended to solve a
wide range of problems.", keywords = "Bees Algorithm (BA); Approximate Solutions;
Blasius Differential Equation.", volume = "5", number = "1", pages = "252-3", }
