Abstract
Differential equations play a noticeable role in engineering, physics, economics, and other disciplines. In this paper, a general approach is suggested to solve a wide variety of linear and nonlinear ordinary differential equations (ODEs) that are independent of their forms, orders, and given conditions. With the aid of certain fundamental concepts of mathematics, Fourier series expansion and metaheuristic methods, ODEs can be represented as an optimization problem. The target is to minimize the weighted residual function (cost function) of the ODEs. To this end, two different approaches, unit weight function and least square weight function, are examined in order to determine the appropriate method. The boundary and initial values of ODEs are considered as constraints for the optimization model. Generational distance metric is used for evaluation and assessment of the approximate solutions versus the exact solutions. Six ODEs and four mechanical problems are approximately solved and compared with their exact solutions. The optimization task is carried out using different optimizers including the particle swarm optimization, the cuckoo search, and the water cycle algorithm. The optimization results obtained show that metaheuristic algorithms can be successfully applied for approximate solving of different types of ODEs. The suggested least square weight function is slightly superior over the unit weight function in terms of accuracy and statistical results for approximate solving of ODEs.
| Original language | English |
|---|---|
| Pages (from-to) | 117-132 |
| Number of pages | 16 |
| Journal | Engineering Applications of Artificial Intelligence |
| Volume | 40 |
| DOIs | |
| Publication status | Published - 2015 Apr 1 |
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Keywords
- Approximate solution
- Fourier series
- Linear/nonlinear differential equation
- Metaheuristics
- Weighted residual function
ASJC Scopus subject areas
- Artificial Intelligence
- Control and Systems Engineering
- Electrical and Electronic Engineering
Cite this
Approximate solving of nonlinear ordinary differential equations using least square weight function and metaheuristic algorithms. / Sadollah, Ali; Eskandar, Hadi; Yoo, Do Guen; Kim, Joong Hoon.
In: Engineering Applications of Artificial Intelligence, Vol. 40, 01.04.2015, p. 117-132.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Approximate solving of nonlinear ordinary differential equations using least square weight function and metaheuristic algorithms
AU - Sadollah, Ali
AU - Eskandar, Hadi
AU - Yoo, Do Guen
AU - Kim, Joong Hoon
PY - 2015/4/1
Y1 - 2015/4/1
N2 - Differential equations play a noticeable role in engineering, physics, economics, and other disciplines. In this paper, a general approach is suggested to solve a wide variety of linear and nonlinear ordinary differential equations (ODEs) that are independent of their forms, orders, and given conditions. With the aid of certain fundamental concepts of mathematics, Fourier series expansion and metaheuristic methods, ODEs can be represented as an optimization problem. The target is to minimize the weighted residual function (cost function) of the ODEs. To this end, two different approaches, unit weight function and least square weight function, are examined in order to determine the appropriate method. The boundary and initial values of ODEs are considered as constraints for the optimization model. Generational distance metric is used for evaluation and assessment of the approximate solutions versus the exact solutions. Six ODEs and four mechanical problems are approximately solved and compared with their exact solutions. The optimization task is carried out using different optimizers including the particle swarm optimization, the cuckoo search, and the water cycle algorithm. The optimization results obtained show that metaheuristic algorithms can be successfully applied for approximate solving of different types of ODEs. The suggested least square weight function is slightly superior over the unit weight function in terms of accuracy and statistical results for approximate solving of ODEs.
AB - Differential equations play a noticeable role in engineering, physics, economics, and other disciplines. In this paper, a general approach is suggested to solve a wide variety of linear and nonlinear ordinary differential equations (ODEs) that are independent of their forms, orders, and given conditions. With the aid of certain fundamental concepts of mathematics, Fourier series expansion and metaheuristic methods, ODEs can be represented as an optimization problem. The target is to minimize the weighted residual function (cost function) of the ODEs. To this end, two different approaches, unit weight function and least square weight function, are examined in order to determine the appropriate method. The boundary and initial values of ODEs are considered as constraints for the optimization model. Generational distance metric is used for evaluation and assessment of the approximate solutions versus the exact solutions. Six ODEs and four mechanical problems are approximately solved and compared with their exact solutions. The optimization task is carried out using different optimizers including the particle swarm optimization, the cuckoo search, and the water cycle algorithm. The optimization results obtained show that metaheuristic algorithms can be successfully applied for approximate solving of different types of ODEs. The suggested least square weight function is slightly superior over the unit weight function in terms of accuracy and statistical results for approximate solving of ODEs.
KW - Approximate solution
KW - Fourier series
KW - Linear/nonlinear differential equation
KW - Metaheuristics
KW - Weighted residual function
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UR - http://www.scopus.com/inward/citedby.url?scp=84923335169&partnerID=8YFLogxK
U2 - 10.1016/j.engappai.2015.01.014
DO - 10.1016/j.engappai.2015.01.014
M3 - Article
AN - SCOPUS:84923335169
VL - 40
SP - 117
EP - 132
JO - Engineering Applications of Artificial Intelligence
JF - Engineering Applications of Artificial Intelligence
SN - 0952-1976
ER -