Approximate solving of nonlinear ordinary differential equations using least square weight function and metaheuristic algorithms

Ali Sadollah, Hadi Eskandar, Do Guen Yoo, Joong Hoon Kim

Research output: Contribution to journalArticlepeer-review

40 Citations (Scopus)


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 languageEnglish
Pages (from-to)117-132
Number of pages16
JournalEngineering Applications of Artificial Intelligence
Publication statusPublished - 2015 Apr 1

Bibliographical note

Funding Information:
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government ( MSIP ) (NRF- 2013R1A2A1A01013886 ).


  • Approximate solution
  • Fourier series
  • Linear/nonlinear differential equation
  • Metaheuristics
  • Weighted residual function

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Artificial Intelligence
  • Electrical and Electronic Engineering


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