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Solving Dirichlet boundary problems for ODEs via swarm intelligence

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Abstract

In this paper, we examine the effects of swarm intelligence techniques on the obtaining of numerical solutions of ordinary differential equations (ODEs) with Dirichlet boundary conditions (DBCs) via feed-forward neural networks. Population-based global optimization methods such as artificial bee colony (ABC), ant colony optimization (ACO), gravitational search algorithm (GSA) and particle swarm optimization (PSO) are utilized in order to solve ODEs with DBCs numerically. Furthermore, we hybridize ACO, ABC and GSA with PSO to improve the numerical solution of optimization algorithms. Unlike the approaches in the literature, we use the personal best and global best solutions of PSO for the updating position of employed and onlooker bees in ABC, in the hybrid method that combines PSO and ABC. We also prefer the adaptive number of swarm scheme on the hybrid system of PSO and ACO. In the experiments, we give some different Dirichlet Boundary Problems to indicate the efficiency of optimization methods. Also the results are compared with the well-known traditional methods as shooting method, finite difference method and Lobatto IIIa.

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  1. Department of Mathematics, Aydın Adnan Menderes University, Aydın, Turkey

    Korhan Günel & İclal Gör

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  1. Korhan Günel
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The research has been supported by the Council of Higher Education in Turkey (YÖK), Coordination of Academic Member Training Program (ÖYP) in Aydın Adnan Menderes University, under Grant No. ADÜ-ÖYP-14011.

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Günel, K., Gör, İ. Solving Dirichlet boundary problems for ODEs via swarm intelligence. Math Sci 16, 325–341 (2022). https://doi.org/10.1007/s40096-021-00424-2

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