Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm

• Published in: Information sciences 2014-09, Vol.279, p.396-415
• Main Authors: Arqub, Omar Abu, Abo-Hammour, Zaer
• Format: Article
• Language: English
• Subjects: 34B15; 34K28; 65K05; Continuous genetic algorithm; Finite difference approximation; System of boundary value problem
• ISSN: 0020-0255; 1872-6291
• DOI: 10.1016/j.ins.2014.03.128

In this paper, continuous genetic algorithm is introduced as an efficient solver for systems of second-order boundary value problems where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values of the unknown variables. The solution methodology is based on representing each derivative in the system of differential equations by its finite difference approximation. After that, the overall residue for all nodes in the given system of differential equations is formulated. The solution to the system of differential equations is then converted into the problem of minimizing the overall residue or maximizing the fitness function based on the nodal values generated from the genetic operators. Three numerical test problems including linear and nonlinear systems were analyzed to illustrate the procedure and confirm the performance of the proposed method. In addition to that, a convergence and sensitivity analysis to genetic operators and control parameters of the algorithm has been carried out. The numerical results show that the proposed algorithm is a robust and accurate procedure for solving systems of second-order boundary value problems. Furthermore, the obtained accuracy for the solutions using CGA is much better than the results obtained using some modern methods.