Title: A new hybrid collocation method for solving nonlinear two-point boundary value problems

Authors: Razieh Delpasand; Mohammad Mehdi Hosseini; Farid Mohammad Maalek Ghaini

Addresses: Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, 76169-14111, Iran; Faculty of Mathematics, Department of Applied Mathematics, Yazd University, Yazd, 89195-74, Iran ' Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, 76169-14111, Iran; Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, 76169-14111, Iran ' Faculty of Mathematics, Department of Applied Mathematics, Yazd University, Yazd, 89195-74, Iran

Abstract: In this paper, numerical solution of boundary value problems (BVPs) of nonlinear ordinary differential equations (ODEs) by the collocation method is considered. Of course, to avoid solving systems of nonlinear algebraic equations resulting from the method, residual function of the boundary value problem is considered and an unconstrained optimisation model is suggested. Particle swarm optimisation (PSO) algorithm is used for solving the unconstrained optimisation problem. In addition, convergence properties of the Chebyshev expansion are studied. The scheme is tested on some interesting examples and the obtained results demonstrate reliability and efficiency of the proposed hybrid method.

Keywords: nonlinear boundary value problems; ordinary differential equations; collocation method; Chebyshev polynomials; particle swarm optimisation; convergence analysis.

DOI: 10.1504/IJDSDE.2022.122527

International Journal of Dynamical Systems and Differential Equations, 2022 Vol.12 No.1, pp.106 - 120

Received: 22 Mar 2019
Accepted: 04 Oct 2019
Published online: 03 May 2022 *