A Particle Swarm Optimization-Based Method for Numerically Solving Ordinary Differential Equations

The Euler method is a typical one for numerically solving initial value problems of ordinary differential equations. Particle swarm optimization (PSO) is an efficient algorithm for obtaining the optimal solution of a nonlinear optimization problem. In this study, a PSO-based Euler-type method is proposed to solve the initial value problem of ordinary differential equations. In the typical Euler method, the equidistant grid points are always used to obtain the approximate solution. The existing shortcoming is that when the iteration number is increasing, the approximate solution could be greatly away from the exact one. Here, it is considered that the distribution of the grid nodes could affect the approximate solution of differential equations on the discrete points. The adopted grid points are assumed to be free and nonequidistant. An optimization problem is constructed and solved by particle swarm optimization (PSO) to determine the distribution of grid points. Through numerical computations, some comparisons are offered to reveal that the proposed method has great advantages and can overcome the existing shortcoming of the typical Euler formulae.