Solving Definite Problem Of Ordinary Differential Equation With An Improved Particle Swarm Optimization

Many mathematical models in the fields of natural sciences,engineering technology and economic management are usually characterized by the solution of ordinary differential equations.It is very important to solve them effectively.Due to the limitations of theoretical methods,many equations have not analytical solutions.Most traditional numerical methods generally have the disadvantages with regard of computational complexity and low accuracy of the solution.Therefore,from the practical point of view,it is not needed to find the analytical formula specifically.People need to find an approximation corresponding to the exact solution.In recent years,with the development of computers,the method of solving ordinary differential equations by intelligent algorithms has become a new direction.Particle swarm optimization is a kind of swarm intelligence algorithm.Based on the research and analysis of particle swarm optimization,in this paper an improved particle swarm optimization algorithm is proposed.And then the improved particle swarm optimization algorithm is applied to abtain the approximate solution of ordinary differential equations.In this paper a new improved particles swarm optimization algorithm is proposed by combining the teaching-learning-based optimization algorithm with the particle swarm optimization algorithm.First of all,the function optimization problem is used to verify the effectiveness of the improved algorithm.The results of numerical experiments show that the improved particle swarm optimization algorithm has better global search ability and higher accuracy compared with the standard particle swarm optimization algorithm.Then,we use the Fourier series to construct the approximate solution of the differential equation,and the differential equation can be transformed into the constrained optimization problem.The standard particle swarm optimization algorithm and the improved particle swarm optimization algorithm are used to calculate the optimization problem respectively,and finally the approximate solutions of the equation are obtained.Based on the above ideas,the error analysis of the results is carried out and the feasibility of the method is illustrated by solving some specific examples.Therefore,in this paper,we not only broaden the application range of particle swarm optimization algorithm,but also provide a new idea for solving the approximate solution of the ordinary differential equation to improve the solution accuary of definite problem of ordinary differential equation.