基于灰狼优化算法的微分方程数值解法

Numerical solution method of differential equations based on grey wolf optimization algorithm

利用灰狼优化(GWO)算法全局优化的优点,提出了基于多项式函数逼近和GWO算法的微分方程数值解法。首先,分别采用最小二乘多项式、勒让德多项式、切比雪夫多项式和伯恩斯坦多项式基函数构造微分方程解的近似函数;再结合加权残差法的思想,令近似函数满足微分方程和定解条件且残差最小,使微分方程转换成以近似函数待定系数为变量的带约束优化问题;然后,利用GWO算法求解该优化问题,进而可得到微分方程多项式近似解。通过对线性和非线性微分方程进行数值模拟,结果表明:与其他几种多项式逼近方法相比,伯恩斯坦多项式逼近所得的数值解与精确解逼近程度最高,验证了所提算法用于求解高阶线性和非线性微分方程的初边值问题的可行性与准确性。研究结果拓宽了GWO算法的应用范围,为求解微分方程初边值问题提供了新方法。

Taking advantage of the global optimization of Grey Wolf Optimizer(GWO),a numerical method was proposed for solving differential equations based on the polynomial function approximation and GWO.Firstly,the Least squares polynomial,Legendre polynomial,Chebyshev polynomial and Bernstein polynomial basis functions were used to develop the approximate functions of the differential equation solutions.Secondly,the approximate function satisfied the differential equation and definite solution conditions by the idea of the weighted residual method,and the differential equation was transformed into the constrained optimization problem by minimizing the residual error.Based on the numerical simulation of linear and nonlinear differential equations,the results show that,the Bernstein polynomial method has the highest approximation degree than the other polynomial approximation methods.The feasibility and accuracy of the proposed algorithm were verified for solving the initial-boundary value problems of high-order linear and nonlinear differential equations.The application range of GWO was broadened and a new method for solving the initial-boundary value problem of differential equation was developed.