毕业论文---微分方程数值解算法分析与Matlab实现.doc

微分方程数值解算法分析与Matlab实现  摘 要 随着社会的发展，微分方程在经济、生物科学、化工等许多科技领域中得到了越来越广泛的应用，而所建立的微分方程模型中，只有极少数的微分方程可以得到解析解，为了了解微分方程的解的性质，不得不进行微分方程数值解的探讨与研究，所以对微分方程数值解的研究有重要的应用价值. 本文首先讨论了常微分方程数值解的几种常见的算法(如欧拉法、改进欧拉法、龙格库塔法等)，分析了算法的具体设计思想，并给出了相应步骤、Matlab程序，及其对算法的误差做了相应的分析，通过实例验证了算法的可行性与有效性.其次对于部分算法的收敛性与稳定性从理论上给出了分析，证明上述算法是收敛的与稳定的.最后针对目前比较前沿的脉冲微分方程与时滞微分方程的数值算法进行了设计，运用经典的四阶龙格库塔方法给出了脉冲微分方程和时滞微分方程以及脉冲时滞微分方程数值解法的计算步骤与相应的程序实现，最后给出了测试实例，证明了该算法是可行的与高效的. 关键词：微分方程；数值解；Matlab实现；仿真实例 Abstract With the development of society, differential equations have been used widely in the economic, biological sciences, chemical and many other fields of technology .But in the differential equation model which was established, only very few analytical solutions of differential equations can be obtained, In order to understand the nature of the solution of differential equations, we have to explore and research differential equations ,so the research on the numerical solution of differential equations have important applications. Firstly，the numerical solution of ODEs of several common algorithms (such as Euler, improved Euler, Runge-Kutta, etc) is discussed, the theoretical algorithms are analyzed in detail, the relevant steps, Matlab programs, and the corresponding analysis of the errors of the theoretical algorithms are given. The feasibility and effectiveness of the algorithms are proved by solving examples. Secondly, the convergence and stability analysis of parts of algorithms are given in theory, the above algorithms are convergence and stable. Finally, the numerical algorithms of the relatively cutting-edge Impulsive differential equations and delay differential equations are designed, the calculation steps, the corresponding program implementation of the delay differential equations ,the impulsive differential equations and the numerical solution of impulsive and delay differential equations are given by using classical Runge-Kutta methods, it shows that the algorithms are feasible and efficient by giving s