Volterra比例延迟积分微分方程Legendre配置法的误差估计

发布时间：2018-07-24 08:42

【摘要】：延迟积分微分方程普遍应用于如生物数学、人口动力学、数控计算等自然科学和工程技术领域,而Volterra延迟积分微分方程是一类特殊的延迟积分微分方程,常常被用于刻画某些生物问题和物理现象。本文主要研究Volterra比例延迟积分微分方程,分别用单步配置法和多域配置法建立方程的数值格式,并分别研究两种方法的误差精度,最后用几个算例的数值计算结果来验证论文中的结论。最小延迟量足够小的情况下,使用单步配置法。首先根据主要不连续点对方程积分区间进行剖分;其次用变换的Legendre多项式作为基函数构建单步配置法的数值格式,并推导出此方法的误差精度;最后用两个算例的数值计算结果来验证理论分析中的结果。当最小延迟量不足够小时,使用多域配置法。首先对单步配置法中的积分区间剖分结果进行再划分,保证主要不连续点在新剖分点集里面;其次构建多域配置法的数值格式,并推导此方法的误差精度;最后用两个算例的数值计算结果来验证理论分析中的结果。理论分析和数值试验结果表明:单步配置法能获得指数级收敛速率,而且在保持变换的Legendre多项式的最高次数不变的情况下,最小延迟量越小,误差精度越高;多域配置法也能获得指数级收敛速率,但相比单步配置法,针对同一类型的Volterra延迟积分微分方程,在变换的Legendre多项式最高次数相同的情况下,其获得误差精度较低。而且多域配置法方法的误差精度不是随着地无限增加而无限升高,而是慢慢趋于稳定值。
[Abstract]:Delay integro-differential equations are widely used in the fields of natural science and engineering, such as biological mathematics, population dynamics, numerical control calculation, etc. Volterra delay integrodifferential equations are a special kind of delayed integro-differential equations. It is often used to depict certain biological problems and physical phenomena. In this paper, the Volterra proportional delay integrodifferential equations are studied. The numerical schemes of the equations are established by using the one-step collocation method and the multi-domain collocation method, and the error accuracy of the two methods is studied respectively. Finally, the numerical results of several numerical examples are used to verify the conclusion of the paper. If the minimum delay is small enough, one-step configuration method is used. Firstly, the integral interval of the equation is divided according to the main discontinuous points, then the transformed Legendre polynomial is used as the basis function to construct the numerical scheme of the one-step collocation method, and the error accuracy of the method is deduced. Finally, the numerical results of two examples are used to verify the results of the theoretical analysis. When the minimum delay is not sufficient, a multi-domain configuration method is used. Firstly, the results of integral interval subdivision in one-step collocation method are redivided to ensure that the main discontinuous points are in the new set of points, secondly, the numerical scheme of multi-domain collocation method is constructed, and the error accuracy of the method is deduced. Finally, the numerical results of two examples are used to verify the results of the theoretical analysis. The results of theoretical analysis and numerical experiments show that the single-step collocation method can obtain exponential convergence rate, and the minimum delay is smaller and the error accuracy is higher under the condition of keeping the highest degree of the transformed Legendre polynomial unchanged. The multi-domain collocation method can also obtain exponential convergence rate, but compared with the single-step collocation method, for the same type of Volterra delay integro-differential equations, the error accuracy is lower when the highest degree of the transformed Legendre polynomials is the same. Moreover, the error accuracy of the multi-domain collocation method does not increase infinitely with the increase of ground infinity, but tends to be stable slowly.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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