终值问题和任意值问题的定义及数值方法

Shixio Wang, Jianhua He, Chen Wang, Xitong Li
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引用次数: 5

摘要

许多工程问题都可以用终值问题来描述,终值问题是初值问题的逆问题。因此,本文研究了ODE问题领域中的终值问题,分析了初值问题与终值问题的区别和联系。提出了更一般的端点值问题的新概念,它既可以描述初始问题,也可以描述最终问题。进一步将这一概念推广到区间内值问题和任意值问题,并指出端点值问题和区间内值问题都是任意值问题的特殊形式。特别地,对离散问题证明了一阶常微分方程终值问题和区间内值问题解的存在唯一性。导出了这些问题的数值计算公式,并给出了每个算法的收敛性和稳定性条件。进一步研究了多变量和高阶终值问题,并讨论了定时滞的条件。最后,通过数值实验验证了所考虑方法的有效性。
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The Definition and Numerical Method of Final Value Problem and Arbitrary Value Problem
Many Engineering Problems could be mathematically described by Final Value Problem, which is the inverse problem of Initial Value Problem. Accordingly, the paper studies the final value problem in the field of ODE problems and analyses the differences and relations between initial and final value problems. The more general new concept of the endpoints-value problem which could describe both initial and final problems is proposed. Further, we extend the concept into inner-interval value problem and arbitrary value problem and point out that both endpoints-value problem and inner-interval value problem are special forms of arbitrary value problem. Particularly, the existence and uniqueness of the solutions of final value problem and inner-interval value problem of first order ordinary differential equation are proved for discrete problems. The numerical calculation formulas of the problems are derived, and for each algorithm, we propose the convergence and stability conditions of them. Furthermore, multivariate and high-order final value problems are further studied, and the condition of fixed delay is also discussed in this paper. At last, the effectiveness of the considered methods is validated by numerical experiment.
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