Interval Techniques for the Numerical Solution of Ordinary Differential Equations.

常微分方程是基本的建模工具 在许多科学和工程应用领域。 模型 通常涉及无法求解的非线性常微分方程组 分析。 这些模型的解释通常 需要计算近似的数值解。 电流 程序计算出解的估计值， 为错误。 然而，目前的守则只提供估计数， 并不能保证这些估计是正确的。 共同研究者将评价现有的间隔技术， 数值求解常微分方程，扩展技术，并编写一套 自我验证程序，计算区间包含的 溶液 通过呼吁适当的 数学定理 对于常微分方程的数值解， 程序的逻辑验证了适当的假设 存在唯一性定理 则程序可以断言 解存在并且包含在计算的区间中。 实际 计算使用区间算术来捕获舍入， 截断误差 为这个项目编写的一套程序将填补一个角色， 区间技术类似于点方法的作用， 70年代初写的几个程序。 它将作为一个 科学家和工程师的基本工具，也是一个测试平台 为进一步研究和改进常微分方程的区间方法。

Ordinary differential equations (ODE's) are fundamental modeling tools in many scientific and engineering application areas. The models typically involve systems of nonlinear ODE's which cannot be solved analytically. This, the interpretation of these models usually requires the computation of approximate numerical solutions. Current programs compute an estimate for the solution and (perhaps) an estimate for the error. However, the current codes only supply estimates; there are no guarantees that the estimates are correct. The co-investigators will evaluate existing interval techniques for numerically solving ODE's, extend the techniques, and write a suite of self-validating programs that compute interval inclusions of the solution. An inclusion is validated by appealing to appropriate mathematical theorems. In the case of numerical solutions of ODE's, the logic of the program verifies the hypothesis of appropriate existence and uniqueness theorems. Then the program can assert that a solution exists and is contained in the computed interval. The actual computations use interval arithmetic to capture round-off and truncation errors. The suite of programs written for this project will fill a role for interval techniques similar to the role filled for point methods by several programs written in the early 1970's. It will serve both as a fundamental tool for scientists and engineers and also as a test-bed for further study and refinement of interval methods for ODE's.