Interval Techniques for the Numerical Solution of Ordinary Differential Equations.

常微分方程（ODE）是许多科学和工程应用领域的基本建模工具。这些模型通常涉及不能解析求解的非线性ODE系统。因此，这些模型的解释通常需要计算近似数值解。当前的程序计算解决方案的估计值和（可能）误差的估计值。然而，目前的规范只提供估计；不能保证这些估计是正确的。共同研究人员将评估现有的用于数值求解ODE的区间技术，扩展该技术，并编写一套计算解的区间包含的自验证程序。通过诉诸适当的数学定理来验证包含。在ODE的数值解的情况下，程序的逻辑验证了适当存在唯一性定理的假设。然后程序可以断言解存在并且包含在计算的区间内。实际的计算使用间隔算术来捕获舍入和截断错误。为这个项目编写的程序套件将填补区间技术的角色，类似于1970年代早期编写的几个程序所填补的点方法的角色。它既可以作为科学家和工程师的基本工具，也可以作为进一步研究和改进ODE区间方法的试验台。

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.