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.
