Enclosing all solutions of two-point boundary value problems for ODEs

The two-point boundary value problem (TPBVP) occurs in a wide variety of problems in engineering and science, including the modeling of chemical reactions, heat transfer, and diffusion, and the solution of optimal control problems. A TPBVP may have no solution, a single solution, or multiple solutions. A new strategy is presented for reliably locating all solutions of a TPBVP. The method determines narrow enclosures of all solutions that occur within a specified search interval. Key features of the method are the use of a new solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of nonlinear dynamic systems with interval-valued parameters and initial states, and the use of a constraint propagation strategy on the Taylor models used to represent the solutions of the dynamic system. Numerical experiments demonstrate the use and computational efficiency of the method.