Solving boundary-value problems using hp-version finite elements in time

An hp-version nite element method for one-dimensional boundary value problems is presented. The method is based on a similar approach developed by the authors for solution of optimal control problems. The primary applications for the methodology include two-point- and multi-point-boundary-value problems, for example, in the time domain. Results presented for a 7-state=3-phase missile problem show that the method is very e cient for time-marching applications. Furthermore, it easily solves time-domain problems with discontinuities in the system equations and=or in the states, where the time at which these jumps (i.e. `eventsʹ) take place is determined by equations that govern the states. An example involving friction with intermittent sticking is presented to illustrate the power of the method.