Expository Research Papers

The two papers in this issue are concerned with analysis of differential equations: the first paper discusses the solution of partial differential equations that describe heat transfer, while the second one analyzes dynamical systems whose behavior alternates between continuous and discrete modes. The paper “Application of Standard and Refined Heat Balance Integral Methods to One-Dimensional Stefan Problems,” by Sarah Mitchell and Tim Myers, deals with a heat transfer problem, on a semi-infinite region. Consider an experiment on a long metal bar occupying the positive real-axis. The experiment starts by heating the origin, thereby raising the bar above the melting temperature. The bar melts near the origin, and as the heat diffuses, the solid-liquid interface propagates slowly but surely towards infinity. The temperature is modeled by heat equations across the two regions, and by a Stefan condition at their interface. The authors heat balance integral methods to solve for the temperature. These methods reduce a partial differential equation to an ordinary differential equation. The authors investigate different boundary conditions, and different approximating functions for the temperature. Readers interested in heat balance integral methods will find this to be a valuable survey with new results that preserve the simplicity of the method. The second paper is concerned with so-called hybrid dynamical systems. A bouncing ball, for instance, is a hybrid dynamical system. The movement of the ball above ground can be described by Newton's law. However, at the very moment the ball hits the ground and bounces, an instantaneous reversal of velocity occurs along with some dissipation of energy. After the bounce, the ball moves again according to Newton's law until the next bounce, and so on. Mathematically one can show that the time points at which the ball bounces represent a convergent sequence. The convergence of this sequence implies that infinitely many bounces occur in a finite amount of time. This is called “Zeno behavior”: infinitely many switches of mode in a finite amount of time. If Zeno behavior occurs in a control system, a numerical simulation of the system is extremely difficult, if not impossible. In the terminology of dynamical systems, the movement of the ball above ground is a “flow” and the bounce is a “jump.” Hybrid dynamical systems alternate between continuous (flow) and discrete (jump) modes. Rafal Goebel and Andrew Teel in their paper “Preasymptotic Stability and Homogeneous Approximations of Hybrid Dynamical Systems” model hybrid dynamical systems and approximate them by simpler systems obtained from linearization and tangent cones. The authors analyze preasymptotic stability, homogeneity, and convergence. A variety of well-chosen simple examples helps us to understand the general concepts and results.