In this Part 2 of our work, we continue the discussion on time-steppers for the cable equation, modeling action potential propagation in cardiac myocyte cells. Here we report results from parallel computations. The cable equation is a diffusion-reaction PDE for transmembrane voltage, whose reaction (source) term depends on several voltage-dependent quantities (gates), each satisfying a first or...

This paper discusses possible solving strategies regarding four ODE problems, all relevant and currently out of the scope of the Maple ODE solver. The problems are: 1. Transformations linearizing non-linear first order ODEs 2. Non-linear 2nd order ODEs for which point symmetries are of no use 3. Hypergeometric solutions for 2nd order linear ODEs 4. Solutions for third and higher order linear OD...

A tool for the order reduction of differential algebraic equations (DAEs) is outlined in this report. Through the use of an equation dependency analysis and nonlinear function approximation, the algebraic equations can be divided into sets that require implicit or explicit solutions. If all of the algebraic variables can be solved or approximated explicitly, the DAE becomes a set of ordinary di...

We study a free boundary problem arising from American put options. In particular we prove existence and uniqueness for this problem and we derive, and prove rigorously, high order asymptotic expansions for the early exercise boundary near expiry. We provide four approximations for the boundary: one is explicit and is valid near expiry (weeks); two others are implicit involving inverse function...

Under some observability assumptions (uniform observability), a high gain observer for a class of implicit dynamical systems is given in this paper. Numerically, the computation of trajectories of such implicit systems usually necessitates the use of an optimization algorithm together with an ODE numerical method. This complicates the synthesis of an observer. The observer design proposed here ...

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational ...