We offer a formulation of linear ordinary differential equations midway between what one encounters in a first undergraduate ODE course and what one encounters in a graduate Differential Geometry course (in the latter instance under the heading of “connections”). Analogies with elementary linear algebra are emphasized; no familiarity with Differential Geometry is assumed.

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the O...

A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or Bäcklund transformations. We describe such a chain for the sixth Painlevé equation (PVI), containing, apart from PVI itself, a Schwarzian version as well as a second-order second-degree ordinary differential equation (ODE). As a byproduct we derive an auto-Bäcklund transformation, relatin...

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...

A short overview of a Matlab–based toolbox developed to support the control-related analysis of a subclass of nD systems called differential linear repetitive processes (LRP) is the subject of this paper. Its main functionality covers two different areas. First the toolbox allows one to build discrete approximations of continuous–time LRPs and then perform analysis/simulation verification studi...

We introduce, analyze, and implement a new method for parameter identification for system of ordinary differential equations that are used to model sets of biochemical reactions. Our method relies on the integral formulation of the ODE system and a method of linear least squares applied to the integral equations. Certain variants of this method are also introduced in this paper.

The unsteady stagnation point flow and heat transfer with prescribed flux towards a stretching and shrinking sheet with viscous dissipation is studied. Similarity transformation is adopted to initially convert the governing differential equations into nonlinear ordinary differential equations. The two-point boundary value ordinary differential equations (ODE) are subsequently converted into par...

Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of nonlinear differential equations. Given a system of nonlinear differential equations, we apply a technique based on finite differences and sparse SDP relaxations for polynomial optimization problems (POP) to obtain a discrete approximatio...

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 ...