There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of ordinary or partial differential equations. Furthermore, we show how a combination of this geometric theory with (differential) algebraic tools allows us to make parts of the theory algorithmic. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for the effective detection of all singularities at a given order or, more precisely, for the determination of a regularity decomposition. Finally, we give a rigorous definition of a regular differential equation, a notoriously difficult notion ubiquitous in the geometric theory of differential equations, and show that our algorithm extracts from each prime component a regular differential equation. Our main tools are on the one hand the algebraic resp. differential Thomas decomposition and on the other hand the Vessiot theory of differential equations.

Singularities of Algebraic Differential Equations

We propose a geometric approach for the numerical integration of singular initial value problems for (systems of) quasi-linear di ﬀ erential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector ﬁeld and thus into one which can be solved in an e ﬃ cient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane–Emden equations and the Thomas–Fermi equation.

On the Numerical Integration of Singular Initial and Boundary Value Problems for Generalised Lane-Emden and Thomas-Fermi Equations

Singularities of algebraic differential equations

We discuss how the geometric theory of differential equations can be used for the numerical integration and visualisation of implicit ordinary differential equations, in particular around singularities of the equation. The Vessiot theory automatically transforms an implicit differential equation into a vector field distribution on a manifold and thus reduces its analysis to standard problems in dynamical systems theory like the integration of a vector field and the determination of invariant manifolds. For the visualisation of low-dimensional situations we adapt the streamlines algorithm of Jobard and Lefer to 2.5 and 3 dimensions. A concrete implementation in Matlab is discussed and some concrete examples are presented.

Elishan Braun

Papers

On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations

On the Numerical Analysis and Visualisation of Implicit Ordinary Differential Equations