During the conference, the talks addressed a wide range of topics related to differential equations, which can be classified in three groups.
Differential Algebraic Equations and Applications
This edition of DART put quite some emphasis on differential-algebraic equations. The theory on which is based the three following talks can be found in the following books
- Peter Kunkel, Volker Mehrmann Differential-Algebraic Equations. Analysis and Numerical Solution, EMS, 2006
- Ricardo Riaza, Differential Algebraic System, 2008, World Scientific
- John Pryce, A simple structural analysis method for DAEs, BIT, 41, 2, 364-394, 2001.
Note also the book:
- R.Lamour, R. Marz, C. Tischendorf, Differential-Algebraic Equations. A projector Based Analysis, 2013, Springer.
Peter Kunkel (slides) treats DAE from the point of view of strangeness index. The main idea is to differentiate the initial equations and to verify if certain assumptions hold concerning the augmented system. The initial and reduced system have the same solutions. This is generated using tools of automatic differentiation. Next a numerical procedure is used to approximate the solution. The idea is to adapt numerical methods as BDF, Gauss-Lobatto or Radau in this context with a control of the error using numerical Gauss-Newton method.
Ricardo Riaza (slides) is studying DAEs in the context of electrical and electronic engineering. The class of DAEs which is mainly considered is that of quasilinear DAEs as many models of "physic world". An important issue of circuit models is the characterization of the index in terms of the circuit topology and of the devices.
Ned Nedialkov (slides) is interested by the implementation to solve numerically structured DAEs. He has produced two codes DAETS and DAESA which are based on a structural analysis given by John Pryce at the beginning of this century. Pryce independently found an analogue of Jacobi method and thus gave a modern analysis for both of them. The talk explains how to adapt Pryce's method in order to obtain a code with good complexity and certified results. Many examples show the behaviour of this method for DAE4 with high index.
In addition to these three talks, Karim Alloula, presented a talk on an index reduction method by deflation for solving DAE with an application to the reactive Rayleigh distillation (slides). The long talk of Wilfrid Perruquetti was devoted to the parameters estimation problem in control theory, using Michel Fliess' state reconstructors, based on the theory of Mikusiński operators (abstract).
Differential Algebra
Long talks were given by Bernard Malgrange on the equivalence problem and bounds for involutivity (abstract), Alban Quadrat, on algebraic D-modules (abstract) and Chun-Ming Yuan, on sparse differential and-or difference resultants (slides). Shorter talks were given by François Boulier, on the symbolic integration of differential fractions (slides), Anja Korporal, on generalized Green's operators and integro-differential operators (slides), Markus Lange-Hegermann, on the Thomas decomposition of differential systems (slides), Brahim Sadik, on an extented Ritt-Lüroth theorem (slides) and Pablo Solernò, on an effective differential Nullstellensatz for ordinary DAE systems over the complex numbers (slides). The talks of Bernard Malgrange and Pablo Solernò were both concerned with bounds of special interest for the complexity of computation: that of Bernard Malgrange with a uniform bound for involutivity and that of Pablo Solernò with the maximal order for the ordinary Nullstellensatz. Both bounds have a doubly exponential behaviour. They therefore give some insight on the talk of Markus Lange-Hegermann. The talk of Anja Korporal extends the classical standpoint of differential algebra by considering integration operators and initial or boundary conditions. The talk of François Boulier is related to it, since it presented a tool which may perhaps be considered as a first step towards a Gröbner basis theory for integro-differential operators. It is also related to the parameters estimation problem (see the talk of Wilfrid Perruquetti).
Differential Galois Theory and Integrability of Dynamical Systems
There were talks more specifically related to the differential Galois theory and the integrability of dynamical systems. Thierry Combot gave a long talk on the application of differential Galois theory to dynamical systems (slides). Shorter talks were given by Carlos Arreche, on a Galois-theoretic proof of the differential transcendence of the incomplete Gamma function (slides), Guillaume Chèze, on an efficient algorithm for computing rational first integral of polynomial vector fields (abstract), Claude Mitschi, on the monodromy of parameterized linear differential systems (slides), Julien Roques on the integrability by discrete quadratures (abstract), Omar León Sánchez, on the Galois theory of logarithmic differential equations (slides), Werner Seiler, on singularities and bifurcation of ordinary differential equations (slides), Jacques-Arthur Weil, on an effective version of the Morales-Ramis-Simo theorem (slides) and Michael Wibmer, on linear differential equations and difference algebraic groups (a related paper). The talk of Guillaume Chèze was concerned with the computation of a Lüroth generator for the algebraic closure of a field extension of transcendence degree 1 in some field of fraction. It is thus related to the talk of Brahim Sadik.