650 A SURVEY OF COMPUTATIONAL DIFFERENTIAL ALGEBRA
B. Mishra, October 1993
In this note, we explore the computational aspects of several problems in differential algebra with concrete applications in dynamics and motion-planning problems in robotics, automatic synthesis of control schemes for nonlinear systems and simulation of physical systems with fixed degrees of freedom.
Our primary aim is to study, compute and structurally describe the solution of a system of differential equations with coefficients in a field (say, the field of complex numbers, ). There seem to have been various approaches in this direction: e.g. ideal theoretic approach of Ritt, Galois theoretic approach of Kolchin and Singer and group theoretic technique of Lie. It is interesting to study their interrelationship and effectivity of various problems they suggest.
In general, these problems are known to be uncomputable; thus, we need to understand under what situations these problems become feasible. As related computer science questions, we also need to study the complexity of these problems, underlying data-structures, effects of the representation (e.g. sparsity).
Of related interest are some closely-related problems such as symbolic integration problem, solving difference equations, integro-differential equations and differential equations with algebraic constraints.