ODEs and DAEs

Ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) arise in a wide variety of scientific and engineering applications. They are essential to scientific computing in physics, in chemistry, and in technical applications. Differential equations are a natural framework in which numerous problems are modeled. In addition to differential equations the models may contain implicit equations, in general purely algebraic equations, to model conservation laws, geometrical or kinematic constraints, Kirchoff's laws, etc. DAEs arise typically in the following situations:

• in models of mechanical systems (multibody systems), e.g., in robotics;
• in electrical circuit simulation;
• in models of chemical processes;
• in partial differential equations (PDEs), e.g., in fluid dynamics;
• in optimal control theory;
• in the study of Hamiltonian systems with constraints;
• in the analysis of stiff differential equations.

The presence of constraints in DAEs leads to theoretical and numerical difficulties which are not present in ODEs.