Delay Differential-Algebraic Equations (DDAEs)

Abstract

Delay differential-algebraic equations (DDAEs) are an important class of mathematical models that broaden standard differential-algebraic equations (DAEs) to incorporate discrete time delays. The time lag terms pose significant analytical and computational challenges. This paper provides a comprehensive overview of current and emerging methods for solving DDAEs and systems of DDAEs. Generalized Taylor series techniques, linear multistep methods, and reduction to ordinary differential equations are examined for numerically integrating DDAEs. Stability, convergence, and accuracy considerations are discussed to assess solver performance. Software libraries and custom implementation tools are also surveyed. Both theoretical analysis and practical application of algorithms are covered. Through definitions, examples, error analyses, and code demonstrations, this paper equips readers to understand key facets of DDAEs and employ advanced techniques to solve them. The topics presented here represent important progress toward addressing real-world systems across science and engineering that fundamentally include time delays.