Report on Recent Developments in Numerical Methods for Differential Equations with Time Delay

General Direction of the Field

The latest research in the field of numerical methods for differential equations with time delay is notably advancing the understanding and computational techniques for handling systems where delays play a critical role. This area is witnessing a shift towards more sophisticated and efficient numerical techniques that can handle both distributed and discrete delays in infinite-dimensional systems, such as those described by partial differential equations (PDEs). The focus is on developing methods that not only ensure stability and accuracy but also enhance computational efficiency, particularly in parallel computing environments.

Researchers are exploring innovative approaches to transform complex delay systems into more manageable forms, either by deriving equivalent systems with fewer delays or by discretizing the delay term using quadrature methods. These transformations facilitate the application of traditional numerical techniques to problems that were previously intractable. Additionally, there is a growing emphasis on extending stability concepts like finite-time input-to-state stability (FTISS) to infinite-dimensional systems, which is crucial for ensuring the robustness of numerical solutions in practical applications.

Domain decomposition methods, particularly waveform relaxation techniques, are being refined to handle PDEs with time delay more effectively. These methods allow for parallel processing and can significantly reduce computational time, making them suitable for large-scale simulations. Furthermore, decoupling techniques are being developed to simplify the solution of coupled systems of evolutionary equations, offering new ways to decompose and solve complex problems by breaking them down into simpler subproblems.

Noteworthy Developments

• Finite-time input-to-state stability for infinite-dimensional systems: This paper significantly extends the FTISS concept to infinite-dimensional systems, providing a theoretical foundation and practical applications in PDEs with sublinear terms and distributed disturbances.
• Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation Methods for PDEs with Time Delay: The introduction and comparison of these methods offer new insights into efficient parallel computing solutions for PDEs involving time delay, enhancing both theoretical understanding and practical implementation.