The phenomena described by fractional differential equations with time delay are ubiquitous and widely used in nature and are an important subject of common concern in the fields of science and engineering. Based on the theoretical achievements and algorithms obtained by researchers, it is essential to construct new algorithms with high performance aimed at several kinds of spatial and time fractional delay differential equations, and to capture the dynamic behaviour of travelling wave solutions in systems based on these algorithms.

There are several challenges facing the field of fraction delay different equations, including the stability analysis of the delay dependence of higher-order numerical time integration schemes for fractional delay differential problems, and the numerical theory of the numerical scheme. Other challenges include the stability and numerical simulation of travelling wave solutions and critical travelling wave solutions of fractional delay differential equations, as well as the design of fourth-order and sixth-order compact schemes for fractional delay equations with strong nonlinearity.

The aim of this Special Issue is to provide a platform for significant contributions to the development and improvement of the theory and application of fractional delay differential equations. We welcome both original research and review articles.

Potential topics include but are not limited to the following:

• Delayed diffusion-wave systems, with and without distributed order in time, and their numerical analysis
• Linear multistep methods or Runge-Kutta methods for fractional or distribution fractional delay differential equations
• Numerical analysis distribution fractional delay diffusion equations based on finite difference methods
• Galerkin spectral schemes for fractional or distribution fractional delay partial differential equations
• Compact difference methods for distribution fractional delay diffusion equations or diffusion-wave equations