The use of fractional partial differential equations in mathematical models has become increasingly popular in the last decade. Unlike the classical derivatives, fractional order derivatives are non-local operators. This property can be interpreted as a type of memory effect which is characteristic for different materials and processes. This explains one of the most significant uses of fractional PDE in applications. The same feature, however, makes difficult the design of fast and accurate numerical methods for such type of equations.

In this paper we present some examples of fractional in time diffusion-wave equation and highlight the main theoretical and numerical problems appearing. In particular, we introduce some interface and transmission problems related with such type of equations.