Exact Solutions of a Time-Delay Advection Equation and a Fractional Time-Delay Advection Equation

Exact solutions are derived for a time-delay advection equation and a fractional-order time-delay advection equation with a time-delay in the spatial derivative. Solutions are obtained, for arbitrary separable initial conditions, by incorporating recently introduced delay functions in a separation of variables approach. Examples are provided showing oscillatory and translatory behaviours fundamentally different to standard propagating wave solutions.