  In this work, exact solutions are derived for an integer- and
fractional-order time-delayed diffusion equation with arbitrary initial
conditions. The solutions are obtained using Fourier transform methods in
conjunction with the known properties of delay functions. It is observed that
the solutions do not exhibit infinite speed of propagation for smooth initial
conditions that are bounded and positive. Sufficient conditions on the initial
condition are also established such that the finite time blowup of the
solutions can be explicitly calculated. Examples are provided that highlight
the contrasting behaviours of these exact solutions with the known dynamics of
solutions to the standard diffusion equation.
