  The Fisher and Burgers' equations with finite memory transport, describing
reaction-diffusion and convection-diffusion processes, respectively have
recently attracted a lot of attention in the context of chemical kinetics,
mathematical biology and turbulence. We show here that they admit exact
solutions. While the speed of the traveling wavefront is dependent on the
relaxation time in Fisher equation, memory effects significantly smoothen out
the shock wave nature of Burgers' solution, without making any influence on the
corresponding wave speed. We numerically analyze the ansatz for the exact
solution and show that for the reaction-diffusion system the strength of the
reaction term must be moderate enough not to exceed a critical limit to allow
travelling wave solution to exist for appreciable finite memory effect.
