Green’s Functions on Various Time Scales for the Time-Fractional Reaction-Diffusion Equation

The time-fractional diffusion equation coupled with a first-order irreversible reaction is investigated by employing integral transforms. We derive Green’s functions for short and long times via approximations of the Mittag-Leffler function. The time value for which the crossover between short- and long-time asymptotic holds is presented in explicit form. Based on the developed Green’s functions, the exact analytic asymptotic solutions of the time-fractional reaction-diffusion equation are obtained. The applicability of the obtained solutions is demonstrated via quantification of the reaction-diffusion kinetics during heterogeneous catalytic chitin conversion to chitosan.