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.

中文翻译：

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时间-分数阶反应-扩散方程的不同时间尺度的格林函数

通过采用积分变换研究了与一阶不可逆反应耦合的时间分数阶扩散方程。我们通过 Mittag-Leffler 函数的近似值推导出短期和长期的格林函数。以显式形式呈现短期和长期渐近线之间交叉成立的时间值。基于所发展的格林函数，得到时间-分数阶反应-扩散方程的精确解析渐近解。通过对多相催化几丁质转化为壳聚糖期间的反应扩散动力学进行量化，证明了所获得溶液的适用性。