APPLICATION OF SCALING AND QUASISTATIC METHODS TO STUDY NONLINEAR SUBDIFFUSION-REACTION EQUATIONS WITH FRACTIONAL TIME DERIVATIVE

摘要

We consider a subdiffusive system where transported particles of spieces A and B chemically react according to the formula A + B → θ. This process is described by the nonlinear subdiffusion-reaction equations with fractional time derivatives. We show that the scaling method, which is commonly used to study diffusion-reaction equations of natural order, is not applicable to the subdiffusion case due to the specific properties of fractional derivatives, unless very special assumptions are taken into account. Contrary to the scaling method, the quasistatic one provides the explicite solutions in the diffusion region and the time evolution of reaction front x_f, which reads x_f = Kt~(α/2), where a is the subdiffusion parameter and K is uniquely determined. We also present the numerical solutions of subdiffusion-reaction equations and show that the numerical results coincide with the analytical ones.