On tempered and substantial fractional calculus

In this paper, we discuss the differences between the tempered fractional calculus and substantial fractional operators in anomalous diffusion modelling, so that people can better understand the two fractional operators. We first introduce the definitions of tempered and substantial fractional operators, and then analyze the properties of two definitions. At last, we prove that the tempered fractional derivative and substantial derivative are equivalent under some conditions. A diffusion problem defined by using tempered derivative is also given to illustrate the slow convergence of an anomalous diffusion process.