The general fractional calculus becomes popular in continuous time random walk recently. However, the boundedness condition of the general fractional integral is one of the fundamental problems. It wasn’t given yet. In this short communication, the classical norm space is used, and a general boundedness theorem is presented. Finally, various long–tailed waiting time probability density functions are suggested in continuous time random walk since the general fractional integral is well defined.

A general fractional calculus is described using fractional operators with respect to another function, and some often used propositions are presented in this framework. Together with the continuous time random walk (CTRW), a general time-fractional Fokker-Planck equation is derived and the governing equation meets the general fractional derivative. Finally, various new probability density functions are proposed in this paper.

The standard definition of the Riemann–Liouville integral is revisited. A new fractional integral is proposed with an exponential kernel. Furthermore, some useful properties such as composition relationship of the new fractional integral and Leibniz integral law are provided. Exact solutions of the fractional homogeneous equation and the non-homogeneous equations are given, respectively. Finally, a finite difference scheme is proposed for solving fractional nonlinear differential equations with exponential memory. The results show the efficiency and convenience of the new fractional derivative.

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