Continuous time random walk to a general fractional Fokker–Planck equation on fractal media

Abstract

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.

Parameter estimation of fractional uncertain differential equations via Adams method

“…Different fractional derivatives may lead to better results. We will consider the general fractional calculus [1,4] and choose the best one in specific real world applications.…”

Terminal Value Problems of Non-homogeneous Fractional Linear Systems with General Memory Kernels

“…Recently, the function space, definitions and numerical methods of a general fractional calculus were developed in [7,11,13]. The physical meaning of the fractional derivative was provided by continuous time random walk [8].…”

A Note on Function Space and Boundedness of the General Fractional Integral in Continuous Time Random Walk

“…Due to the new features in comparison with the standard fractional derivatives, much attention has been paid to theoretical research and applications, for example, fractional calculus of variations [4], the Laplace transform [5,6], exact solution [7,8] and numerical methods [9]. Motivated by the continuous time random walk understood by means of the standard fractional calculus [10,11], suppose a long-tailed waiting time probability density function [12] which is more general than the power law function Then, one comes across a general time-fractional Fokker-Planck equation with the general fractional integral where T is the temperature, P(x, t) is the probability density function, K is the diffusion coefficient, F(x) represents external force field and k b represents the Boltzmann constant. So the physical meaning of the general fractional derivative or the kernel function g(t) was provided.…”

“…So the physical meaning of the general fractional derivative or the kernel function g(t) was provided. More details were shown in [12]. However, one of the fundamental problems is still not addressed yet.…”