Well-posedness and numerical algorithm for the tempered fractional differential equations

Abstract

Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cytoplasm. The famous continuous time random walk (CTRW) model with power law waiting time distribution (having diverging first moment) describes this phenomenon. Because of the finite lifetime of biological particles, sometimes it is necessary to temper the power law measure such that the waiting time measure has convergent first moment. Then the time operator of the Fokker-Planck equation corresponding to the CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. This paper focus on discussing the properties of the time tempered fractional derivative, and studying the wellposedness and the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation. By adjusting the parameter of the proposed algorithm, any desired convergence order can be obtained and the computational cost linearly increases with time. And the effectiveness of the algorithm is numerically confirmed.

On some analytic properties of tempered fractional calculus

“…The integral transform which is now called a tempered fractional integral appears to have been first analysed in [6], but the associated model of fractional calculus has been described more explicitly in e.g. [21,22]. Both these papers and the references therein contain a number of applications of tempered fractional calculus to stochastic processes, random walks, Brownian motion, diffusion, turbulence, etc.…”

“…Proposition 2.6 (The semigroup property [21,17]). Let [a, b] be a real interval and α 1 , α 2 , β ∈ C be parameters with Re(α i ) > 0, Re(β) > 0.…”

Numerical Algorithms of the Two-dimensional Feynman–Kac Equation for Reaction and Diffusion Processes

“…Definition 1 (Riemann-Liouville tempered fractional integral [6,10,21]) Suppose that the real function v(t) is piecewise continuous on (a, b) and α > 0, λ ≥ 0, v(t) ∈ L[a, b]. The Riemann-Liouville tempered fractional integral of order α is defined to be…”

Stability and convergence of difference methods for two‐dimensional Riesz space fractional advection‐dispersion equations with delay

“…In consequence, fractional differential equations (FDEs) are gaining significant importance and have been analyzed by several researchers, for example, see References . Some of the latest works related with the high‐order numerical methods for FDEs have been worked in References .…”