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

TL;DR

This paper investigates the properties of tempered fractional derivatives, establishes the well-posedness of related differential equations, and develops an efficient numerical algorithm with adjustable accuracy for solving these equations.

Contribution

It introduces a new analysis of tempered fractional derivatives, proves well-posedness, and proposes a flexible, linearly scalable numerical algorithm for tempered fractional ODEs.

Findings

The algorithm achieves adjustable convergence order.

Numerical results confirm the algorithm's effectiveness.

Computational cost increases linearly with time.

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 ({\em 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 well-posedness 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.