Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE

Abstract

The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem.

Inverse problems of determining parameters of the fractional partial differential equations

“…, p ℓ ). For the numerical treatment, in Li, Zhang, Jia, and Yamamoto [13], Sun, Li, and Jia [29], the authors reformulated the inverse problem into an optimization problem…”

“…, ℓ in (3) or (4) follows as byproduct from the uniqueness results (e.g., Cheng, Nakagawa, Yamazaki, and Yamamoto [4], Li, Imanuvilov, and Yamamoto [16], Kian, Okasanen, Soccorsi, and Yamamoto [9]) for inverse coefficient problems which are surveyed in the chapter "Inverse problems of determining coefficients of the fractional partial differential equations" of this handbook. Here as a supplement to this chapter, we give a partial list of numerical researches on the reconstruction for fractional orders and related coefficients for fractional differential equations: Chen, Liu, Jiang, Turner, and Burrage [3], Lukashchuk [23], Sun, Li, and Jia [29], Tatar, Tinaztepe, and Ulusoy [30], Yu, Jiang, and Qi [32], Yu, Jiang, and Wang [33], Zhang, Li, Chi, Jia, and Li [35], Zheng and Wei [36] and the references therein.…”

Uniqueness of determining the variable fractional order in variable-order time-fractional diffusion equations

“…A simultaneous inversion of the space-dependent diffusivity coefficient and the fractional order in one-dimensional tFDEs based on observations from the end points of the spatial interval was developed in [18]. A numerical inversion of the fractional orders of the multi-term tFDEs in multiple space dimensions was studied in [31].…”