On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

Abstract

In this paper, we consider an inverse problem of recovering the initial value for a generalization of time‐fractional diffusion equation, where the time derivative is replaced by a regularized hyper‐Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill‐posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

Identifying initial value problem for time‐fractional diffusion equation with Caputo‐like counterpart hyper‐Bessel operator: Optimal error bound analysis and regularization method

“…where 0 , 1 , … , m are arbitrary parameters such that their sum is less than m. And the authors gave the form of the solution based on appropriate eigenfunction expansions in Al-Musalhi et al 27 and Zhang 29 but established only the existence and uniqueness of the solution. Tuan et al 30 studied a terminal value problem for a generalization of the fractional diffusion equation with Caputo-like counterpart hyper-Bessel operator by using fractional Tikhonov regular method and gave the error estimates between the regularized solution and the exact solution under two parameter choice rules, but the error estimates have the saturation phenomena.…”

Identifying the space source term problem for time-space-fractional diffusion equation

“…Fractional calculus has many applications in mechanics, physics and engineering science, etc. We present to the reader much of the published work on these issues, such as [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references cited therein. This makes it attractive to study this model.…”

“…Recently, the simple source problem, i.e, ϕ(t) = 1 and γ = 1 in Eq. (1.1) has been considered by Fan Yang, Zhang and Li, see [20,21,[25][26][27]; the authors used the Landweber iterative regularization, Truncation regularization and Tikhonov regularization methods solve this problem and achieved the results of convergence results to the order of p p+1 for 0 < p < 2 and 1 2 for p > 2, respectively.…”

High-order compact finite volume scheme for the 2D multi-term time fractional sub-diffusion equation

“…Tuan et al solved final value problem for nonlinear time fractional reaction-diffusion equation with discrete data [38]. Tuan et al handled a terminal value problem for a generalization of fractional diffusion equation with hyper-Bessel operator [39]. Nguyen et al settled a terminal value problem for time fractional diffusion equation by using regularization [26].…”