Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator

Abstract

In this paper, we consider an inverse problem of identifying the source term for a generalization of the time-fractional diffusion equation, where regularized hyper-Bessel operator is used instead of the time derivative. First, we investigate the existence of our source term; the conditional stability for the inverse source problem is also investigated. Then, we show that the backward problem is ill-posed; the fractional Landweber method and the fractional Tikhonov method are used to deal with this inverse problem, and the regularized solution is also obtained. We present convergence rates for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Finally, we present a numerical example to illustrate the proposed method.

Stability of a nonlinear fractional pseudo-parabolic equation system regarding fractional order of the time

“…Fractional PDEs are important in a variety of domains, including physics and engineering of the memory effect, viscoelasticity, porous media, and other fields [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Viscosity has an essential role in the research of the material characteristics of constructions and biological materials.…”

On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems

“…Therefore, PDEs with fractional derivatives are a generalization equation with integer-order partial derivatives and a strong theoretical and practical interest. There have been many authors researching this field, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Inverse problem for a nonlinear third order in time partial differential equation

“…Moreover, for this case, if the time derivative in Equation () is a fractional derivative with the order 0 < α < 1 and 1 < α < 2 , Equation () is called a fractional diffusion and diffusion‐wave equation. Direct and inverse problems for fractional partial differential equations have attracted much attention in various fields of the applied science; see previous studies 29–36 …”