Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to e...

In this article, we study an inverse problem to determine an unknown source term in a space time fractional diffusion equation, whereby the data are obtained at a certain time. In general, this problem is ill-posed in the sense of Hadamard, so the Fourier truncation method is proposed to solve the problem. In the theoretical results, we propose a priori and a posteriori parameter choice rules a...

In this paper, we investigate an inverse problem to determine an unknown source term that has a separable-variable form in the time-fractional diffusion equation, whereby the data is obtained at a certain time. This problem is ill-posed, and we use the Landweber iterative regularization method to solve this inverse source problem. Two kinds of convergence rates are obtained by using an a priori...

This article deals with the initial-boundary value problem for a moderately coupled system of time-fractional diffusion equations. Defining mild solution, we establish fundamental unique existence, limited smoothing property and long-time asymptotic behavior which mostly inherit those single equation. Owing to coupling effect, also obtain uniqueness an inverse on determining all fractional orde...

Abstract This paper is concerned with the inverse random source problem for a stochastic time fractional diffusion equation, where assumed to be driven by Gaussian field. The direct shown well-posed examining well-posedness and regularity of solution equivalent two-point boundary value in frequency domain. For problem, Fourier modulus coefficient proved uniquely determined variance transform da...

In this paper, we obtain the sharp uniqueness for an inverse $ x $-source problem a one-dimensional time-fractional diffusion equation with zeroth-order term by minimum possible lateral Cauchy data. The key ingredient is unique continuation which holds weak solutions.