Identifying the initial state for a parabolic diffusion from their time averages with fractional derivative

The main purpose of this paper is to study a problem of recovering a parabolic equation with fractional derivative from its time averaging. This problem can be established as a new boundary value problem where a Cauchy condition is replaced by a prescribed time average of the solution. By applying some properties of the Mittag-Leffler function, we set some of the results above existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space. Moreover, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given, and convergence rate between the regularized solution and the exact solution in L-p space is also derived.

Automated summary

The main purpose of this paper is to study the problem of recovering a parabolic equation with fractional derivative from its time averaging. By applying the properties of the Mittag-Leffler function, the existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space are established. The ill-posedness of the problem in the sense of Hadamard is also shown, and the convergence rate between the regularized solution and the exact solution in L-p space is derived.