On backward problem for fractional spherically symmetric diffusion equation with observation data of nonlocal type

Abstract

The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundary value problem where a Cauchy condition is replaced with a prescribed time average of the solution. In this work, we set some of the results above existence and regularity of the mild solutions of the proposed problem in some suitable space. Next, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given by the fractional Tikhonov method and convergence rate between the regularized solution and the exact solution under a priori parameter choice rule and under a posteriori parameter choice rule.

Galerkin Method for a Backward Problem of Time-Space Fractional Symmetric Diffusion Equation

“…Theorem 2. Assume that the solution u(•, 0) given by ( 18) satisfies a prior condition (20), g(x) ∈ L 2 (Ω) , f (x, t) ∈ L ∞ (0, T; L 2 (Ω)); then, we have:…”

“…In addition, we need to estimate the second item on the right of ( 49). According to Lemma 6, Theorem 3, ( 50), and a priori condition (20), we have:…”

“…We know that the backward problem of the time-fractional parabolic equation (0 < α < 1, β = 2) has been studied by many researchers, and some meaningful regularization methods have been presented and used to overcome the ill-posedness-for instance, the quasi-reversible method [13,14], total variation method [15], iterative fractional Tikhonov method [16], projection method [17], truncation method [18], simplified Tikhonov method [19], and fractional Tikhonov method [20]. For the backward problem of the space-fractional parabolic equation (α = 1, 1 < β < 2), there also have been some research works in which some regularization methods have been developed, such as the convolution and spectral methods [21], iteration method [22], simplified Tikhonov method [23], the logarithmic, negative exponential and fractional Tikhonov methods [24][25][26], optimal method [27], modified kernel method [28], and generalized Tikhonov method [29].…”

The Landweber Iterative Regularization Method for Identifying the Unknown Source of Caputo-Fabrizio Time Fractional Diffusion Equation on Spherically Symmetric Domain

“…In [25][26][27], the inverse problem of the fractional diffusion equation on a columnar symmetric domain are studied. In [28][29][30][31], the inverse problem of the fractional diffusion equation on a spherically symmetric domain are studied. In Figures 1 and 2, the grain of a spherically symmetric domain diffusion geometry is shown, which is actually consistent with laboratory measurements of helium diffusion from a physical point of view from apatite.…”