Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems

Abstract

We consider initial value/boundary value problems for fractional diffusion-wave equation:where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to ∞ and the proofs are based on the eigenfunction expansions. Second for α ∈ (0, 1), we apply the eigenfunction expansions and prove (i) stability in the backward problem in time, (ii) the uniqueness in determining an initial value and (iii) the uniqueness of solution by the decay rate as t → ∞, (iv) stability in an inverse source problem of determining t-dependent factor in the source by observation at one point over (0, T ).

The quasi-boundary value regularization method for identifying the initial value with discrete random noise

“…However, for some practical situations, the part of the diffusion coefficient, or initial data, or boundary data, or source term may not be known, we need to find them using some additional measurement data, which will lead to the inverse problem of the fractional diffusion equation, such as [11][12][13]. Recently, many researchers have presented results of the initial value problem and boundary value problem on fractional differential equations, such as [14][15][16].…”

Space-dependent source determination in a time-fractional diffusion equation using a local discontinuous Galerkin method

“…[18,6,24,19,21,17,2,30,25,29,22,27,28,31,8]. In contrast to the classical diffusion equations, the fractional diffusion equations can be used to describe the anomalous diffusion phenomena such as super-diffusion or sub-diffusion.…”

“…Obviously for α = 1, D α t u = u t . The existence of a unique weak solution for the direct problem (1.1) has been studied in [22] and the only uniqueness theorem for the inverse source problem mentioned above is the following.…”

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

“…The regularity of the solution of (1.1) is restrictive. For example, for the homogeneous equation with initial data u 0 ∈ L 2 (Ω), we have the following stability estimate [44] …”

“…However the solutions of (1.1) has low regularity [44]. We only obtain firstorder accuracy when solving (1.1) by using the approximation scheme (1.6) with p ≥ 2.…”