We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet $ \partial_t^{\alpha} u(x, t) = -Au(x, $, where -A \sum_{i, j 1}^d \partial_i(a_{ij}(x) \partial_j) + \sum_{j b_j(x) \partial_j c(x) $. establish uniqueness for an inverse problem determining order \alpha fractional derivatives by data u(x_0, 0<t<T at one point x_0 in a spa...

We deal with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the source magnitude in the time fractional diffusion equation from viewpoint of numerics. Such simultaneous inversion problem is often of severe ill-posedness as compared with that of determining a single coefficient function. The forward problem is solved by employing an implicit finite...

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated. On the basis of the optimal control framework, the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established, and a time-space spectral method is proposed to numerically solve the resulting minimization problem. The co...

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by fractional-order derivatives. This article considers a nonlocal inverse problem and shows that the exponents of the fractional time and space derivatives are determined uniquely by the data u(t, 0) = g(t), 0 < t < T . The uniqueness result is a theoretical background for determining experimental...

This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes timechanged by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well-posed Dirichlet problem for space-time fractional diffusions...