We investigate a backward problem of the time-space fractional symmetric diffusion equation with source term, wherein negative Laplace operator −Δ contained in main belongs to category uniformly elliptic operators. The is ill-posed because solution does not depend continuously on measured data. In this paper, existence and uniqueness conditional stability for inverse are given proven. Based lea...

Ultrafast subordinators are nondecreasing Lévy processes obtained as the limit of suitably normalized sums of independent random variables with slowly varying probability tails. They occur in a physical model of ultraslow diffusion, where the inverse or hitting time process randomizes the time variable. In this paper, we use regular variation arguments to prove that a wide class of ultrafast su...

We consider initial value/boundary value problems for fractional diffusion-wave equation: ∂ α t u(x, t) = Lu(x, t), where 0 < α ≤ 2, where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of ths weak solutions and the asymptotic behaviour as the time t goes to ∞ and the proofs are based on the eigenfunction expansions. ...

We develop an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs). We first introduce a new family of interpolants, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points. We perform such a construction following a spectral theory recently developed in [M. Zayernouri ...

The model is a subdiffusion-type fractional degenerate parabolic equation. We consider two inverse source problems with power vertical diffusion coefficients, including the well-known Monin-Obukhov atmospheric model. Final-time over-determination condition posed. Solutions to both are expressed in series expansion. Numerical examples discussed.

We study a linear-quadratic optimal control problem involving a parabolic equation with fractional diffusion and Caputo fractional time derivative of orders s ∈ (0, 1) and γ ∈ (0, 1], respectively. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator. Thus, we consider an equivalent formulation with a quasi-stationary elliptic problem...

The article presents a method for solving the inverse problem of two-dimensional anomalous diffusion equation with Riemann–Liouville fractional-order derivative. In first part present study, authors numerical solution direct problem. For this purpose, differential scheme was developed based on alternating direction implicit method. presented accompanied by examples illustrating its accuracy. se...

In this article, the electro-osmotic flow of Oldroyd-B fluid in a circular micro-channel with slip boundary condition is considered. The corresponding fractional system is represented by using a newly defined time-fractional Caputo-Fabrizio derivative without singular kernel. Closed form solutions for the velocity field are acquired by means of Laplace and finite Hankel transforms. Additionally...

It is known that if a stochastic process is a solution to a classical Itô stochastic differential equation (SDE), then its transition probabilities satisfy in the weak sense the associated Cauchy problem for the forward Kolmogorov equation. The forward Kolmogorov equation is a parabolic partial differential equation with coefficients determined by the corresponding SDE. Stochastic processes whi...