We derive a computable a posteriori error estimator for the αharmonic extension problem, which localizes the fractional powers of elliptic operators supplemented with Dirichlet boundary conditions. Our a posteriori error estimator relies on the solution of small discrete problems on anisotropic cylindrical stars. It exhibits built-in flux equilibration and is equivalent to the energy error up t...

in this paper, a new numerical method for solving time-fractional diffusion equations is introduced. for this purpose, finite difference scheme for discretization in time and chebyshev collocation method is applied. also, to simplify application of the method, the matrix form of the suggested method is obtained. illustrative examples show that the proposed method is very efficient and accurate.

We prove the existence of a positive SOLA (Solutions Obtained as Limits Approximations) to following PDE involving fractional power Laplacian ( ? ? ) s u = 1 ? + ? 2 ? ? in ? , > 0 ? N ? . Here, is bounded domain ?N, s?(0,1), 2s<N, ?,??(0,1), 2s?=2N?N?2s critical Sobolev exponent and nonnegative Radon measure ?.

The question about behavior of gaps between zeros polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics roots differentiation. In this paper, we connect rigorously solutions Steinerberger’s PDE evolution for class trigonometric polynomials. Namely...

We consider the focusing L2-critical half-wave equation in one space dimension i∂tu = Du− |u|u, where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold M∗ > 0 such that all H1/2 solutions with ‖u‖L2 < M∗ extend globally in time, while solutions with ‖u‖L2 > M∗ may develop singularities in finite time. In this paper, we first prove the ex...