A generalised fractional derivative (the ψ-Caputo derivative) is studied. Generalisations of
standard discretisations are constructed for this derivative: L1, L1-2, L2-1 σ for derivatives of …

Nonlocal operators of fractional type are a popular modeling choice for applications that do
not adhere to classical diffusive behavior; however, one major challenge in nonlocal …

We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic
optimal control problem for the so-called fractional heat equation; control constraints are …

In this paper, we develop a conditional Monte Carlo method for solving PDEs involving an
integral fractional Laplacian on any bounded domain in arbitrary dimensions. We first …

We provide the convergence analysis for a-Galerkin method to solve the fractional Dirichlet
problem. This can be understood as a follow-up of [H. Antil, P. Dondl, and L. Striet, SIAM J …

This survey hinges on the interplay between regularity and approximation for linear and
quasilinear fractional elliptic problems on Lipschitz domains. For the linear Dirichlet integral …

We propose a monotone discretization for the integral fractional Laplace equation on
bounded Lipschitz domains with the homogeneous Dirichlet boundary condition. The …

This paper addresses the approximation of fractional harmonic maps. Besides a unit-length
constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness …

For first-order discretizations of the integral fractional Laplacian, we provide sharp local error
estimates on proper subdomains in both the local H^1-norm and the localized energy norm …

The numerical solution of a 1D fractional Laplacian boundary value problem is studied.
Although the fractional Laplacian is one of the most important and prominent nonlocal …