A PDE approach to fractional diffusion: a space-fractional wave equation

Abstract

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order s ∈ (0, 1), of symmetric, coercive, linear, elliptic, second-order operators in bounded domains Ω. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder C = Ω × (0, ∞). We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space-time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in Ω with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates.

Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative

“…In this paper, let W γ,p (Ω), γ ∈ R + be the Sobolev-Slobodecki space (see Subsection 3.1.3 in [9], Subsection 2.1. in [10]). Let W 1 2 ,2 00 (Ω) be the Lions-Magenes space which was introduced in Subsection 2.1 in [4]. Then, from [9,10,4] the space H γ (Ω) is defined by…”

“…Let W 1 2 ,2 00 (Ω) be the Lions-Magenes space which was introduced in Subsection 2.1 in [4]. Then, from [9,10,4] the space H γ (Ω) is defined by…”

A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

“…Some effects of fractional diffusion on the metastability of the solutions has been studied in [30]. Our main aim is to test the accuracy of the general splitting scheme ( 29)- (32) in resolving these metastability effects. We note that the proposed discrete scheme gives at least two additional advantages-first, it can be used on non-uniform space meshes, second, even for strong nonlinear reactions only one non-local subproblem is solved at each time step.…”

“…We restrict to application models based on nonlinear reaction and nonlocal diffusion models. As examples of such methods we can mention the approach based on the extension method [32,33], the algorithms based on matrix function vector product f (A h )b computation [34,35]. A special attention is given for development of splitting techniques to solve efficiently nonlinear reaction problems, see [36,37].…”

On a nonlinear Volterra integrodifferential equation involving fractional derivative with Mittag-Leffler kernel

“…The above settings can be found in [5] (Section 3), [6] (Section 2), [7] (Section 2) and therein. In the next lemmas, we present some useful embeddings between the spaces mentioned above.…”