This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.

The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in R d is considered. The differential operator is given by the fractional power L β , β ∈ (0, 1), of an integer order elliptic differential operator L and is therefore non-local. Its inverse L −β is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation, the inverse fractional power operator L −β is approximated by a weighted sum of non-fractional resolvents (I + t 2 j L) −1 at certain quadrature nodes t j > 0. The resolvents are then discretized in space by a standard finite element method.This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way, an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation, the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for L = κ 2 − ∆, κ > 0, with homogeneous Dirichlet boundary conditions on the unit cube (0, 1) d in d = 1, 2, 3 spatial dimensions for varying β ∈ (0, 1) attest the theoretical results.where · is the Euclidean norm on R d and Γ, K ν denote the gamma function and the modified Bessel function of the second kind, respectively. Via the positive parameters σ, ν, and κ the most important characteristics of the random field u

Abstract. A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear stochastic partial differential equations driven by additive noise. An error representation formula is found in an abstract setting based on the semigroup formulation of stochastic evolution equations. This is then applied to the stochastic heat, linearized Cahn-Hilliard, and wave equations. In all cases it is found that the rate of weak convergence is twice the rate of strong convergence, sometimes up to a logarithmic factor, under the same or, essentially the same, regularity requirements.

Abstract. Semidiscrete finite element approximation of the linear stochastic wave equation with additive noise is studied in a semigroup framework. Optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions. These are used to prove strong convergence estimates for the stochastic problem. The theory presented here applies to multi-dimensional domains and spatially correlated noise. Numerical examples illustrate the theory.

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation.

Fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation. In a fractional diffusion equation, the second derivative in the spatial variable is replaced by a fractional derivative of order less than two. The resulting solutions spread faster than the classical solutions and may exhibit asymmetry, depending on the fractional derivative used. Fractional reaction-diffusion equations combine the fractional diffusion with a classical reaction term. In this paper, we develop a practical method for numerical solution of fractional reaction-diffusion equations, based on operator splitting. Then we present results of numerical simulations to illustrate the method, and investigate properties of numerical solutions. We also discuss applications to biology, where the reaction term models species growth and the diffusion term accounts for movements.

We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.If (S t ) t∈[0,T ] is the analytic semigroup generated by −A, then (1.1) holds withwhere b : (0, ∞) → R is the Riesz kernel b t = t ρ−2 /Γ(ρ − 1) for some ρ ∈ (1, 2), then (S t ) t∈[0,T ] satisfies (1.1). The latter example is the main motivation of the present paper. In Subsection 5.2 we verify (1.1) for slightly more general kernels b.2010 Mathematics Subject Classification. 60H15, 60H07, 65C30, 65M60. Key words and phrases. Stochastic Volterra equation, finite element method, backward Euler, convolution quadrature, strong and weak convergence, Malliavin calculus, regularity, duality. 1 2 A. ANDERSSON, M. KOVÁCS, AND S. LARSSONThe main object of study in this paper is the stochastic evolution equation

Reproduction-Dispersal equations, called reaction-diffusion equations in the physics literature, model the growth and spreading of biological species. IntegroDifference equations were introduced to address the shortcomings of this model, since the dispersal of invasive species is often more widespread than what the classical RD model predicts. In this paper, we extend the RD model, replacing the classical second derivative dispersal term by a fractional derivative of order 1 < α ≤ 2. Fractional derivative models are used in physics to model anomalous super-diffusion, where a cloud of particles spreads faster than the classical diffusion model predicts. This paper also establishes a connection between the new RD model and a corresponding ID equation with a heavy tail dispersal kernel. The general theory developed here accommodates a wide variety of infinitely divisible dispersal kernels that adapt to any scale. Each one corresponds to a generalised RD model with a different dispersal operator. The connection established here between RD and ID equations can also be exploited to generate convergent numerical solutions of RD equations along with explicit error bounds.