150 research outputs found
Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions
In this work, we propose novel discretisations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretisations with order of convergence depending on the regularity of the domain and the function on which the fractional Laplacian is acting. Unlike other existing approaches in literature, our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results.FdT acknowledges support of
Toppforsk project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway.
ERCIM ``Alain Benoussan" Fellowship programm
Numerical continuation for fractional PDEs: sharp teeth and bloated snakes
Partial differential equations (PDEs) involving fractional Laplace operators
have been increasingly used to model non-local diffusion processes and are
actively investigated using both analytical and numerical approaches. The
purpose of this work is to study the effects of the spectral fractional
Laplacian on the bifurcation structure of reaction-diffusion systems on bounded
domains. In order to do this we use advanced numerical continuation techniques
to compute the solution branches. Since current available continuation packages
only support systems involving the standard Laplacian, we first extend the
pde2path software to treat fractional PDEs. The new capabilities are then
applied to the study of the Allen-Cahn equation, the Swift-Hohenberg equation
and the Schnakenberg system (in which the standard Laplacian is each replaced
by the spectral fractional Laplacian). Our study reveals some common effects,
which contributes to a better understanding of fractional diffusion in generic
reaction-diffusion systems. In particular, we investigate the changes in
snaking bifurcation diagrams and also study the spatial structure of
non-trivial steady states upon variation of the order of the fractional
Laplacian. Our results show that the fractional order can induce very
significant qualitative and quantitative changes in global bifurcation
structures
Modeling cardiac structural heterogeneity via space-fractional differential equations
We discuss here the use of non-local models in space and fractional order operators in the characterisation of structural complexity and the modeling of propagation in heterogeneous biological tissues. In the specific, we consider the application of space-fractional operators in the context of cardiac electrophysiology, where the lack of clear separation of scales of the highly heterogeneous myocardium triggers peculiar features such as the dispersion of action potential duration, that have been observed experimentally, but cannot be described by the standard monodomain or bidomain models. We describe the methodology and compare the results of a standard monodomain model with results of a model with a non-local component in space
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