7,017 research outputs found
Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations
We study a class of degenerate convection diffusion equations with a
fractional nonlinear diffusion term. These equations are natural
generalizations of anomalous diffusion equations, fractional conservations
laws, local convection diffusion equations, and some fractional Porous medium
equations. In this paper we define weak entropy solutions for this class of
equations and prove well-posedness under weak regularity assumptions on the
solutions, e.g. uniqueness is obtained in the class of bounded integrable
functions. Then we introduce a monotone conservative numerical scheme and prove
convergence toward an Entropy solution in the class of bounded integrable
functions of bounded variation. We then extend the well-posedness results to
non-local terms based on general L\'evy type operators, and establish some
connections to fully non-linear HJB equations. Finally, we present some
numerical experiments to give the reader an idea about the qualitative behavior
of solutions of these equations
Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
The fractional Laplacian is a non-local operator which
depends on the parameter and recovers the usual Laplacian as . A numerical method for the fractional Laplacian is proposed, based on
the singular integral representation for the operator. The method combines
finite difference with numerical quadrature, to obtain a discrete convolution
operator with positive weights. The accuracy of the method is shown to be
. Convergence of the method is proven. The treatment of far
field boundary conditions using an asymptotic approximation to the integral is
used to obtain an accurate method. Numerical experiments on known exact
solutions validate the predicted convergence rates. Computational examples
include exponentially and algebraically decaying solution with varying
regularity. The generalization to nonlinear equations involving the operator is
discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure
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