Finite difference method for a general fractional porous medium equation

Abstract

We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[ \frac{\partial u}{\partial t}+(-Δ)^{σ/2} (u^m)=0 \] posed for $x\in {\mathbb{R}}^N$, $t>0$, with $m\ge 1$, $\sigma \in (0,2)$, and nonnegative initial data $u(x,0)$. We prove existence and uniqueness of the solution of the numerical method and also the convergence to the theoretical solution of the equation with an order depending on $\sigma$. We also propose a two points approximation to a $\sigma$-derivative with order $O(h^{2-\sigma })$.

26 pages

References in corpus (4)

• A PDE approach to fractional diffusion in general domains: a priori error analysis
• Classical solutions for a logarithmic fractional diffusion equation
• Finite difference method for a fractional porous medium equation
• Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

Cited by in corpus (7)

• The Fisher-KPP problem with doubly nonlinear diffusion
• Multilevel methods for nonuniformly elliptic operators
• Finite and infinite speed of propagation for porous medium equations with fractional pressure
• The Mesa Problem for the Fractional Porous Medium Equation
• Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
• Transformations of Self-Similar Solutions for porous medium equations of fractional type
• An extension problem related to inverse fractional operators