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Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method. (English) Zbl 1112.65006

Consider the non-random Lévy-Feller advection-dispersion equation (LFADE) \[ \frac{\partial u(x,t)}{\partial t} = a D^\alpha_\theta u(x,t) - b \frac{\partial u(x,t)}{\partial x} \] where \(a > 0\), \(b \geq 0\), \(x \in \mathbb{R}\) (or \(0 < x < L\)), \(t > 0\), and \(D^\alpha_\theta\) is the Riesz-Feller fractional derivative (in space) of order \(\alpha\) (\(1 < \alpha \leq 2\)) and skewness \(\theta\) (\(| \theta| \leq 2 - \alpha\)), subject to initial condition \(u(x,0)=\varphi (x)\).
A random walk model for approximating the solution \(u\) governed by (LFADE) is presented. This random walk model converges to model (LFADE) by use of a properly scaled transition to vanishing equidistant space and time steps. An explicit finite difference approximation (EFDA) for (LFADE), resulting from the Grünwald-Letnikov discretization of fractional derivatives, is proposed. As a result of the interpretation of the random walk model, the stability and convergence of (EFDA) for (LFADE) in a bounded domain are discussed. Finally, some numerical examples show the application of the presented techniques.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
60H30 Applications of stochastic analysis (to PDEs, etc.)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
26A33 Fractional derivatives and integrals
60G50 Sums of independent random variables; random walks
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