Fractional Partial Differential Equations with Boundary Conditions

Abstract

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\Omega )$ and $L_1(\Omega )$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax-Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.

References in corpus (1)

• Matrix approach to discrete fractional calculus II: partial fractional differential equations

Cited by in corpus (5)

• Space-Time Duality and High-Order Fractional Diffusion
• A compounded random walk for space-fractional diffusion on finite domains
• On the fractional version of Leibniz rule
• On viscosity solutions of space-fractional diffusion equations of Caputo type
• Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes II