Boundary Problems for the Fractional and Tempered Fractional Operators

Abstract

To characterize the Brownian motion in a bounded domain $\Omega$, it is well known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; in contrast, for the Lévy flights or tempered Lévy flights in a bounded domain, the boundary conditions involve the information of a solution in the complementary set of $\Omega$, i.e., $\Bbb{R}^n\backslash \Omega$, with the potential reason that paths of the corresponding stochastic process are discontinuous. Guided by probability intuitions and the stochastic perspectives of anomalous diffusion, we show the reasonable ways, ensuring the clear physical meaning and well-posedness of the partial differential equations (PDEs), of specifying “boundary” conditions for space fractional PDEs modeling the anomalous diffusion. Some properties of the operators are discussed, and the well-posednesses of the PDEs with generalized boundary conditions are proved.