Boundary conditions at a thin membrane for normal diffusion equation which generate subdiffusion

Abstract

We consider a particle transport process in a one-dimensional system with a thin membrane, described by a normal diffusion equation. We consider two boundary conditions at the membrane that are linear combinations of integral operators, with time dependent kernels, which act on the functions and their spatial derivatives define on both membrane surfaces. We show how boundary conditions at the membrane change the temporal evolution of the first and second moments of particle position distribution (the Green's function) which is a solution to normal diffusion equation. As these moments define the kind of diffusion, an appropriate choice of boundary conditions generates the moments characteristic for subdiffusion. The interpretation of the process is based on a particle random walk model in which the subdiffusion effect is caused by anomalously long stays of the particle in the membrane.

arXiv comment10 pages, 6 figures

References in corpus (19)

• Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations
• Fractional Fokker-Planck Equation for Ultraslow Kinetics
• Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary
• Fractional diffusion modeling of ion channel gating
• Fractional Langevin Equation of Distributed Order
• Paradoxical diffusion: Discriminating between normal and anomalous random walks
• Paradoxes of Subdiffusive Infiltration in Disordered Systems
• Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces
• Aging dynamics in interacting many-body systems
• From the solutions of diffusion equation to the solutions of subdiffusive one
• Diffusion of antibiotics through a biofilm in the presence of diffusion and absorption barriers
• Continuous-time random walk with a superheavy-tailed distribution of waiting times
• Asymptotic solutions of decoupled continuous-time random walks with superheavy-tailed waiting time and heavy-tailed jump length distributions
• Random walk model of subdiffusion in a system with a thin membrane
• Model of anomalous diffusion--absorption process in a system consisting of two different media separated by a thin membrane
• A Tutorial on the Basic Special Functions of Fractional Calculus
• How to determine a boundary condition at a thin membrane for diffusion from experimental data
• First passage time for subdiffusion: The nonextensive entropy approach versus the fractional model
• Boundary conditions at a thin membrane that generate non--Markovian normal diffusion

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