Fractional anisotropic diffusion equation in cylindrical brush model

Fractional nutrient uptake model of plant roots

Most nutrient uptake problems are modeled by the convection-diffusion equation (CDE) abiding by Fick's law. Because nutrients needed by plants exist in the soil solution as a form of ions and the soil is a typical fractal structure of heterogeneity, it makes the solute transport appear anomalous diffusion in soil. Taking anomalous diffusion as a transport process, we propose time and space fractional nutrient uptake models based on the classic Nye-Tinker-Barber model. There does not appear apparent sub-diffusion of nitrate in the time fractional model until four months and the time fractional models are appropriate for describing long-term dynamics and slow sorption reaction; the space fractional model can capture super-diffusion in short term and it is suitable for describing nonlocal phenomena and daily variations driven by transpiration and metabolism; the anomalous diffusion more apparently appears near the root surface in the modeling simulation.