Finite domain anomalous spreading consistent with first and second laws

After reviewing the problematic behavior of some previously suggested finite-interval spatial operators of the symmetric Riesz type, we create a wish list leading toward a new spatial operator suitable to use in the space–time fractional differential equation of anomalous diffusion when the transport of material is strictly restricted to a bounded domain. Based on recent studies of wall effects, we introduce a new definition of the spatial operator and illustrate its favorable characteristics. We provide two numerical methods to solve the modified space–time fractional differential equation and show particular results illustrating compliance to our established list of requirements, most important to the conservation principle and the second law of thermodynamics.