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Finite Domain Anomalous Spreading Consistent with First and Second Law
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.Comment: 14 figure
Memory effects in measure transport equations
Transport equations with a nonlocal velocity field have been introduced as a
continuum model for interacting particle systems arising in physics, chemistry
and biology. Fractional time derivatives, given by convolution integrals of the
time-derivative with power-law kernels, are typical for memory effects in
complex systems. In this paper we consider a nonlinear transport equation with
a fractional time-derivative. We provide a well-posedness theory for weak
measure solutions of the problem and an integral formula which generalizes the
classical push-forward representation formula to this setting
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