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.

💡 Research Summary

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The paper addresses a fundamental difficulty in modeling anomalous (non‑Gaussian) diffusion within a bounded domain. While the space‑time fractional diffusion equation with a symmetric Riesz spatial derivative and a Caputo time derivative accurately describes anomalous spreading on an infinite line, naïve extensions to a finite interval violate basic physical principles: the conservation of mass (the “first law”) and the monotonic increase of entropy (the “second law”). The authors first review the standard formulation

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