These analyses collectively demonstrate that the fundamental structure and parameters of partial differential equations (PDEs) precisely govern the ultimate physical fate of mass, energy, or potential fields. Specifically, the long-term behavior of a system is determined by its boundary conditions: boundaries lead to a stable steady state, enforcing either total loss (Dirichlet to zero equilibrium) or perfect conservation (Neumann to an average equilibrium), in contrast to infinite domains where spreading is perpetual. For transport, convection dictates the overall displacement of the material, while diffusion controls its immediate shape and spreading, but the decay rate ultimately limits the material's total existence time. Finally, the dimensionality of a concentrated source dictates the field's behavior, proving that distributing a charge eliminates the dramatic singularity characteristic of a 3D point source.

🫧Cue Column

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1. how the shape and peak height change for pure diffusion and pure decay and the combined scenario
2. Point Source Diffusion-From Transient Pulse to Steady Source
3. Electrostatic Potentials-Numerical Validation of Point vs Distributed Charge
4. How the delta function is used to model charge distributions concentrated on a line or a surface instead of a single point
5. the concentration profile over time for three scenario-pure diffusion and pure convection and the combined case
6. Fick's second law is used to Chemical Mixing and Heat Transfer and Semiconductor Doping </aside>