A Parabolic Partial Differential Equation in Three Different Geometries

Partial differential equations occur in diverse areas of engineering and physical sciences, for example, acoustics, aerodynamics, elasticity, electrodynamics, fluid dynamics, geophysics, heat transfer, meteorology, oceanography, optics, plasma physics, and quantum mechanics.

This Demonstration simulates solutions of the following parabolic equation:

,

where applies to a rectangle, cylinder, or sphere, with the boundary conditions at and and the initial condition

The problem has applications in transport phenomena, specifically, heat, mass, and momentum transfer.