One can show that this is the only solution to the heat equation with the given initial condition. The normal modes tend to zero (exponentially) as t ! 1. Overall, u(x; t) ! 0 (exponentially) uniformly …

Three physical principles are used here. 1. Heat (or thermal) energy of a body with uniform properties: where m is the body mass, u is the temperature, c is the specific heat, units [c] = …

To specify the boundary conditions, we might simply give speci c values or rules for the left and right endpoints. If we no longer are restricted to problems for which we have a simple formula …

What about non‐steady‐state problems (quite common in heat transfer)? A very long, very wide, very tall slab is initially at a temperature To. At time t = 0, the left face of the slab is exposed to …

General Structure Applies for any 1D problem: flame, fuel cell stack, CVD stagnation flow, ... A 1D problem is partitioned into domains

D D ii. Cylindrical and spherical solutions involve Bessel functions, but here are the equations:

1. Derivation. Imagine a dilute material species free to di use along one dimension; a gas in a cylindrical cavity, for example. To model this mathematically, we consider the concentration of …