Heat conduction in a uniform solid occupying a bounded region with boundary surface…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter
Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions
Problem
Heat conduction in a uniform solid occupying a bounded region V with boundary surface S obeys the diffusion equation ∂tT=D∇2T for t>0, with D a positive constant.
((a)) Prove that at most one solution has a given initial temperature throughout V and a given, possibly time-dependent, temperature on S for every t>0.
((b)) Now let S be insulated everywhere instead, so that ∂T/∂n=0 on S for every t>0, with the initial temperature given as before. Decide whether your argument from (a) still gives at most one solution, and name the step that changes.
((c)) To recover an earlier temperature from a profile measured now, one would solve ∂tu=−D∇2u forward from the measured profile, with t now counting back into the past and u given on S. Run the argument from (a) on this equation, find the step where it fails, and decide whether that failure by itself shows that two different solutions share the same data.