A metal bar of length

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Conditions of Validity, Ch The three equations, their characteristics and what makes a problem well posed

Problem

1.0\,\mathrm{m}

, initially at a uniform

20^\circ\text{C}

throughout, has one end suddenly raised to

100^\circ\text{C}

t=0

and held there; the other end is kept at

20^\circ\text{C}

. The temperature obeys the diffusion equation

u_t = Du_{xx}

. ((a)) Without solving the equation, predict whether the temperature at a point just inside the far end, say

1\,\mathrm{cm}

from it, changes at

t=0^+

, and if so by how much (qualitatively). What happens at the far end itself? ((b)) State what data are needed to pose this problem, reading the answer from the characteristics of

u_t=Du_{xx}

found earlier, and contrast this with what a wave equation on the same bar would need.

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